7.E: Utambulisho wa Trigonometric na Ulinganisho (Mazoezi)

7.1: Kutatua Ulinganisho wa Trigonometric na Utambulisho

Katika sehemu hii, tutaanza uchunguzi wa utambulisho wa msingi wa trigonometric, ikiwa ni pamoja na jinsi tunavyoweza kuthibitisha na jinsi tunavyoweza kuitumia ili kurahisisha maneno ya trigonometric.

Maneno

1) Tunajua\(g(x)=\cos x\) ni hata kazi,\(f(x)=\sin x\) na\(h(x)=\tan x\) ni kazi isiyo ya kawaida. Nini kuhusu\(G(x)=\cos ^2 x\),\(F(x)=\sin ^2 x\) na\(H(x)=\tan ^2 x\)? Je, wao hata, isiyo ya kawaida, au wala? Kwa nini?

Jibu

Kazi zote tatu,\(F,G,\) na\(H\),ni hata.

Hii ni kwa sababu

\(F(-x)=\sin(-x)\sin(-x)=(-\sin x)(-\sin x)=\sin^2 x=F(x),G(-x)=\cos(-x)\cos(-x)=\cos x\cos x= cos^2 x=H(-x)=\tan(-x)\tan(-x)=(-\tan x)(-\tan x)=\tan2x=H(x)\)

2) Kuchunguza grafu\(f(x)=\sec x\) ya wakati\([-\pi ,\pi ]\). Tunawezaje kujua kama kazi ni hata au isiyo ya kawaida kwa kuchunguza tu grafu ya\(f(x)=\sec x\)?

3) Baada ya kuchunguza utambulisho wa usawa\(\sec t\),kueleza kwa nini kazi haijulikani katika pointi fulani.

Jibu

Wakati\(\cos t = 0\),basi\(\sec t = 10\),ambayo ni undefined.

4) Utambulisho wote wa Pythagorean ni kuhusiana. Eleza jinsi ya kuendesha equations\(\sin^2t+\cos^2t=1\) kupata kutoka kwa aina nyingine.

Kialjebra

Kwa mazoezi 5-15, tumia utambulisho wa msingi ili kurahisisha kikamilifu kujieleza.

5)\(\sin x \cos x \sec x\)

Jibu

\(\sin x\)

6)\(\sin(-x)\cos(-x)\csc(-x)\)

7)\(\tan x\sin x+\sec x\cos^2x\)

Jibu

\(\sec x\)

8)\(\csc x+\cos x\cot(-x)\)

9)\(\dfrac{\cot t+\tan t}{\sec (-t)}\)

Jibu

\(\csc x\)

10)\(3\sin^3 t\csc t+\cos^2 t+2\cos(-t)\cos t\)

11)\(-\tan(-x)\cot(-x)\)

Jibu

\(-1\)

12)\(\dfrac{-\sin (-x)\cos x\sec x\csc x\tan x}{\cot x}\)

13)\(\dfrac{1+\tan ^2\theta }{\csc ^2\theta }+\sin ^2\theta +\dfrac{1}{\sec ^2 \theta }\)

Jibu

\(\sec^2 x\)

14)\(\left (\dfrac{\tan x}{\csc ^2 x}+\dfrac{\tan x}{\sec ^2 x} \right )\left (\dfrac{1+\tan x}{1+\cot x} \right )-\dfrac{1}{\cos ^2 x}\)

15)\(\dfrac{1-\cos ^2 x}{\tan ^2 x}+2\sin ^2 x\)

Jibu

\(\sin^2 x+1\)

Kwa mazoezi 16-28, kurahisisha kujieleza kwanza trigonometric kwa kuandika fomu rahisi katika suala la kujieleza pili.

16)\(\dfrac{\tan x+\cot x}{\csc x}; \cos x\)

17)\(\dfrac{\sec x+\csc x}{1+\tan x}; \sin x\)

Jibu

\(\dfrac{1}{\sin x}\)

18)\(\dfrac{\cos x}{1+\sin x}+\tan x; \cos x\)

19)\(\dfrac{1}{\sin x\cos x}-\cot x; \cot x\)

Jibu

\(\dfrac{1}{\cot x}\)

20)\(\dfrac{1}{1-\cos x}-\dfrac{\cos x}{1+\cos x}; \csc x\)

21)\((\sec x+\csc x)(\sin x+\cos x)-2-\cot x; \tan x\)

Jibu

\(\tan x\)

22)\(\dfrac{1}{\csc x-\sin x}; \sec x\) na\(\tan x\)

23)\(\dfrac{1-\sin x}{1+\sin x}-\dfrac{1+\sin x}{1-\sin x}; \sec x\) na\(\tan x\)

Jibu

\(-4\sec x \tan x\)

24)\(\tan x; \sec x\)

25)\(\sec x; \cot x\)

Jibu

\(\pm \sqrt{\dfrac{1}{\cot ^2 x}+1}\)

26)\(\sec x; \sin x\)

27)\(\cot x; \sin x\)

Jibu

\(\dfrac{\pm \sqrt{1-\sin ^2 x}}{\sin x}\)

28)\(\cot x; \csc x\)

Kwa mazoezi 29-33, thibitisha utambulisho.

29)\(\cos x-\cos^3x=\cos x \sin^2 x\)

Jibu

Majibu yatatofautiana. Mfano wa ushahidi:

\(\begin{align*} \cos x-\cos^3x &= \cos x (1-\cos^2 x)\\ &= \cos x\sin ^x \end{align*}\)

30)\(\cos x(\tan x-\sec(-x))=\sin x-1\)

31)\(\dfrac{1+\sin ^2x}{\cos ^2 x}=\dfrac{1}{\cos ^2 x}+\dfrac{\sin ^2x}{\cos ^2 x}=1+2\tan ^2x\)

Jibu

Majibu yatatofautiana. Mfano wa ushahidi:

\(\begin{align*} \dfrac{1+\sin ^2x}{\cos ^2 x} &= \dfrac{1}{\cos ^2 x}+\dfrac{\sin ^2x}{\cos ^2 x}\\ &= \sec ^2x+\tan ^2x\\ &= \tan ^2x+1+\tan ^2x\\ &= 1+2\tan ^2x \end{align*}\)

32)\((\sin x+\cos x)^2=1+2 \sin x\cos x\)

33)\(\cos^2x-\tan^2x=2-\sin^2x-\sec^2x\)

Jibu

Majibu yatatofautiana. Mfano wa ushahidi:

\(\begin{align*} \cos^2x-\tan^2x &= 1-\sin^2x-\left (\sec^2x -1 \right )\\ &= 1-\sin^2x-\sec^2x +1\\ &= 2-\sin^2x-\sec^2x \end{align*}\)

Upanuzi

Kwa mazoezi 34-39, kuthibitisha au kupinga utambulisho.

34)\(\dfrac{1}{1+\cos x}-\dfrac{1}{1-\cos (-x)}=-2\cot x\csc x\)

35)\(\csc^2x(1+\sin^2x)=\cot^2x\)

Jibu

Uongo

36)\(\left (\dfrac{\sec ^2(-x)-\tan ^2x}{\tan x} \right )\left (\dfrac{2+2\tan x}{2+2\cot x} \right )-2\sin ^2 x=\cos (2x) \)

37)\(\dfrac{\tan x}{\sec x}\sin (-x)=\cos ^2x\)

Jibu

Uongo

38)\(\dfrac{\sec (-x)}{\tan x+\cot x}=-\sin (-x)\)

39)\(\dfrac{1+\sin x}{\cos x}=\dfrac{\cos x}{1+\sin (-x)}\)

Jibu

Imeonekana na utambulisho hasi na Pythagorean

Kwa mazoezi 40-, kuamua kama utambulisho ni kweli au uongo. Ikiwa uongo, pata kujieleza sawa sawa.

40)\(\dfrac{\cos ^2 \theta -\sin ^2 \theta }{1-\tan ^2 \theta }=\sin ^2 \theta\)

41)\(3\sin^2\theta + 4\cos^2\theta =3+\cos^2\theta\)

Jibu

Kweli

\(\begin{align*} 3\sin^2\theta + 4\cos^2\theta &= 3\sin ^2\theta +3\cos ^2\theta +\cos^2\theta \\ &= 3\left ( \sin ^2\theta +\cos ^2\theta \right )+\cos^2\theta \\ &= 3+\cos^2\theta \end{align*}\)

42)\(\dfrac{\sec \theta +\tan \theta }{\cot \theta+\cos ^\theta }=\sec ^2 \theta\)

7.2: Utambulisho wa Jumla na Tofauti

Katika sehemu hii, tutajifunza mbinu ambazo zitatuwezesha kutatua matatizo muhimu. Fomu zinazofuata zitapunguza maneno mengi ya trigonometric na equations. Kumbuka kwamba, katika sehemu hii, fomu ya neno hutumiwa sawa na utambulisho wa neno.

Maneno

1) Eleza msingi wa utambulisho wa cofunction na wakati wao kuomba.

Jibu

Utambulisho wa cofunction hutumika kwa pembe za ziada. Kuangalia pembe mbili za papo hapo za pembetatu ya kulia, ikiwa moja ya hatua hizo za pembe\(x\),hatua ya pili ya angle\(\dfrac{\pi }{2}-x\). Kisha\(\sin x=\cos \left (\dfrac{\pi }{2}-x \right )\). Hiyo inashikilia kwa utambulisho mwingine wa cofunction. Jambo muhimu ni kwamba pembe ni nyongeza.

2) Je, kuna njia moja tu ya kutathmini\(\cos \left (\dfrac{5\pi }{4} \right )\)? Eleza jinsi ya kuanzisha suluhisho kwa njia mbili tofauti, na kisha uhakikishe kuhakikisha wanatoa jibu sawa.

3) Eleza kwa mtu ambaye amesahau mali isiyo ya kawaida ya kazi za sinusoidal jinsi fomu za kuongeza na kuondoa zinaweza kuamua tabia hii\(f(x)=\sin (x)\) na\(g(x)=\cos (x)\). (Kidokezo:\(0-x=-x\))

Jibu

\(\sin (-x)=-\sin x\), hivyo\(\sin x\) ni isiyo ya kawaida. \(\cos (-x)=\cos (0-x)=\cos x\), hivyo\(\cos x\) ni hata.

Kialjebra

Kwa mazoezi 4-9, pata thamani halisi.

4)\(\cos \left (\dfrac{7\pi }{12} \right)\)

5)\(\cos \left (\dfrac{\pi }{12} \right)\)

Jibu

\(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)

6)\(\sin \left (\dfrac{5\pi }{12} \right)\)

7)\(\sin \left (\dfrac{11\pi }{12} \right)\)

Jibu

\(\dfrac{\sqrt{6}-\sqrt{2}}{4}\)

8)\(\tan \left (-\dfrac{\pi }{12} \right)\)

9)\(\tan \left (\dfrac{19\pi }{12} \right)\)

Jibu

\(-2-\sqrt{3}\)

Kwa ajili ya mazoezi 10-13, kuandika upya kwa suala la\(\sin x\) na\(\cos x\)

10)\(\sin \left (x+\dfrac{11\pi }{6} \right)\)

11)\(\sin \left (x-\dfrac{3\pi }{4} \right)\)

Jibu

\(-\dfrac{\sqrt{2}}{2}\sin x-\dfrac{\sqrt{2}}{2}\cos x\)

12)\(\cos \left (x-\dfrac{5\pi }{6} \right)\)

13)\(\cos \left (x+\dfrac{2\pi }{3} \right)\)

Jibu

\(-\dfrac{1}{2}\cos x-\dfrac{\sqrt{3}}{2}\sin x\)

Kwa mazoezi 14-19, kurahisisha maneno yaliyotolewa.

14)\(\csc \left (\dfrac{\pi }{2}-t \right)\)

15)\(\sec \left (\dfrac{\pi }{2}-\theta \right)\)

Jibu

\(\csc \theta\)

16)\(\cot \left (\dfrac{\pi }{2}-x \right)\)

17)\(\tan \left (\dfrac{\pi }{2}-x \right)\)

Jibu

\(\cot x\)

18)\(\sin(2x)\cos(5x)-\sin(5x)\cos(2x)\)

19)\(\dfrac{\tan \left (\dfrac{3}{2}x \right)-\tan \left (\dfrac{7}{5}x \right)}{1+\tan \left (\dfrac{3}{2}x \right)\tan \left (\dfrac{7}{5}x \right)}\)

Jibu

\(\tan \left (\dfrac{x}{10} \right)\)

Kwa mazoezi 20-21, pata maelezo yaliyoombwa.

20) Kutokana\(\sin a=\dfrac{2}{3}\) na kwamba na\(\cos b=-\dfrac{1}{4}\),na\(a\) na\(b\) wote katika kipindi\(\left [ \dfrac{\pi }{2}, \pi \right )\),kupata\(\sin (a+b)\) na\(\cos (a-b)\).

21) Kutokana\(\sin a=\dfrac{4}{5}\) na hilo na\(\cos b=\dfrac{1}{3}\), pamoja\(a\) na\(b\) wote katika kipindi\(\left [ 0, \dfrac{\pi }{2} \right )\), tafuta\(\sin (a-b)\) na\(\cos (a+b)\).

Jibu

\(\sin (a-b)=\left ( \dfrac{4}{5} \right )\left ( \dfrac{1}{3} \right )-\left ( \dfrac{3}{5} \right )\left ( \dfrac{2\sqrt{2}}{3} \right )=\dfrac{4-6\sqrt{2}}{15}\)

\(\cos (a+b)=\left ( \dfrac{3}{5} \right )\left ( \dfrac{1}{3} \right )-\left ( \dfrac{4}{5} \right )\left ( \dfrac{2\sqrt{2}}{3} \right )=\dfrac{3-8\sqrt{2}}{15}\)

Kwa mazoezi 22-24, pata thamani halisi ya kila kujieleza.

22)\(\sin \left ( \cos^{-1}\left ( 0 \right )- \cos^{-1}\left ( \dfrac{1}{2} \right )\right )\)

23)\(\cos \left ( \cos^{-1}\left ( \dfrac{\sqrt{2}}{2} \right )+ \sin^{-1}\left ( \dfrac{\sqrt{3}}{2} \right )\right )\)

Jibu

\(\dfrac{\sqrt{2}-\sqrt{6}}{4}\)

24)\(\tan \left ( \sin^{-1}\left ( \dfrac{1}{2} \right )- \cos^{-1}\left ( \dfrac{1}{2} \right )\right )\)

Graphic

Kwa mazoezi 25-32, kurahisisha maneno, na kisha graph maneno yote kama kazi ili kuthibitisha grafu ni sawa.

25)\(\cos \left ( \dfrac{\pi }{2}-x \right )\)

Jibu

\(\sin x\)

26)\(\sin (\pi -x)\)

27)\(\tan \left ( \dfrac{\pi }{3}+x \right )\)

Jibu

\(\cot \left ( \dfrac{\pi }{6}-x \right )\)

28)\(\sin \left ( \dfrac{\pi }{3}+x \right )\)

29)\(\tan \left ( \dfrac{\pi }{4}-x \right )\)

Jibu

\(\cot \left ( \dfrac{\pi }{4}+x \right )\)

30)\(\cos \left ( \dfrac{7\pi }{6}+x \right )\)

31)\(\sin \left ( \dfrac{\pi }{4}+x \right )\)

Jibu

\(\dfrac{\sin x}{\sqrt{2}}+\dfrac{\cos x}{\sqrt{2}}\)

32)\(\cos \left ( \dfrac{5\pi }{4}+x \right )\)

Kwa mazoezi 33-41, tumia grafu ili uone kama kazi ni sawa au tofauti. Ikiwa ni sawa, onyesha kwa nini. Ikiwa ni tofauti, badala ya kazi ya pili na moja ambayo inafanana na ya kwanza. (Kidokezo: fikiria\(2x=x+x\))

33)\(f(x)=\sin(4x)-\sin(3x)\cos x, g(x)=\sin x \cos(3x)\)

Jibu

Wao ni sawa.

34)\(f(x)=\cos(4x)+\sin x \sin(3x), g(x)=-\cos x \cos(3x)\)

35)\(f(x)=\sin(3x)\cos(6x), g(x)=-\sin(3x)\cos(6x)\)

Jibu

Wao ni tofauti, jaribu\(g(x)=\sin(9x)-\cos(3x)\sin(6x)\)

36)\(f(x)=\sin(4x), g(x)=\sin(5x)\cos x-\cos(5x)\sin x\)

37)\(f(x)=\sin(2x), g(x)=2 \sin x \cos x\)

Jibu

Wao ni sawa.

38)\(f(\theta )=\cos(2\theta ), g(\theta )=\cos^2\theta -\sin^2\theta\)

39)\(f(\theta )=\tan(2\theta ), g(\theta )=\dfrac{\tan \theta }{1+\tan^2\theta }\)

Jibu

Wao ni tofauti, jaribu\(g(\theta )=\dfrac{2\tan \theta }{1-\tan^2\theta }\)

40)\(f(x)=\sin(3x)\sin x, g(x)=\sin^2(2x)\cos^2x-\cos^2(2x)\sin2x\)

41)\(f(x)=\tan(-x), g(x)=\dfrac{\tan x-\tan(2x)}{1-\tan x \tan(2x)}\)

Jibu

Wao ni tofauti, jaribu\(g(x)=\dfrac{\tan x-\tan(2x)}{1+\tan x \tan(2x)}\)

Teknolojia

Kwa mazoezi 42-46, pata thamani halisi algebraically, na kisha kuthibitisha jibu kwa calculator kwa uhakika wa nne decimal.

42)\(\sin (75^{\circ})\)

43)\(\sin (195^{\circ})\)

Jibu

\(-\dfrac{\sqrt{3}-1}{2\sqrt{2}}\), au\(-0.2588\)

44)\(\cos (165^{\circ})\)

45)\(\cos (345^{\circ})\)

Jibu

\(\dfrac{1+\sqrt{3}}{2\sqrt{2}}\), au\(-0.9659\)

46)\(\tan (-15^{\circ})\)

Upanuzi

Kwa ajili ya mazoezi 47-51, kuthibitisha utambulisho zinazotolewa.

47)\(\tan \left ( x+\dfrac{\pi }{4} \right )=\dfrac{\tan x+1}{1-\tan x}\)

Jibu

\(\begin{align*} \tan \left ( x+\dfrac{\pi }{4} \right ) &= \\ \dfrac{\tan x + \tan\left (\tfrac{\pi}{4} \right )}{1-\tan x \tan\left (\tfrac{\pi}{4} \right )} &= \\ \dfrac{\tan x+1}{1-\tan x(1)} &= \dfrac{\tan x+1}{1-\tan x} \end{align*}\)

48)\(\dfrac{\tan (a+b)}{\tan (a-b)}=\dfrac{\sin a \cos a + \sin b \cos b}{\sin a \cos a - \sin b \cos b}\)

49)\(\dfrac{\cos (a+b)}{\cos a \cos b}=1-\tan a \tan b\)

Jibu

\(\begin{align*} \dfrac{\cos (a+b)}{\cos a \cos b} &= \\ \dfrac{\cos a \cos b}{\cos a \cos b}- \dfrac{\sin a \sin b}{\cos a \cos b} &= 1-\tan a \tan b \end{align*}\)

50)\(\cos(x+y)\cos(x-y)=\cos^2x-\sin^2y\)

51)\(\dfrac{\cos(x+h)-\cos(x)}{h}=\cos x\dfrac{\cos h-1}{h}-\sin x \dfrac{\sin h}{h}\)

Jibu

\(\begin{align*} \dfrac{\cos(x+h)-\cos(x)}{h} &= \\ \dfrac{\cos x\cosh - \sin x\sinh -\cos x}{h} &= \\ \dfrac{\cos x(\cosh-1) - \sin x(\sinh-1)}{h} &= \cos x\dfrac{\cos h-1}{h}-\sin x \dfrac{\sin h}{h} \end{align*}\)

Kwa mazoezi 52-, kuthibitisha au kupinga kauli.

52)\(\tan (u+v)=\dfrac{\tan u+\tan v}{1-\tan u \tan v}\)

53)\(\tan (u-v)=\dfrac{\tan u-\tan v}{1+\tan u \tan v}\)

Jibu

Kweli

54)\(\dfrac{\tan (x+y)}{1+\tan x \tan x}=\dfrac{\tan x + \tan y}{1-\tan^2 x \tan^2 y}\)

55) Kama\(\alpha ,\beta\),na\(\gamma\) ni pembe katika pembetatu moja, kisha kuthibitisha au kukanusha\(\sin(α+β)=\sin γ\).

Jibu

Kweli. Kumbuka kuwa\(\sin (\alpha +\beta )=\sin (\pi -\gamma )\) na kupanua upande wa kulia.

56) Ikiwa\(\alpha ,\beta\), na\(\gamma\) ni pembe katika pembetatu sawa, basi kuthibitisha au kupinga\(\tan \alpha +\tan \beta +\tan \gamma =\tan \alpha \tan \beta \tan \gamma\).

7.3: Mbili-Angle, Nusu-Angle, na Kupunguza Formula

Katika sehemu hii, tutachunguza makundi matatu ya ziada ya utambulisho. Utambulisho wa pembeni mbili hutokana na kanuni za jumla za kazi za msingi za trigonometric: sine, cosine, na tangent. Njia za kupunguza ni muhimu hasa katika calculus, kwa vile zinatuwezesha kupunguza nguvu ya muda wa trigonometric. Njia za nusu za angle zinatuwezesha kupata thamani ya kazi za trigonometric zinazohusisha pembe za nusu, ikiwa angle ya awali inajulikana au la.

Maneno

1) Eleza jinsi ya kuamua utambulisho wa kupunguza kutoka utambulisho wa pembeni mbili\(\cos(2x)=\cos^2x-\sin^2x\)

Jibu

Tumia utambulisho wa Pythagorean na utenganishe muda wa mraba.

2) Eleza jinsi ya kuamua formula mbili-angle kwa\(\tan(2x)\) kutumia formula mbili-angle kwa\(\cos(2x)\) na\(\sin (2x)\).

3) Tunaweza kuamua formula ya nusu ya angle\(\tan \left ( \dfrac{x}{2} \right )=\dfrac{\sqrt{1-\cos x}}{\sqrt{1+\cos x}}\) kwa kugawanya formula\(\sin \left ( \dfrac{x}{2} \right )\) kwa\(\cos \left ( \dfrac{x}{2} \right )\). Eleza jinsi ya kuamua formula mbili\(\tan \left ( \dfrac{x}{2} \right )\) ambazo hazihusishi mizizi yoyote ya mraba.

Jibu

\(\dfrac{1-\cos x}{\sin x}\)\(\dfrac{\sin x}{1+\cos x}\), kuzidisha juu na chini\(\sqrt{1-\cos x}\) na\(\sqrt{1+\cos x}\), kwa mtiririko huo.

4) Kwa formula ya nusu ya angle iliyotolewa katika zoezi la awali\(\tan \left ( \dfrac{x}{2} \right )\),kueleza kwa nini kugawa na\(0\) si wasiwasi. (Kidokezo: kuchunguza maadili ya\(\cos x\) muhimu kwa denominator kuwa\(0\).)

Kialjebra

Kwa mazoezi 5-8, tafuta maadili halisi ya a)\(\sin (2x)\), b)\(\cos(2x)\),na c)\(\tan(2x)\) bila kutatua kwa\(x\).

5) Kama\(\sin x =\dfrac{1}{8}\),na\(x\) ni katika roboduara\(\mathrm{I}\).

Jibu

1. \(\dfrac{3\sqrt{7}}{32}\)
2. \(\dfrac{31}{32}\)
3. \(\dfrac{3\sqrt{7}}{31}\)

6) Kama\(\cos x =\dfrac{2}{3}\), na\(x\) ni katika roboduara\(\mathrm{I}\).

7) Kama\(\cos x =-\dfrac{1}{2}\), na\(x\) ni katika roboduara\(\mathrm{III}\).

Jibu

1. \(\dfrac{\sqrt{3}}{2}\)
2. \(-\dfrac{1}{2}\)
3. \(-\sqrt{3}\)

8) Kama\(\tan x =-8\), na\(x\) ni katika roboduara\(\mathrm{IV}\).

Kwa mazoezi 9-10, tafuta maadili ya kazi sita za trigonometric ikiwa hali zinazotolewa zinashikilia.

9)\(\cos(2\theta )=\dfrac{3}{5}\) na\(90^{\circ}\leq \theta \leq 180^{\circ}\)

Jibu

\(\cos \theta =-\frac{2\sqrt{5}}{5},\sin \theta =\frac{\sqrt{5}}{5},\tan \theta =-\frac{1}{2},\csc \theta =\sqrt{5},\sec \theta =-\frac{\sqrt{5}}{2},\cot \theta =-2\)

10)\(\cos(2\theta )=\dfrac{1}{\sqrt{2}}\) na\(180^{\circ}\leq \theta \leq 270^{\circ}\)

Kwa mazoezi 11-12, kurahisisha kwa kujieleza moja ya trigonometric.

11)\(2\sin \left ( \dfrac{\pi }{4} \right )2\cos \left ( \dfrac{\pi }{4} \right )\)

Jibu

\(2\sin \left ( \dfrac{\pi }{2} \right )\)

12)\(4\sin \left ( \dfrac{\pi }{8} \right )\cos \left ( \dfrac{\pi }{8} \right )\)

Kwa mazoezi 13-19, pata thamani halisi kwa kutumia formula za nusu-angle.

13)\(\sin \left ( \dfrac{\pi }{8} \right )\)

Jibu

\(\dfrac{\sqrt{2-\sqrt{2}}}{2}\)

14)\(\cos \left ( -\dfrac{11\pi }{12} \right )\)

15)\(\sin \left ( \dfrac{11\pi }{12} \right )\)

Jibu

\(\dfrac{\sqrt{2-\sqrt{3}}}{2}\)

16)\(\cos \left ( \dfrac{7\pi }{8} \right )\)

17)\(\tan \left ( \dfrac{5\pi }{12} \right )\)

Jibu

\(2+\sqrt{3}\)

18)\(\tan \left ( -\dfrac{3\pi }{12} \right )\)

19)\(\tan \left ( -\dfrac{3\pi }{8} \right )\)

Jibu

\(-1-\sqrt{2}\)

Kwa mazoezi 20-23, tafuta maadili halisi ya a)\(\sin \left ( \dfrac{x}{2} \right )\) b)\(\cos \left ( \dfrac{x}{2} \right )\),na c)\(\tan \left ( \dfrac{x}{2} \right )\) bila kutatua\(x\),lini\(0^{\circ}\leq \theta \leq 360^{\circ}\)

20) Kama\(\tan x =-\dfrac{4}{3}\), na\(x\) ni katika roboduara\(\mathrm{IV}\).

21) Kama\(\sin x =-\dfrac{12}{13}\), na\(x\) ni katika roboduara\(\mathrm{III}\).

Jibu

1. \(\dfrac{3\sqrt{13}}{13}\)
2. \(-\dfrac{2\sqrt{13}}{13}\)
3. \(-\dfrac{3}{2}\)

22) Kama\(\csc x =7\), na\(x\) ni katika roboduara\(\mathrm{II}\).

23) Kama\(\sec x =-4\), na\(x\) ni katika roboduara\(\mathrm{II}\).

Jibu

1. \(\dfrac{\sqrt{10}}{4}\)
2. \(\dfrac{\sqrt{6}}{4}\)
3. \(\dfrac{\sqrt{15}}{3}\)

Kwa mazoezi 24-27, tumia Kielelezo hapa chini ili kupata nusu iliyoombwa na pembe mbili.

24) Kupata\(\sin (2\theta )\),\(\cos (2\theta )\),na\(\tan (2\theta )\).

25) Kupata\(\sin (2\alpha )\),\(\cos (2\alpha )\), na\(\tan (2\alpha )\).

Jibu

\(\dfrac{120}{169}, -\dfrac{119}{169}, -\dfrac{120}{119}\)

26) Kupata\(\sin \left (\dfrac{\theta }{2} \right )\),\(\cos \left (\dfrac{\theta }{2} \right )\), na\(\tan \left (\dfrac{\theta }{2} \right )\).

27) Kupata\(\sin \left (\dfrac{\alpha }{2} \right )\),\(\cos \left (\dfrac{\alpha }{2} \right )\), na\(\tan \left (\dfrac{\alpha }{2} \right )\).

Jibu

\(\dfrac{2\sqrt{13}}{13}, \dfrac{3\sqrt{13}}{13}, \dfrac{2}{3}\)

Kwa mazoezi 28-33, kurahisisha kila kujieleza. Je, si kutathmini.

28)\(\cos ^2(28^{\circ})-\sin ^2(28^{\circ})\)

29)\(2\cos ^2(37^{\circ})-1\)

Jibu

\(\cos (74^{\circ})\)

30)\(1-2\sin ^2(17^{\circ})\)

31)\(\cos ^2(9x)-\sin ^2(9x)\)

Jibu

\(\cos (18x)\)

32)\(4\sin (8x)\cos (8x)\)

33)\(6\sin (5x)\cos (5x)\)

Jibu

\(3\sin (10x)\)

Kwa mazoezi 34-37, kuthibitisha utambulisho uliotolewa.

34)\((\sin t-\cos t)^2=1-\sin(2t)\)

35)\(\sin(2x)=-2 \sin(-x) \cos(-x)\)

Jibu

\(-2 \sin(-x)\cos(-x)=-2(-\sin(x)\cos(x))=\sin(2x)\)

36)\(\cot x-\tan x=2 \cot(2x)\)

37)\(\dfrac{1+\cos (2\theta )}{\sin (2\theta )}\tan ^2\theta =\tan \theta\)

Jibu

\(\dfrac{\sin (2\theta )}{1+\cos (2\theta )}\tan ^2\theta =\dfrac{2\sin (\theta )\cos (\theta )}{1+\cos ^2\theta -\sin ^2\theta }\tan ^2\theta=\)

\(\dfrac{2\sin (\theta )\cos (\theta )}{2\cos ^2\theta }\tan ^2\theta=\dfrac{\sin (\theta )}{\cos (\theta )}\tan ^2\theta=\)

\(\cot (\theta )\tan ^2\theta=\tan \theta\)

Kwa mazoezi 38-44, fungua upya maneno na kielelezo hakuna zaidi ya 1.

38)\(\cos ^2 (5x)\)

39)\(\cos ^2 (6x)\)

Jibu

\(\dfrac{1+\cos (12x)}{2}\)

40)\(\sin ^4 (8x)\)

41)\(\sin ^4 (3x)\)

Jibu

\(\dfrac{3+\cos(12x)-4\cos(6x)}{8}\)

42)\(\cos^2x \sin^4x\)

43)\(\cos^4x \sin^2x\)

Jibu

\(\dfrac{2+\cos(2x)-2\cos(4x)-\cos(6x)}{32}\)

44)\(\tan^2x \sin^2x\)

Teknolojia

Kwa mazoezi 45-52, kupunguza usawa kwa nguvu za moja, na kisha angalia jibu graphically.

45)\(\tan^4x\)

Jibu

\(\dfrac{3+\cos(4x)-4\cos(2x)}{3+\cos(4x)+4\cos(2x)}\)

46)\(\sin^2(2x)\)

47)\(\sin^2x \cos^2x\)

Jibu

\(\dfrac{1-\cos(4x)}{8}\)

48)\(\tan^2x \sin x\)

49)\(\tan^4x \cos^2 x\)

Jibu

\(\dfrac{3+\cos(4x)-4\cos(2x)}{4(\cos(2x)+1)}\)

50)\(\cos^2x \sin (2x)\)

51)\(\cos^2(2x) \sin x\)

Jibu

\(\dfrac{(1+\cos(4x))\sin x}{2}\)

52)\(\tan ^2\left ( \dfrac{x}{2} \right )\sin x\)

Kwa mazoezi 53-54, algebraically kupata kazi sawa, tu kwa suala la\(\sin x\) na/au\(\cos x\),na kisha angalia jibu kwa kuchora equations zote mbili.

53)\(\sin (4x)\)

Jibu

\(4\sin x\cos x(\cos^2x-\sin^2x)\)

54)\(\cos (4x)\)

Upanuzi

Kwa mazoezi 55-63, kuthibitisha utambulisho.

55)\(\sin (2x)=\dfrac{2\tan x}{1+\tan ^2x}\)

Jibu

\(\dfrac{2\tan x}{1+\tan ^2x}=\dfrac{\tfrac{2\sin x}{\cos x}}{1+\tfrac{\sin ^2x}{\cos ^2x}}=\dfrac{\tfrac{2\sin x}{\cos x}}{\tfrac{\cos ^2x+\sin ^2x}{\cos ^2x}}=\dfrac{2\sin x}{\cos x}\cdot \dfrac{\cos ^2x}{1}=2\sin x \cos x=\sin (2x)\)

56)\(\cos (2\alpha )=\dfrac{1-\tan ^2\alpha }{1+\tan ^2\alpha }\)

57)\(\tan (2x)=\dfrac{2\sin x \cos x }{2\cos ^2 x-1}\)

Jibu

\(\dfrac{2\sin x \cos x }{2\cos ^2 x-1}=\dfrac{\sin (2x)}{ \cos (2x)}=\tan (2x)\)

58)\((\sin^2x-1)^2=\cos(2x)+\sin^4x\)

59)\(\sin(3x)=3\sin x \cos^2x-\sin^3x\)

Jibu

\(\begin{align*} \sin (x+2x) &= \sin x \cos (2x)+\sin (2x) \cos x\\ &= \sin x(\cos ^2 x - \sin ^2 x)+2\sin x \cos x \cos x\\ &= \sin x \cos ^2 x-\sin ^3 x + 2\sin x\cos ^2 x\\ &= 3\sin x\cos ^2 x - \sin ^3 x \end{align*}\)

60)\(\cos(3x)=\cos^3x-3\sin^2x\cos x\)

61)\(\dfrac{1+\cos (2t)}{\sin (2t)-\cos t}=\dfrac{2\cos t}{2\sin t-1}\)

Jibu

\(\begin{align*} \dfrac{1+\cos (2t)}{\sin (2t)-\cos t} &= \dfrac{1+2\cos ^2t-1}{2\sin t\cos t-\cos t}\\ &= \dfrac{2\cos ^2t}{\cos t(2\sin t-1)}\\ &= \dfrac{2\cos t}{2\sin t-1} \end{align*}\)

62)\(\sin(16x)=16 \sin x \cos x \cos(2x)\cos(4x)\cos(8x)\)

63)\(\cos(16x)=(\cos^2(4x)-\sin^2(4x)-\sin(8x))(\cos^2(4x)-\sin^2(4x)+\sin(8x))\)

Jibu

\(\begin{align*} (\cos^2(4x)-\sin^2(4x)-\sin(8x))(\cos^2(4x)-\sin^2(4x)+\sin(8x)) &= (\cos(8x)-\sin(8x))(\cos(8x)+\sin(8x))\\ &= \cos ^2 (8x)-\sin ^2 (8x)\\ &= \cos(16x) \end{align*}\)

7.4: Jumla ya bidhaa na Bidhaa kwa-Jumla Formula

Kutoka kwa utambulisho wa jumla na tofauti, tunaweza kupata fomu za bidhaa hadi jumla na fomu za jumla kwa bidhaa kwa sine na cosine. Fomu za bidhaa kwa jumla zinaweza kuandika tena bidhaa za sines, bidhaa za cosines, na bidhaa za sine na cosine kama kiasi cha tofauti za sines na cosines. Tunaweza pia kupata utambulisho wa jumla hadi bidhaa kutoka kwa utambulisho wa bidhaa hadi jumla kwa kutumia mbadala. Fomu za jumla hadi bidhaa hutumiwa kuandika upya jumla au tofauti kama bidhaa za sines na cosines.

Maneno

1) Kuanzia na bidhaa kwa jumla formula\(\sin \alpha \cos \beta =\dfrac{1}{2} \left[\sin(\alpha +\beta )+\sin(\alpha -\beta ) \right]\),kueleza jinsi ya kuamua formula kwa\(\cos \alpha \sin \beta\).

Jibu

\(\alpha \)Kuingiza katika cosine na\(\beta \) ndani ya sine na kutathmini.

2) Eleza njia mbili tofauti za kuhesabu\(\cos (195^{\circ}) \cos (105^{\circ})\),moja ambayo inatumia bidhaa kwa jumla. Njia ipi ni rahisi?

3) Eleza hali ambapo tunataka kubadilisha equation kutoka jumla ya bidhaa na kutoa mfano.

Jibu

Majibu yatatofautiana. Kuna baadhi ya milinganyo zinazohusisha jumla ya maneno mawili trig ambapo wakati kubadilishwa kuwa bidhaa ni rahisi kutatua. Kwa mfano:\(\dfrac{\sin (3x)+\sin x}{\cos x}=1\). Wakati wa kubadili nambari kwa bidhaa equation inakuwa:\(\dfrac{2\sin (2x)\cos x}{\cos x}=1\)

4) Eleza hali ambapo tunataka kubadilisha equation kutoka bidhaa kwa jumla, na kutoa mfano.

Kialjebra

Kwa mazoezi 5-10, rejesha tena bidhaa kama jumla au tofauti.

5)\(16\sin(16x)\sin(11x)\)

Jibu

\(8(\cos(5x)-\cos(27x))\)

6)\(2\cos(36t)\cos(6t)\)

7)\(2\sin(5x)\cos(3x)\)

Jibu

\(\sin(2x)+\sin(8x)\)

8)\(10\cos(5x)\sin(10x)\)

9)\(\sin(-x)\sin(5x)\)

Jibu

\(\dfrac{1}{2}(\cos(6x)-\cos(4x))\)

10)\(\sin(3x)\cos(5x)\)

Kwa mazoezi 11-16, rejesha tena jumla au tofauti kama bidhaa.

11)\(\cos(6t)+\cos(4t)\)

Jibu

\(2\cos(5t)\cos t\)

12)\(\sin(3x)+\sin(7x)\)

13)\(\cos(7x)+\cos(-7x)\)

Jibu

\(2\cos(7x)\)

14)\(\sin(3x)-\sin(-3x)\)

15)\(\cos(3x)+\cos(9x)\)

Jibu

\(2\cos(6x)\cos(3x)\)

16)\(\sin h-\sin(3h)\)

Kwa mazoezi 17-21, tathmini bidhaa kwa zifuatazo kwa kutumia jumla au tofauti ya kazi mbili.

17)\(\cos (45^{\circ}) \cos (15^{\circ})\)

Jibu

\(\dfrac{1}{4}(1+\sqrt{3})\)

18)\(\cos (45^{\circ}) \sin (15^{\circ})\)

19)\(\sin (-345^{\circ}) \sin (-15^{\circ})\)

Jibu

\(\dfrac{1}{4}(\sqrt{3}-2)\)

20)\(\sin (195^{\circ}) \cos (15^{\circ})\)

21)\(\sin (-45^{\circ}) \sin (-15^{\circ})\)

Jibu

\(\dfrac{1}{4}(\sqrt{3}-1)\)

Kwa mazoezi 22-26, tathmini bidhaa kwa kutumia jumla au tofauti ya kazi mbili. Acha kwa suala la sine na cosine.

22)\(\cos (23^{\circ}) \sin (17^{\circ})\)

23)\(2\sin (100^{\circ}) \sin (20^{\circ})\)

Jibu

\(\cos (80^{\circ})-\cos (120^{\circ})\)

24)\(2\sin (-100^{\circ})\sin (-20^{\circ})\)

25)\(\sin (213^{\circ})\cos (8^{\circ})\)

Jibu

\(\dfrac{1}{2}\left (\sin (221^{\circ})+\sin (205^{\circ}) \right )\)

26)\(2\cos (56^{\circ})\cos (47^{\circ})\)

Kwa mazoezi 27-31, rejesha tena jumla kama bidhaa ya kazi mbili. Acha kwa suala la sine na cosine.

27)\(\sin (76^{\circ})+\sin (14^{\circ})\)

Jibu

\(\sqrt{3}\cos (31^{\circ})\)

28)\(\cos (58^{\circ})-\cos (12^{\circ})\)

29)\(\sin (101^{\circ})-\sin (32^{\circ})\)

Jibu

\(2\cos (66.5^{\circ})\sin (34.5^{\circ})\)

30)\(\cos (100^{\circ})+\cos (200^{\circ})\)

31)\(\sin (-1^{\circ})+\sin (-2^{\circ})\)

Jibu

\(2\sin (-1.5^{\circ})\cos (0.5^{\circ})\)

Kwa mazoezi 32-38, kuthibitisha utambulisho.

32)\(\dfrac{\cos (a-b)}{\cos (a+b)}=\dfrac{1-\tan a \tan b}{1+\tan a \tan b}\)

33)\(4\sin(3x)\cos(4x)=2\sin(7x)-2\sin x\)

Jibu

\(\begin{align*} 2\sin(7x)-2\sin x &= 2\sin(4x+3x)-2\sin(4x-3x)\\ &= 2(\sin(4x)\cos(3x)+\sin(3x)\cos(4x))-2(\sin(4x)\cos(3x)-\sin(3x)\cos(4x))\\ &= 2\sin(4x)\cos(3x)+2\sin(3x)\cos(4x))-2\sin(4x)\cos(3x)+2\sin(3x)\cos(4x))\\ &= 4\sin(3x)\cos(4x) \end{align*}\)

34)\(\dfrac{6\cos (8x)\sin (2x)}{\sin (-6x)}=-3\sin(10x)\csc(6x)+3\)

35)\(\sin x + \sin(3x)=4\sin x\cos^2 x\)

Jibu

\(\begin{align*} \sin x + \sin(3x) &= 2\sin\left ( \dfrac{4x}{2} \right )\cos\left ( -\dfrac{2x}{2} \right )\\ &= 2\sin(2x)\cos x\\ &= 2(2\sin x \cos x)\cos x\\ &= 4\sin x\cos^2 x \end{align*}\)

36)\(2(\cos^3x-\cos x \sin^2x)=\cos(3x)+\cos x\)

37)\(2\tan x \cos(3x)=\sec x(\sin(4x)-\sin(2x))\)

Jibu

\(\begin{align*} 2\tan x \cos(3x) &= \dfrac{2\sin x\cos (3x)}{\cos x}\\ &= \dfrac{2(.5(\sin (4x)-\sin (2x)))}{\cos x}\\ &= \dfrac{1}{\cos x}(\sin(4x)-\sin(2x))\\ &= \sec x(\sin(4x)-\sin(2x)) \end{align*}\)

38)\(\cos(a+b)+\cos(a-b)=2\cos a \cos b\)

Numeric

Kwa mazoezi 39-43, rejesha tena jumla kama bidhaa ya kazi mbili au bidhaa kama jumla ya kazi mbili. Kutoa jibu lako kwa suala la dhambi na cosines. Kisha tathmini jibu la mwisho kwa nambari, lililozunguka kwenye maeneo manne ya decimal.

39)\(\cos (58^{\circ})+\cos (12^{\circ})\)

Jibu

\(2\cos (35^{\circ})\cos (23^{\circ}),1.5081\)

40)\(\sin (2^{\circ})-\sin (3^{\circ})\)

41)\(\cos (44^{\circ})-\cos (22^{\circ})\)

Jibu

\(-2\sin (33^{\circ})\sin (11^{\circ}),-0.2078\)

42)\(\cos (176^{\circ})\sin (9^{\circ})\)

43)\(\sin (-14^{\circ})\sin (85^{\circ})\)

Jibu

\(\dfrac{1}{2}\left (\cos (99^{\circ})-\cos (71^{\circ}) \right ),-0.2410\)

Teknolojia

Kwa mazoezi 44-48, algebraically kuamua kama kila moja ya maneno yaliyotolewa ni utambulisho wa kweli. Ikiwa sio utambulisho, badala ya upande wa kulia na maneno sawa na upande wa kushoto. Thibitisha matokeo kwa kuchora maneno yote kwenye calculator.

44)\(2\sin(2x)\sin(3x)=\cos x-\cos(5x)\)

45)\(\dfrac{\cos(10\theta )+\cos(6\theta )}{\cos(6\theta )-\cos(10\theta )}=\cot(2\theta )\cot(\theta )\)

Jibu

Ni utambulisho.

46)\(\dfrac{\sin(3x)-\sin(5x)}{\cos(3x)+\cos(5x)}=\tan x\)

47)\(2\cos(2x)\cos x+\sin(2x)\sin x=2\sin x\)

Jibu

Si utambulisho, lakini\(2\cos ^3 x\) ni.

48)\(\dfrac{\sin(2x)+\sin(4x)}{\sin(2x)-\sin(4x)}=-\tan (3x)\cot x\)

Kwa mazoezi 49-53, kurahisisha maneno kwa muda mmoja, kisha graph kazi ya awali na toleo lako kilichorahisishwa ili kuthibitisha kuwa ni sawa.

49)\(\dfrac{\sin(9t)-\sin(3t)}{\cos(9t)+\cos(3t)}\)

Jibu

\(\tan (3t)\)

50)\(2\sin(8x)\cos(6x)-\sin(2x)\)

51)\(\dfrac{\sin(3x)-\sin(x)}{\sin x}\)

Jibu

\(2\cos (2x)\)

52)\(\dfrac{\cos(5x)+\cos(3x)}{\sin(5x)+\sin(3x)}\)

53)\(\sin x \cos(15x)-\cos x \sin(15x)\)

Jibu

\(-\sin (14x)\)

Upanuzi

Kwa mazoezi 54-55, kuthibitisha kanuni zifuatazo za jumla hadi bidhaa.

54)\(\sin x - \sin y = 2\sin \left ( \dfrac{x-y}{2} \right )\cos \left ( \dfrac{x+y}{2} \right )\)

55)\(\cos x + \cos y = 2\cos \left ( \dfrac{x+y}{2} \right )\cos \left ( \dfrac{x-y}{2} \right )\)

Jibu

Anza na\(\cos x + \cos y\). Kufanya badala na basi\(x=\alpha +\beta\) na basi\(y=\alpha -\beta\),hivyo\(\cos x + \cos y\) inakuwa:

\(\cos(\alpha +\beta )+\cos(\alpha -\beta )=\cos\alpha \cos\beta -\sin\alpha \sin\beta +\cos\alpha \cos\beta +\sin\alpha \sin\beta =2\cos\alpha \cos \beta\)

Tangu\(x=\alpha +\beta\) na\(y=\alpha -\beta\),tunaweza kutatua kwa\(\alpha \) na\(\beta \) katika suala la\(x\) na\(y\) na mbadala katika kwa\(2\cos\alpha \cos \beta\) na kupata

\(2\cos \left ( \dfrac{x+y}{2} \right )\cos \left ( \dfrac{x-y}{2} \right )\)

Kwa mazoezi 56-63, kuthibitisha utambulisho.

56)\(\dfrac{\sin(6x)+\sin(4x)}{\sin(6x)-\sin(4x)}=\tan (5x)\cot x\)

57)\(\dfrac{\cos(3x)+\cos x}{\cos(3x)-\cos(4x)}=-\cot (2x)\cot x\)

Jibu

\(\dfrac{\cos(3x)+\cos x}{\cos(3x)-\cos(4x)}=\dfrac{2\cos(2x)\cos x}{-2\sin(2x)\sin x}=-\cot (2x)\cot x\)

58)\(\dfrac{\cos(6y)+\cos(8y)}{\sin(6y)-\sin(4y)}=\cot y\cos(7y)\sec(5y)\)

59)\(\dfrac{\cos(2y)-\cos(4y)}{\sin(2y)+\sin(4y)}=\tan y\)

Jibu

\(\begin{align*} \dfrac{\cos(2y)-\cos(4y)}{\sin(2y)+\sin(4y)} &= \dfrac{-2\sin(3y)\sin(-y)}{2\sin(3y)\cos y}\\ &= \dfrac{2\sin(3y)\sin(y)}{2\sin(3y)\cos y}\\ &= \tan y \end{align*}\)

60)\(\dfrac{\sin(10x)-\sin(2x)}{\cos(10x)+\cos(2x)}=\tan(4x)\)

61)\(\cos x-\cos(3x)=4\sin^2x \cos x\)

Jibu

\(\begin{align*} \cos x-\cos(3x) &= -2\sin(2x)\sin(-x)\\ &= 2(2\sin x \cos x)\sin x\\ &= 4\sin^2x \cos x \end{align*}\)

62)\((\cos(2x)-\cos(4x))^2+(\sin(4x)+\sin(2x))^2=4\sin^2(3x)\)

63)\(\tan \left ( \dfrac{\pi }{4}-t \right )=\dfrac{1-\tan t}{1+\tan t}\)

Jibu

\(\tan \left ( \dfrac{\pi }{4}-t \right )=\dfrac{\tan \left ( \dfrac{\pi }{4} \right )-\tan t}{1+\tan \left ( \dfrac{\pi }{4} \right )\tan t}=\dfrac{1-\tan t}{1+\tan t}\)

7.5: Kutatua equations ya Trigonometric

Katika sehemu za awali za sura hii, tuliangalia utambulisho wa trigonometric. Utambulisho ni kweli kwa maadili yote katika uwanja wa kutofautiana. Katika sehemu hii, tunaanza utafiti wetu wa equations trigonometric kujifunza matukio halisi ya dunia kama vile kutafuta vipimo vya piramidi.

Maneno

1) Je! Kuna daima kuwa na ufumbuzi wa usawa wa kazi ya trigonometric? Kama siyo, kuelezea equation kwamba bila kuwa na ufumbuzi. Eleza kwa nini au kwa nini.

Jibu

Hakutakuwa na ufumbuzi wa usawa wa kazi ya trigonometric. Kwa mfano wa msingi,\(\cos(x)=-5\).

2) Wakati wa kutatua equation ya trigonometric inayohusisha kazi zaidi ya moja ya trig, je, tunataka kujaribu kuandika upya equation hivyo inaelezwa kwa suala la kazi moja ya trigonometric? Kwa nini au kwa nini?

3) Wakati wa kutatua equations linear trig kwa suala la sine tu au cosine, tunajuaje kama kutakuwa na ufumbuzi?

Jibu

Ikiwa kazi ya sine au cosine ina mgawo wa moja, jitenga neno kwa upande mmoja wa ishara sawa. Ikiwa nambari imewekwa sawa na ina thamani kamili chini ya au sawa na moja, equation ina ufumbuzi, vinginevyo haifai. Ikiwa sine au cosine haina mgawo sawa na moja, bado hutenganisha neno lakini kisha ugawanye pande zote mbili za equation na mgawo wa kuongoza. Kisha, kama idadi ni kuweka sawa na ina thamani kamili zaidi ya moja, equation haina ufumbuzi.

Kialjebra

Kwa mazoezi 4-12, pata ufumbuzi wote hasa wakati \(0\leq \theta < 2\pi\)

4)\(2\sin \theta=-\sqrt{2}\)

5)\(2\sin \theta=\sqrt{3}\)

Jibu

\(\dfrac{\pi }{3}, \dfrac{2\pi }{3}\)

6)\(2\cos \theta=1\)

7)\(2\cos \theta=-\sqrt{2}\)

Jibu

\(\dfrac{3\pi }{4}, \dfrac{5\pi }{4}\)

8)\(\tan \theta=-1\)

9)\(\tan x=1\)

Jibu

\(\dfrac{\pi }{4}, \dfrac{5\pi }{4}\)

10)\(\cot x+1=0\)

11)\(4\sin^2 x-2=0\)

Jibu

\(\dfrac{\pi }{4}, \dfrac{3\pi }{4}, \dfrac{5\pi }{4}, \dfrac{7\pi }{4}\)

12)\(\csc^2 x-4=0\)

Kwa mazoezi 13-22, tatua hasa\([0,2\pi )\)

13)\(2\cos \theta=\sqrt{2}\)

Jibu

\(\dfrac{\pi }{4}, \dfrac{7\pi }{4}\)

14)\(2\cos \theta=-1\)

15)\(2\sin \theta=-1\)

Jibu

\(\dfrac{7\pi }{6}, \dfrac{11\pi }{6}\)

16)\(2\sin \theta=-\sqrt{3}\)

17)\(2\sin (3\theta )=1\)

Jibu

\(\dfrac{\pi }{18}, \dfrac{5\pi }{18}, \dfrac{13\pi }{18}, \dfrac{17\pi }{18}, \dfrac{25\pi }{18}, \dfrac{29\pi }{18}\)

18)\(2\sin (2\theta )=\sqrt{3}\)

19)\(2\cos (3\theta )=-\sqrt{2}\)

Jibu

\(\dfrac{3\pi }{12}, \dfrac{5\pi }{12}, \dfrac{11\pi }{12}, \dfrac{13\pi }{12}, \dfrac{19\pi }{12}, \dfrac{21\pi }{12}\)

20)\(\cos (2\theta )=-\dfrac{\sqrt{3}}{2}\)

21)\(2\sin(\pi \theta )=1\)

Jibu

\(\dfrac{1}{6}, \dfrac{5}{6}, \dfrac{13}{6}, \dfrac{17}{6}, \dfrac{25}{6}, \dfrac{29}{6}, \dfrac{37}{6}\)

22)\(2\cos \left(\dfrac{\pi }{5}\theta \right)=\sqrt{3}\)

Kwa mazoezi 23-32, pata ufumbuzi wote halisi\([0,2\pi )\)

23)\(\sec(x)\sin(x)-2\sin(x)=0\)

Jibu

\(0, \dfrac{\pi }{3}, \pi , \dfrac{5\pi }{3}\)

24)\(\tan(x)-2\sin(x)\tan(x)=0\)

25)\(2\cos^2 t+\cos(t)=1\)

Jibu

\(\dfrac{\pi }{3}, \pi , \dfrac{5\pi }{3}\)

26)\(2\tan^2(t)=3\sec(t)\)

27)\(2\sin(x)\cos(x)-\sin(x)+2\cos(x)-1=0\)

Jibu

\(\dfrac{\pi }{3}, \dfrac{3\pi }{2}, \dfrac{5\pi }{3}\)

28)\(\cos^2\theta =\dfrac{1}{2}\)

29)\(\sec^2 x =1\)

Jibu

\(0, \pi \)

30)\(\tan^2(x)=-1+2\tan(-x)\)

31)\(8\sin^2(x)+6\sin(x)+1=0\)

Jibu

\(\pi -\sin^{-1}\left ( -\dfrac{1}{4} \right ), \dfrac{7\pi }{6}, \dfrac{11\pi }{6}, 2\pi +\sin^{-1}\left ( -\dfrac{1}{4} \right )\)

32)\(\tan^5(x)=\tan(x)\)

Kwa mazoezi 33-40, tatua kwa njia zilizoonyeshwa katika sehemu hii hasa wakati\([0,2\pi )\)

33)\(\sin(3x)\cos(6x)-\cos(3x)\sin(6x)=-0.9\)

Jibu

\(\dfrac{1}{3}\left (\sin^{-1}\left ( \dfrac{9}{10} \right ) \right ), \dfrac{\pi }{3}-\dfrac{1}{3}\left (\sin^{-1}\left ( \dfrac{9}{10} \right ) \right ), \dfrac{2\pi }{3}+\dfrac{1}{3}\left (\sin^{-1}\left ( \dfrac{9}{10} \right ) \right ), \pi -\dfrac{1}{3}\left (\sin^{-1}\left ( \dfrac{9}{10} \right ) \right ), \dfrac{4\pi }{3}+\dfrac{1}{3}\left (\sin^{-1}\left ( \dfrac{9}{10} \right ) \right ), \dfrac{5\pi }{3}-\)

34)\(\sin(6x)\cos(11x)-\cos(6x)\sin(11x)=-0.1\)

35)\(\cos(2x)\cos x+\sin(2x)\sin x=1\)

Jibu

\(0\)

36)\(6\sin(2t)+9\sin t=0\)

37)\(9\cos(2\theta )=9\cos^2\theta -4\)

Jibu

\(\dfrac{\pi }{6}, \dfrac{5\pi }{6}, \dfrac{7\pi }{6}, \dfrac{11\pi }{6}\)

38)\(\sin(2t)=\cos t\)

39)\(\cos(2t)=\sin t\)

Jibu

\(\dfrac{3\pi }{2}, \dfrac{\pi }{6}, \dfrac{5\pi }{6}\)

40)\(\cos(6x)-\cos(3x)=0\)

Kwa mazoezi 41-49, tatua hasa wakati\([0,2\pi )\). Tumia formula ya quadratic ikiwa equations haifai.

41)\(\tan^2 x-\sqrt{3}\tan x=0\)

Jibu

\(0, \dfrac{\pi }{3}, \pi , \dfrac{4\pi }{3}\)

42)\(\sin^2 x+\sin x-2=0\)

43)\(\sin^2 x-2\sin x-4=0\)

Jibu

Hakuna ufumbuzi.

44)\(5\cos^2 x+3\cos x-1=0\)

45)\(3\cos^2 x-3\cos x-2=0\)

Jibu

\(\cos^{-1}\left (\dfrac{1}{3}\left ( 1-\sqrt{7} \right ) \right ), 2\pi -\cos^{-1}\left (\dfrac{1}{3}\left ( 1-\sqrt{7} \right ) \right )\)

46)\(5\sin^2 x+2\sin x-1=0\)

47)\(\tan^2 x+5\tan x-1=0\)

Jibu

\(\tan^{-1}\left (\dfrac{1}{2}\left ( \sqrt{29}-5 \right ) \right ), \pi +\tan^{-1}\left (\dfrac{1}{2}\left ( -\sqrt{29}-5 \right ) \right ), \pi +\tan^{-1}\left (\dfrac{1}{2}\left ( \sqrt{29}-5 \right ) \right ), 2\pi +\tan^{-1}\left (\dfrac{1}{2}\left ( -\sqrt{29}-5 \right ) \right )\)

48)\(\cot^2 x=-\cot x\)

49)\(-\tan^2 x-\tan x-2=0\)

Jibu

Hakuna ufumbuzi.

Kwa mazoezi 50-65, pata ufumbuzi halisi kwa muda\([0,2\pi )\). Angalia fursa za kutumia utambulisho wa trigonometric.

50)\(\sin^2 x-\cos^2 x-\sin x=0\)

51)\(\sin^2 x+\cos^2 x=0\)

Jibu

Hakuna ufumbuzi.

52)\(\sin(2x)-\sin x=0\)

53)\(\cos(2x)-\cos x=0\)

Jibu

\(0, \dfrac{2\pi }{3}, \dfrac{4\pi }{3}\)

54)\(\dfrac{2\tan x}{2-\sec ^2 x}-\sin^2 x=\cos^2 x\)

55)\(1-\cos(2x)=1+\cos(2x)\)

Jibu

\(\dfrac{\pi }{4}, \dfrac{3\pi }{4}, \dfrac{5\pi }{4}, \dfrac{7\pi }{4}\)

56)\(\sec^2 x=7\)

57)\(10\sin x \cos x=6 \cos x\)

Jibu

\(\sin^{-1}\left ( \dfrac{3}{5} \right ), \dfrac{\pi }{2}, \pi -\sin^{-1}\left ( \dfrac{3}{5} \right ), \dfrac{3\pi }{2}\)

58)\(-3\sin t=15\cos t \sin t\)

59)\(4\cos^2x - 4 = 15\cos x\)

Jibu

\(\cos^{-1}\left ( -\dfrac{1}{4} \right ), 2\pi -\cos^{-1}\left ( -\dfrac{1}{4} \right )\)

60)\(8\sin^2 x+6\sin x+1=0\)

61)\(8\cos^2 \theta =3-2\cos \theta\)

Jibu

\(\dfrac{\pi }{3}, \cos^{-1}\left ( -\dfrac{3}{4} \right ), 2\pi -\cos^{-1}\left ( -\dfrac{3}{4} \right ), \dfrac{5\pi }{3}\)

62)\(6\cos^2 x+7\sin x-8=0\)

63)\(12\sin^2 t+\cos t-6=0\)

Jibu

\(\cos^{-1}\left ( \dfrac{3}{4} \right ), \cos^{-1}\left ( -\dfrac{2}{3} \right ), 2\pi -\cos^{-1}\left ( -\dfrac{2}{3} \right ), 2\pi -\cos^{-1}\left ( \dfrac{3}{4} \right )\)

64)\(\tan x=3\sin x\)

65)\(\cos^3 t=\cos t\)

Jibu

\(0, \dfrac{\pi }{2}, \pi , \dfrac{3\pi }{2}\)

Graphic

Kwa mazoezi 66-72, algebraically kuamua ufumbuzi wote wa equation trigonometric hasa, kisha kuthibitisha matokeo kwa kuchora equation na kutafuta zero.

66)\(6\sin^2 x-5\sin x+1=0\)

67)\(8\cos^2 x-2\cos x-1=0\)

Jibu

\(\dfrac{\pi }{3}, \cos^{-1}\left ( -\dfrac{1}{4} \right ), 2\pi -\cos^{-1}\left ( -\dfrac{1}{4} \right ), \dfrac{5\pi }{3}\)

68)\(100\tan^2x+20\tan x-3=0\)

69)\(2\cos^2 x-\cos x+15=0\)

Jibu

Hakuna ufumbuzi.

70)\(20\sin^2 x-27\sin x+7=0\)

71)\(2\tan^2 x+7\tan x+6=0\)

Jibu

\(\pi +\tan^{-1}(-2), \pi +\tan^{-1}\left (-\dfrac{3}{2}\right ), 2\pi +\tan^{-1}(-2), 2\pi +\tan^{-1}\left (-\dfrac{3}{2} \right )\)

72)\(130\tan^2 x+69\tan x-130=0\)

Teknolojia

Kwa mazoezi 73-76, tumia calculator ili kupata ufumbuzi wote kwa maeneo manne ya decimal.

73)\(\sin x=0.27\)

Jibu

\(2\pi k+0.2734, 2\pi k+2.8682\)

74)\(\sin x=-0.55\)

75)\(\tan x=-0.34\)

Jibu

\(\pi k-0.3277\)

76)\(\cos x=0.71\)

77)\(\tan^2 x+3\tan x-3=0\)

Jibu

\(0.6694,1.8287,3.8110,4.9703\)

78)\(6\tan^2 x+13\tan x=-6\)

79)\(\tan^2 x-\sec x=1\)

Jibu

\(1.0472,3.1416,5.2360\)

80)\(\sin^2 x-2\cos^2 x=0\)

81)\(2\tan^2 x+9\tan x-6=0\)

Jibu

\(0.5326,1.7648,3.6742,4.9064\)

82)\(4\sin^2 x+\sin(2x)\sec x-3=0\)

Upanuzi

Kwa mazoezi 83-92, tafuta ufumbuzi wote hasa kwa equations juu ya muda\([0,2\pi )\).

83)\(\csc^2 x-3\csc x-4=0\)

Jibu

\(\sin^{-1}\left ( \dfrac{1}{4} \right ), \pi -\sin^{-1}\left ( \dfrac{1}{4} \right ), \dfrac{3\pi }{2}\)

84)\(\sin^2 x-\cos^2 x-1=0\)

85)\(\sin^2 x(1-\sin^2 x)+\cos^2 x(1-\sin^2 x)=0\)

Jibu

\(\dfrac{\pi }{2}, \dfrac{3\pi }{2}\)

86)\(3\sec^2 x+2+\sin^2 x-\tan^2 x+\cos^2 x=0\)

87)\(\sin^2 x-1+2\cos(2x)-\cos^2 x=1\)

Jibu

Hakuna ufumbuzi.

88)\(\tan^2 x-1-\sec^3 x \cos x=0\)

89)\(\dfrac{\sin (2x)}{\sec ^2 x}=0\)

Jibu

\(0, \dfrac{\pi }{2}, \pi , \dfrac{3\pi }{2}\)

90)\(\dfrac{\sin (2x)}{2\csc ^2 x}=0\)

91)\(2\cos^2 x-\sin^2 x-\cos x-5=0\)

Jibu

Hakuna ufumbuzi.

92)\(\dfrac{1}{\sec ^2 x}+2+\sin ^2 x+4\cos ^2 x=4\)

Real-World Matumizi

93) Ndege ina gesi ya kutosha tu kuruka hadi mji\(200\) maili kaskazini mashariki mwa eneo lake la sasa. Kama majaribio anajua kwamba mji ni\(25\) maili kaskazini, ngapi digrii kaskazini ya mashariki lazima ndege kuruka?

Jibu

\(7.2^{\circ}\)

94) Ikiwa barabara ya upakiaji imewekwa karibu na lori, kwa urefu wa\(4\) miguu, na barabara ni\(15\) miguu ndefu, ni pembe gani inayofanya barabara na ardhi?

95) Ikiwa barabara ya upakiaji imewekwa karibu na lori, kwa urefu wa\(2\) miguu, na barabara ni\(20\) miguu ndefu, ni pembe gani inayofanya barabara na ardhi?

Jibu

\(5.7^{\circ}\)

96) mwanamke ni kuangalia roketi ilizinduliwa sasa\(11\) maili katika urefu. Ikiwa yeye amesimama\(4\) maili kutoka pedi ya uzinduzi, ni pembe gani anaangalia kutoka usawa?

97) mwanaanga ni katika roketi ilizindua sasa\(15\) maili katika urefu. Ikiwa mtu amesimama\(2\) maili kutoka pedi ya uzinduzi, kwa pembe gani anamtazama kutoka usawa? (Kidokezo: hii inaitwa angle ya unyogovu.)

Jibu

\(82.4^{\circ}\)

98) Mwanamke amesimama\(8\) mita mbali na jengo la urefu wa\(10\) mita. Ni pembe gani anaangalia juu ya jengo?

99) Mtu amesimama\(10\) mita mbali na jengo la urefu wa\(6\) mita. Mtu aliye juu ya jengo anamtazama. Ni pembe gani mtu anayemtazama?

Jibu

\(31.0^{\circ}\)

100) Jengo refu la\(20\) mguu lina kivuli ambacho ni\(55\) miguu ndefu. Je! Ni pembe gani ya mwinuko wa jua?

101) Jengo la urefu wa\(90\) mguu lina kivuli ambacho ni\(2\) miguu ndefu. Je! Ni pembe gani ya mwinuko wa jua?

Jibu

\(88.7^{\circ}\)

102) Uangalizi juu ya\(3\) mita za ardhi kutoka kwa mtu mrefu wa\(2\) mita hutupa kivuli cha\(6\) mita kwenye\(6\) mita za ukuta kutoka kwa mtu huyo. Kwa nuru gani ni mwanga?

103) Uangalizi juu ya\(3\) miguu ya ardhi kutoka kwa mwanamke mrefu wa\(5\) mguu hupiga kivuli cha\(15\) mguu mrefu juu ya\(6\) miguu ya ukuta kutoka kwa mwanamke. Kwa nuru gani ni mwanga?

Jibu

\(59.0^{\circ}\)

Kwa mazoezi 104-106, pata suluhisho la tatizo la neno algebraically. Kisha tumia calculator ili kuthibitisha matokeo. Pindua jibu kwa kumi ya karibu ya shahada.

104) Mtu anafanya mkono na miguu yake kugusa ukuta na mikono yake\(1.5\) miguu mbali na ukuta. Ikiwa mtu huyo ni\(6\) miguu mirefu, miguu yake hufanya pembe gani na ukuta?

105) Mtu anafanya mkono na miguu yake kugusa ukuta na mikono yake\(3\) miguu mbali na ukuta. Ikiwa mtu huyo ni\(5\) miguu mirefu, miguu yake hufanya pembe gani na ukuta?

Jibu

\(36.9^{\circ}\)

106) Ngazi ya\(23\) mguu imewekwa karibu na nyumba. Ikiwa ngazi inakwenda kwa\(7\) miguu kutoka nyumbani wakati hakuna traction ya kutosha, ni angle gani lazima ngazi kufanya na ardhi ili kuepuka kuacha?

7.6: Mfano na Ulinganifu wa Trigonometric

Matukio mengi ya asili pia ni mara kwa mara. Kwa mfano, awamu za mwezi zina kipindi cha takriban\(28\) siku, na ndege wanajua kuruka kusini takriban wakati uleule kila mwaka. Hivyo jinsi gani tunaweza mfano equation kutafakari tabia ya mara kwa mara? Kwanza, tunapaswa kukusanya na kurekodi data. Kisha tunapata kazi inayofanana na muundo uliozingatiwa na kubadilisha kazi ili kupata mfano wa kutegemewa. Hapa. tutaangalia zaidi aina maalum za tabia ya mara kwa mara na usawa wa mfano ili kufaa data.

Maneno

1) Eleza aina gani za matukio ya kimwili ni bora kulingana na kazi za sinusoidal. Je! Ni sifa gani zinazohitajika?

Jibu

Tabia ya kimwili inapaswa kuwa mara kwa mara, au mzunguko.

2) Ni habari gani ni muhimu kujenga mfano wa trigonometric wa joto la kila siku? Kutoa mifano ya seti mbili tofauti ya habari ambayo itawezesha modeling na equation.

3) Ikiwa tunataka kutengeneza mvua za nyongeza zaidi ya mwaka, je, kazi ya sinusoidal itakuwa mfano mzuri? Kwa nini au kwa nini?

Jibu

Kwa kuwa mvua nyingi zinaongezeka, kazi ya sinusoidal haiwezi kuwa bora hapa.

4) Eleza athari za sababu ya uchafu kwenye grafu za kazi za mwendo wa harmonic.

Kialjebra

Kwa mazoezi 5-13, pata fomu inayowezekana ya kazi ya trigonometric iliyowakilishwa na meza iliyotolewa ya maadili.

5)

\(x\)	\(y\)
\ (x\) ">\(0\)	\ (y\) ">\(−4\)
\ (x\) ">\(3\)	\ (y\) ">\(−1\)
\ (x\) ">\(6\)	\ (y\) ">\(2\)
\ (x\) ">\(9\)	\ (y\) ">\(−1\)
\ (x\) ">\(12\)	\ (y\) ">\(−4\)
\ (x\) ">\(15\)	\ (y\) ">\(−1\)
\ (x\) ">\(18\)	\ (y\) ">\(2\)

 

Jibu

\(y=−3 \cos \left(\dfrac{π}{6}x \right)−1\)

6)

\(x\)	\(y\)
\ (x\) ">\(0\)	\ (y\) ">\(5\)
\ (x\) ">\(2\)	\ (y\) ">\(1\)
\ (x\) ">\(4\)	\ (y\) ">\(−3\)
\ (x\) ">\(6\)	\ (y\) ">\(1\)
\ (x\) ">\(8\)	\ (y\) ">\(5\)
\ (x\) ">\(10\)	\ (y\) ">\(1\)
\ (x\) ">\(12\)	\ (y\) ">\(−3\)

7)

\(x\)	\(y\)
\ (x\) ">\(0\)	\ (y\) ">\(2\)
\ (x\) ">\(\frac{π}{4}\)	\ (y\) ">\(7\)
\ (x\) ">\(\frac{π}{2}\)	\ (y\) ">\(2\)
\ (x\) ">\(\frac{3π}{4}\)	\ (y\) ">\(−3\)
\ (x\) ">\(π\)	\ (y\) ">\(2\)
\ (x\) ">\(\frac{5π}{4}\)	\ (y\) ">\(7\)
\ (x\) ">\(\frac{3π}{2}\)	\ (y\) ">\(2\)

Jibu

\(5 \sin (2x)+2\)

8)

\(x\)	\(y\)
\ (x\) ">\(0\)	\ (y\) ">\(2\)
\ (x\) ">\(\frac{π}{4}\)	\ (y\) ">\(7\)
\ (x\) ">\(\frac{π}{2}\)	\ (y\) ">\(2\)
\ (x\) ">\(\frac{3π}{4}\)	\ (y\) ">\(−3\)
\ (x\) ">\(π\)	\ (y\) ">\(2\)
\ (x\) ">\(\frac{5π}{4}\)	\ (y\) ">\(7\)
\ (x\) ">\(\frac{3π}{2}\)	\ (y\) ">\(2\)

9)

\(x\)	\(y\)
\ (x\) ">\(0\)	\ (y\) ">\(1\)
\ (x\) ">\(1\)	\ (y\) ">\(−3\)
\ (x\) ">\(2\)	\ (y\) ">\(−7\)
\ (x\) ">\(3\)	\ (y\) ">\(−3\)
\ (x\) ">\(4\)	\ (y\) ">\(1\)
\ (x\) ">\(5\)	\ (y\) ">\(−3\)
\ (x\) ">\(6\)	\ (y\) ">\(−7\)

Jibu

\(4\cos \left(\dfrac{xπ}{2} \right)−3\)

10)

\(x\)	\(y\)
\ (x\) ">\(0\)	\ (y\) ">\(−2\)
\ (x\) ">\(1\)	\ (y\) ">\(4\)
\ (x\) ">\(2\)	\ (y\) ">\(10\)
\ (x\) ">\(3\)	\ (y\) ">\(4\)
\ (x\) ">\(4\)	\ (y\) ">\(−2\)
\ (x\) ">\(5\)	\ (y\) ">\(4\)
\ (x\) ">\(6\)	\ (y\) ">\(10\)

11)

\(x\)	\(y\)
\ (x\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; picha ya background:hakuna; asili ya asili: sanduku la padding; background-msimamo-x: 0%; background-kurudia:kurudia; background-ukubwa: auto; mpaka chini-rangi: rgb (204, 204, 204); mpaka chini-style: dashed; mpaka wa chini-upana: 1px; mpaka -picha-outset: 0; mpakana-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpaka picha-upana-upana: 1; mpaka-kushoto-rangi: rgb (204, 204, 204); mpaka-kushoto-style: dashed; - upana wa kulia:1px; mpako-juu-rangi: rgb (204, 204 , 204); mpako-juu-style: dashed; upana-juu-upana: 1px; sanduku-sizing: sanduku la mpaka; padding-chini-chini: 16px; padding-kulia: 16px; padding-juu-juu: 16px; maandishi-align: katikati; wima align: juu; neno-wrap: kuvunja-neno; ">\(0\)	\ (y\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; picha ya background:hakuna; asili ya asili: sanduku la padding; background-msimamo-x: 0%; background-kurudia:kurudia; background-ukubwa: auto; mpaka chini-rangi: rgb (204, 204, 204); mpaka chini-style: dashed; mpaka wa chini-upana: 1px; mpaka -picha-outset: 0; mpakana-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpaka picha-upana-upana: 1; mpaka-kushoto-rangi: rgb (204, 204, 204); mpaka-kushoto-style: dashed; - upana wa kulia:1px; mpako-juu-rangi: rgb (204, 204 , 204); mpako-juu-style: dashed; upana-juu-upana: 1px; sanduku-sizing: sanduku la mpaka; padding-chini-chini: 16px; padding-kulia: 16px; padding-juu-juu: 16px; maandishi-align: katikati; wima align: juu; neno-wrap: kuvunja-neno; ">\(5\)
\ (x\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; rangi ya background: rgb (243, 251, 255); picha ya background:hakuna; asili ya asili:sanduku; background-msimamo-x: 0%; background-kurudia: kurudia; background: auto; mpaka chini-rangi: rgb: rgb (204, 204, 204); Mpaka-chini-style: dashed; mpaka-chini-upana: 1px; mpaka-picha-outset: 0; mpaka-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpako-picha-upana: 1; mpaka-kushoto-rangi: rgb (204, 204); mpaka-kushoto-style: dashed; mpaka-kushoto-upana: 1px; mpaka-haki-rangi: rgb (204, 204), 204); mpaka-haki-style: dashed; mpaka-haki- upana: 1px; mpaka-juu-rangi: rgb (204, 204, 204); mpako-juu-style: dashed; upana-upana-upana: 1px; sanduku-sizing: sanduku mpaka; padding-chini: 16px; padding-juu-juu: 16px; Nakala-align: juu; neno-wrap: kuvunja neno; ">\(1\)	\ (y\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; rangi ya background: rgb (243, 251, 255); picha ya background:hakuna; asili ya asili:sanduku; background-msimamo-x: 0%; background-kurudia: kurudia; background: auto; mpaka chini-rangi: rgb: rgb (204, 204, 204); Mpaka-chini-style: dashed; mpaka-chini-upana: 1px; mpaka-picha-outset: 0; mpaka-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpako-picha-upana: 1; mpaka-kushoto-rangi: rgb (204, 204); mpaka-kushoto-style: dashed; mpaka-kushoto-upana: 1px; mpaka-haki-rangi: rgb (204, 204), 204); mpaka-haki-style: dashed; mpaka-haki- upana: 1px; mpaka-juu-rangi: rgb (204, 204, 204); mpako-juu-style: dashed; upana-upana-upana: 1px; sanduku-sizing: sanduku mpaka; padding-chini: 16px; padding-juu-juu: 16px; Nakala-align: juu; neno-wrap: kuvunja neno; ">\(-3\)
\ (x\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; picha ya background:hakuna; asili ya asili: sanduku la padding; background-msimamo-x: 0%; background-kurudia:kurudia; background-ukubwa: auto; mpaka chini-rangi: rgb (204, 204, 204); mpaka chini-style: dashed; mpaka wa chini-upana: 1px; mpaka -picha-outset: 0; mpakana-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpaka picha-upana-upana: 1; mpaka-kushoto-rangi: rgb (204, 204, 204); mpaka-kushoto-style: dashed; - upana wa kulia:1px; mpako-juu-rangi: rgb (204, 204 , 204); mpako-juu-style: dashed; upana-juu-upana: 1px; sanduku-sizing: sanduku la mpaka; padding-chini-chini: 16px; padding-kulia: 16px; padding-juu-juu: 16px; maandishi-align: katikati; wima align: juu; neno-wrap: kuvunja-neno; ">\(2\)	\ (y\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; picha ya background:hakuna; asili ya asili: sanduku la padding; background-msimamo-x: 0%; background-kurudia:kurudia; background-ukubwa: auto; mpaka chini-rangi: rgb (204, 204, 204); mpaka chini-style: dashed; mpaka wa chini-upana: 1px; mpaka -picha-outset: 0; mpakana-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpaka picha-upana-upana: 1; mpaka-kushoto-rangi: rgb (204, 204, 204); mpaka-kushoto-style: dashed; - upana wa kulia:1px; mpako-juu-rangi: rgb (204, 204 , 204); mpako-juu-style: dashed; upana-juu-upana: 1px; sanduku-sizing: sanduku la mpaka; padding-chini-chini: 16px; padding-kulia: 16px; padding-juu-juu: 16px; maandishi-align: katikati; wima align: juu; neno-wrap: kuvunja-neno; ">\(5\)
\ (x\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; rangi ya background: rgb (239, 239, 239); picha ya background:hakuna; asili ya asili:sanduku; background-msimamo-x: 0%; background-kurudia: kurudia; background: auto; mpaka chini-rangi: rgb: rgb (204, 204, 204); Mpaka-chini-style: dashed; mpaka-chini-upana: 1px; mpaka-picha-outset: 0; mpaka-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpako-picha-upana: 1; mpaka-kushoto-rangi: rgb (204, 204); mpaka-kushoto-style: dashed; mpaka-kushoto-upana: 1px; mpaka-haki-rangi: rgb (204, 204), 204); mpaka-haki-style: dashed; mpaka-haki- upana: 1px; mpaka-juu-rangi: rgb (204, 204, 204); mpako-juu-style: dashed; upana-upana-upana: 1px; sanduku-sizing: sanduku mpaka; padding-chini: 16px; padding-juu-juu: 16px; Nakala-align: juu; neno-wrap: kuvunja neno; ">\(3\)	\ (y\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; rangi ya background: rgb (239, 239, 239); picha ya background:hakuna; asili ya asili:sanduku; background-msimamo-x: 0%; background-kurudia: kurudia; background: auto; mpaka chini-rangi: rgb: rgb (204, 204, 204); Mpaka-chini-style: dashed; mpaka-chini-upana: 1px; mpaka-picha-outset: 0; mpaka-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpako-picha-upana: 1; mpaka-kushoto-rangi: rgb (204, 204); mpaka-kushoto-style: dashed; mpaka-kushoto-upana: 1px; mpaka-haki-rangi: rgb (204, 204), 204); mpaka-haki-style: dashed; mpaka-haki- upana: 1px; mpaka-juu-rangi: rgb (204, 204, 204); mpako-juu-style: dashed; upana-upana-upana: 1px; sanduku-sizing: sanduku mpaka; padding-chini: 16px; padding-juu-juu: 16px; Nakala-align: juu; neno-wrap: kuvunja neno; ">\(13\)
\ (x\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; picha ya background:hakuna; asili ya asili: sanduku la padding; background-msimamo-x: 0%; background-kurudia:kurudia; background-ukubwa: auto; mpaka chini-rangi: rgb (204, 204, 204); mpaka chini-style: dashed; mpaka wa chini-upana: 1px; mpaka -picha-outset: 0; mpakana-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpaka picha-upana-upana: 1; mpaka-kushoto-rangi: rgb (204, 204, 204); mpaka-kushoto-style: dashed; - upana wa kulia:1px; mpako-juu-rangi: rgb (204, 204 , 204); mpako-juu-style: dashed; upana-juu-upana: 1px; sanduku-sizing: sanduku la mpaka; padding-chini-chini: 16px; padding-kulia: 16px; padding-juu-juu: 16px; maandishi-align: katikati; wima align: juu; neno-wrap: kuvunja-neno; ">\(4\)	\ (y\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; picha ya background:hakuna; asili ya asili: sanduku la padding; background-msimamo-x: 0%; background-kurudia:kurudia; background-ukubwa: auto; mpaka chini-rangi: rgb (204, 204, 204); mpaka chini-style: dashed; mpaka wa chini-upana: 1px; mpaka -picha-outset: 0; mpakana-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpaka picha-upana-upana: 1; mpaka-kushoto-rangi: rgb (204, 204, 204); mpaka-kushoto-style: dashed; - upana wa kulia:1px; mpako-juu-rangi: rgb (204, 204 , 204); mpako-juu-style: dashed; upana-juu-upana: 1px; sanduku-sizing: sanduku la mpaka; padding-chini-chini: 16px; padding-kulia: 16px; padding-juu-juu: 16px; maandishi-align: katikati; wima align: juu; neno-wrap: kuvunja-neno; ">\(5\)
\ (x\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; rangi ya background: rgb (239, 239, 239); picha ya background:hakuna; asili ya asili:sanduku; background-msimamo-x: 0%; background-kurudia: kurudia; background: auto; mpaka chini-rangi: rgb: rgb (204, 204, 204); Mpaka-chini-style: dashed; mpaka-chini-upana: 1px; mpaka-picha-outset: 0; mpaka-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpako-picha-upana: 1; mpaka-kushoto-rangi: rgb (204, 204); mpaka-kushoto-style: dashed; mpaka-kushoto-upana: 1px; mpaka-haki-rangi: rgb (204, 204), 204); mpaka-haki-style: dashed; mpaka-haki- upana: 1px; mpaka-juu-rangi: rgb (204, 204, 204); mpako-juu-style: dashed; upana-upana-upana: 1px; sanduku-sizing: sanduku mpaka; padding-chini: 16px; padding-juu-juu: 16px; Nakala-align: juu; neno-wrap: kuvunja neno; ">\(5\)	\ (y\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; rangi ya background: rgb (239, 239, 239); picha ya background:hakuna; asili ya asili:sanduku; background-msimamo-x: 0%; background-kurudia: kurudia; background: auto; mpaka chini-rangi: rgb: rgb (204, 204, 204); Mpaka-chini-style: dashed; mpaka-chini-upana: 1px; mpaka-picha-outset: 0; mpaka-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpako-picha-upana: 1; mpaka-kushoto-rangi: rgb (204, 204); mpaka-kushoto-style: dashed; mpaka-kushoto-upana: 1px; mpaka-haki-rangi: rgb (204, 204), 204); mpaka-haki-style: dashed; mpaka-haki- upana: 1px; mpaka-juu-rangi: rgb (204, 204, 204); mpako-juu-style: dashed; upana-upana-upana: 1px; sanduku-sizing: sanduku mpaka; padding-chini: 16px; padding-juu-juu: 16px; Nakala-align: juu; neno-wrap: kuvunja neno; ">\(-3\)
\ (x\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; picha ya background:hakuna; asili ya asili: sanduku la padding; background-msimamo-x: 0%; background-kurudia:kurudia; background-ukubwa: auto; mpaka chini-rangi: rgb (204, 204, 204); mpaka chini-style: dashed; mpaka wa chini-upana: 1px; mpaka -picha-outset: 0; mpako-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpaka picha-picha-upana: 1; mpaka-kushoto-rangi: rgb (204, 204, 204); mpaka-kushoto-style: dashed; - upana wa kulia:1px; mpako-juu-rangi: rgb (204, 204 , 204); mpako-juu-style: dashed; upana-juu-upana: 1px; sanduku-sizing: sanduku la mpaka; padding-chini-chini: 16px; padding-kulia: 16px; padding-juu-juu: 16px; maandishi-align: katikati; wima align: juu; neno-wrap: kuvunja-neno; ">\(6\)	\ (y\)” style="background-attachment:kitabu; background-clip:sanduku la mpaka; picha ya background:hakuna; asili ya asili: sanduku la padding; background-msimamo-x: 0%; background-kurudia:kurudia; background-ukubwa: auto; mpaka chini-rangi: rgb (204, 204, 204); mpaka chini-style: dashed; mpaka wa chini-upana: 1px; mpaka -picha-outset: 0; mpako-picha-kurudia: kunyoosha; mpaka-picha-kipande: 100%; mpaka picha-picha-upana: 1; mpaka-kushoto-rangi: rgb (204, 204, 204); mpaka-kushoto-style: dashed; - upana wa kulia:1px; mpako-juu-rangi: rgb (204, 204 , 204); mpako-juu-style: dashed; upana-juu-upana: 1px; sanduku-sizing: sanduku la mpaka; padding-chini-chini: 16px; padding-kulia: 16px; padding-juu-juu: 16px; maandishi-align: katikati; wima align: juu; neno-wrap: kuvunja-neno; ">\(5\)

Jibu

\(5−8 \sin \left(\dfrac{xπ}{2} \right)\)

12)

\(x\)	\(y\)
\ (x\) ">\(−3\)	\ (y\) ">\(−1−\sqrt{2}\)
\ (x\) ">\(−2\)	\ (y\) ">\(−1\)
\ (x\) ">\(−1\)	\ (y\) ">\(1−\sqrt{2}\)
\ (x\) ">\(0\)	\ (y\) ">\(0\)
\ (x\) ">\(1\)	\ (y\) ">\(\sqrt{2}−1\)
\ (x\) ">\(2\)	\ (y\) ">\(1\)
\ (x\) ">\(3\)	\ (y\) ">\(\sqrt{2}+1\)

13)

\(x\)	\(y\)
\ (x\) ">\(−1\)	\ (y\) ">\(\sqrt{3}−2\)
\ (x\) ">\(0\)	\ (y\) ">\(0\)
\ (x\) ">\(1\)	\ (y\) ">\(2−\sqrt{3}\)
\ (x\) ">\(2\)	\ (y\) ">\(\frac{\sqrt{3}}{3}\)
\ (x\) ">\(3\)	\ (y\) ">\(1\)
\ (x\) ">\(4\)	\ (y\) ">\(\sqrt{3}\)
\ (x\) ">\(5\)	\ (y\) ">\(2+\sqrt{3}\)

Jibu

\(\tan \left(\dfrac{xπ}{12} \right)\)

Graphic

Kwa mazoezi 14-16, graph kazi iliyotolewa, na kisha kupata iwezekanavyo mchakato wa kimwili kwamba equation inaweza mfano.

14)\(f(x)=−30 \cos \left(\dfrac{xπ}{6} \right)−20 \cos ^2 \left(\dfrac{xπ}{6} \right)+80 \; [0,12]\)

15)\(f(x)=−18 \cos \left(\dfrac{xπ}{12} \right)−5 \sin \left(\dfrac{xπ}{12} \right)+100\) juu ya muda\([0,24]\)

Jibu

Majibu yatatofautiana. Jibu la sampuli: Kazi hii inaweza kuiga mabadiliko ya joto wakati wa siku moja ya moto sana huko Phoenix, Arizona.

16)\(f(x)=10−\sin \left(\dfrac{xπ}{6} \right)+24 \tan \left(\dfrac{xπ}{240} \right)\) kwa muda\([0,80]\)

Teknolojia

Kwa ajili ya zoezi 17, kujenga kazi modeling tabia na kutumia calculator kupata matokeo taka.

17) Mji wa wastani wa mvua kwa sasa ni\(20\) inchi na inatofautiana msimu kwa\(5\) inchi. Kutokana na hali zisizotarajiwa, mvua inaonekana kupungua kwa\(15\%\) kila mwaka. Ni miaka ngapi kuanzia sasa tunatarajia mvua kufikia\(0\) inchi za awali? Kumbuka, mfano huo ni batili mara moja unatabiri mvua mbaya, hivyo chagua hatua ya kwanza ambayo inakwenda chini\(0\).

Jibu

\(9\)miaka kuanzia sasa

Real-World Matumizi

Kwa mazoezi 18-29, jenga kazi ya sinusoidal na taarifa iliyotolewa, na kisha kutatua equation kwa maadili yaliyotakiwa.

18) Nje ya joto juu ya kipindi cha siku inaweza kuonyeshwa kama kazi ya sinusoidal. Tuseme joto la juu la\(105°F\) hutokea saa 5:00 na joto la wastani kwa siku ni\(85°F\). Pata joto, kwa kiwango cha karibu, saa 9AM.

19) Nje ya joto juu ya kipindi cha siku inaweza kuonyeshwa kama kazi ya sinusoidal. Tuseme joto la juu la\(84°F\) hutokea saa 6:00 na joto la wastani kwa siku ni\(70°F.\) Kupata joto, kwa kiwango cha karibu, saa 7AM.

Jibu

\(56 °F\)

20) Nje ya joto juu ya kipindi cha siku inaweza kuonyeshwa kama kazi ya sinusoidal. Tuseme joto linatofautiana kati\(47°F\) na\(63°F\) wakati wa mchana na wastani wa joto la kila siku hutokea saa 10 asubuhi. Ni saa ngapi baada ya usiku wa manane joto la kwanza linafikia\(51°F\)?

21) Nje ya joto juu ya kipindi cha siku inaweza kuonyeshwa kama kazi ya sinusoidal. Tuseme joto linatofautiana kati\(64°F\) na\(86°F\) wakati wa mchana na wastani wa joto la kila siku hutokea saa 12 asubuhi. Ni saa ngapi baada ya usiku wa manane joto la kwanza linafikia\(70°F\)?

Jibu

\(1.8024\)masaa

22) Gurudumu la Ferris ni\(20\) mita mduara na limepanda kutoka jukwaa ambalo ni\(2\) mita juu ya ardhi. Msimamo wa saa sita kwenye gurudumu la Ferris ni kiwango na jukwaa la upakiaji. Gurudumu hukamilisha mapinduzi\(1\) kamili kwa\(6\) dakika. Ni kiasi gani cha safari, kwa dakika na sekunde, hutumiwa zaidi kuliko\(13\) mita juu ya ardhi?

23) Gurudumu la Ferris ni\(45\) mita za kipenyo na limepanda kutoka kwenye jukwaa ambalo ni\(1\) mita juu ya ardhi. Msimamo wa saa sita kwenye gurudumu la Ferris ni kiwango na jukwaa la upakiaji. Gurudumu hukamilisha mapinduzi\(1\) kamili kwa\(10\) dakika. Ni dakika ngapi za safari zinazotumiwa zaidi kuliko\(27\) mita juu ya ardhi? Pande zote kwa pili ya pili.

Jibu

\(4:30\)

24) Eneo la barafu la bahari karibu na Ncha ya Kaskazini linabadilika kati ya kilomita za mraba\(6\) milioni mnamo Septemba 1 hadi kilomita za mraba\(14\) milioni Machi 1. Kutokana na kushuka kwa sinusoidal, wakati kuna kilomita za mraba milioni 9 za barafu la bahari? Kutoa jibu lako kama tarehe mbalimbali, kwa siku ya karibu.

25) Eneo la barafu la bahari karibu na Pole la Kusini linabadilika kati ya kilomita za mraba\(18\) milioni mnamo Septemba hadi kilomita za mraba\(3\) milioni mwezi Machi. Kutokana na kushuka kwa sinusoidal, wakati kuna zaidi ya kilomita za mraba milioni 15 za barafu la bahari? Kutoa jibu lako kama tarehe mbalimbali, kwa siku ya karibu.

Jibu

Kuanzia Julai 8 hadi Oktoba 23

26) Wakati wa msimu wa msimu wa\(90\) siku, mvua ya kila siku inaweza kuonyeshwa na kazi za sinusoidal. Kama mvua fluctuates kati ya chini ya\(2\) inchi siku\(10\) na\(12\) inchi siku\(55\), wakati gani ni mvua ya kila siku zaidi ya\(10\) inchi?

27) Wakati wa msimu wa msimu wa\(90\) siku, mvua ya kila siku inaweza kuonyeshwa na kazi za sinusoidal. Chini ya\(4\) inchi ya mvua ilirekodiwa siku\(30\), na kwa ujumla wastani wa mvua ya kila siku ilikuwa\(8\) inchi. Katika kipindi gani ilikuwa mvua ya kila siku chini ya\(5\) inchi?

Jibu

Kutoka siku\(19\) hadi siku\(40\)

28) Katika eneo fulani, mvua ya kila mwezi inakaribia\(8\) inchi Juni 1 na huanguka chini ya\(1\) inchi mnamo Desemba 1. Tambua vipindi ambapo eneo liko chini ya hali ya mafuriko (zaidi ya\(7\) inchi) na hali ya ukame (chini ya\(2\) inchi). Kutoa jibu lako katika suala la siku ya karibu.

29) Katika eneo fulani, mvua ya kila mwezi inakaribia\(24\) inchi mnamo Septemba na huanguka chini ya\(4\) inchi mwezi Machi. Tambua vipindi ambapo eneo liko chini ya hali ya mafuriko (zaidi ya\(22\) inchi) na hali ya ukame (chini ya\(5\) inchi). Kutoa jibu lako katika suala la siku ya karibu.

Jibu

Mafuriko: Julai 24 hadi Oktoba 7. Ukame: Februari 4 hadi Machi 27

Kwa mazoezi 30-32, pata amplitude, kipindi, na mzunguko wa kazi iliyotolewa.

30) Uhamisho\(h(t)\) kwa sentimita ya wingi uliosimamishwa na chemchemi unatokana na kazi\(h(t)=8 \sin (6πt),\) ambapo\(t\) hupimwa kwa sekunde. Pata amplitude, kipindi, na mzunguko wa makazi haya.

31) Uhamisho\(h(t)\) kwa sentimita ya wingi uliosimamishwa na chemchemi unatokana na kazi\(h(t)=11 \sin (12πt),\) ambapo\(t\) hupimwa kwa sekunde. Pata amplitude, kipindi, na mzunguko wa makazi haya.

Jibu

Amplitude:\(11\), kipindi:\(\dfrac{1}{6}\), mzunguko:\(6\) Hz

32) Uhamisho\(h(t)\) kwa sentimita ya wingi uliosimamishwa na chemchemi unatokana na kazi\(h(t)=4 \cos \left(\dfrac{π}{2}t \right)\), ambapo\(t\) hupimwa kwa sekunde. Pata amplitude, kipindi, na mzunguko wa makazi haya.

Kwa ajili ya zoezi 33, kujenga equation kwamba mifano ya tabia ilivyoelezwa.

33) Uhamisho\(h(t)\), kwa sentimita, wa wingi uliosimamishwa na chemchemi unatokana na kazi\(h(t)=−5 \cos (60πt)\), ambapo\(t\) hupimwa kwa sekunde. Pata amplitude, kipindi, na mzunguko wa makazi haya.

Jibu

Amplitude:\(5\), kipindi:\(\dfrac{1}{30}\), mzunguko:\(30\) Hz

Kwa ajili ya mazoezi 34-41, kujenga equation kwamba mfano tabia ilivyoelezwa.

34) Idadi ya kulungu oscillates\(19\) juu na chini ya wastani wakati wa mwaka, kufikia thamani ya chini kabisa mwezi Januari. Wastani wa idadi ya watu huanza kwa\(800\) kulungu na kuongezeka kwa\(160\) kila mwaka. Kupata kazi ambayo mifano ya idadi ya watu,\(P\), katika suala la miezi tangu Januari,\(t\).

35) Idadi ya sungura oscillates\(15\) juu na chini ya wastani wakati wa mwaka, kufikia thamani ya chini kabisa mwezi Januari. Wastani wa idadi ya watu huanza kwenye\(650\) sungura na huongezeka kwa\(110\) kila mwaka. Kupata kazi ambayo mifano ya idadi ya watu,\(P\), katika suala la miezi tangu Januari,\(t\).

Jibu

\(P(t)=−15 \cos \left(\dfrac{π}{6}t \right)+650+\dfrac{55}{6}t\)

36) idadi ya watu muskrat oscillates\(33\) juu na chini ya wastani wakati wa mwaka, kufikia thamani ya chini katika Januari. Idadi ya watu wastani huanza kwenye\(900\) muskrats na huongezeka kwa\(7\%\) kila mwezi. Kupata kazi ambayo mifano ya idadi ya watu,\(P\), katika suala la miezi tangu Januari,\(t\).

37) Idadi ya samaki oscillates\(40\) juu na chini ya wastani wakati wa mwaka, kufikia thamani ya chini kabisa mwezi Januari. Wastani wa idadi ya watu huanza kwa\(800\) samaki na huongezeka kwa\(4\%\) kila mwezi. Kupata kazi ambayo mifano ya idadi ya watu,\(P\), katika suala la miezi tangu Januari,\(t\).

Jibu

\(P(t)=−40 \cos \left(\dfrac{π}{6}t \right)+800(1.04)^t\)

38) Spring masharti ya dari ni vunjwa\(10\) cm chini kutoka usawa na kutolewa. Amplitude hupungua kwa\(15\%\) kila pili. Spring oscillates\(18\) mara kila pili. Pata kazi inayoonyesha umbali\(D\), mwisho wa spring unatoka kwa usawa kwa suala la sekunde\(t\), tangu chemchemi ilitolewa.

39) Spring iliyounganishwa na dari ni vunjwa\(7\) cm chini kutoka usawa na kutolewa. Amplitude hupungua kwa\(11\%\) kila pili. Spring oscillates\(20\) mara kila pili. Pata kazi inayoonyesha umbali\(D\), mwisho wa spring unatoka kwa usawa kwa suala la sekunde,\(t,\) tangu chemchemi ilitolewa.

Jibu

\(D(t)=7(0.89)^t \cos (40πt)\)

40) Spring masharti ya dari ni vunjwa\(17\) cm chini kutoka usawa na kutolewa. Baada ya\(3\) sekunde, amplitude imepungua hadi\(13\) cm. Spring oscillates\(14\) mara kila pili. Pata kazi inayoonyesha umbali, mwisho\(D,\) wa chemchemi hutoka kwa usawa kwa suala la sekunde\(t\), tangu chemchemi ilitolewa.

41) Spring masharti ya dari ni vunjwa\(19\) cm chini kutoka usawa na kutolewa. Baada ya\(4\) sekunde, amplitude imepungua hadi\(14\) cm. Spring oscillates\(13\) mara kila pili. Pata kazi inayoonyesha umbali\(D\), mwisho wa spring unatoka kwa usawa kwa suala la sekunde\(t\), tangu chemchemi ilitolewa.

Jibu

\(D(t)=19(0.9265)^t \cos (26πt)\)

Kwa mazoezi 42-47, fanya kazi ya kuimarisha tabia iliyoelezwa. Kisha, hesabu matokeo yaliyohitajika kwa kutumia calculator.

42) ziwa fulani kwa sasa ina wastani trout idadi ya\(20,000\). Idadi ya watu kwa kawaida oscillates juu na chini ya wastani kwa\(2,000\) kila mwaka. Mwaka huu, ziwa lilifunguliwa kwa wavuvi. Ikiwa wavuvi hupata\(3,000\) samaki kila mwaka, itachukua muda gani kwa ziwa kuwa na shimo tena?

43) wakazi Whitefish sasa\(500\) katika ziwa. Idadi ya watu kwa kawaida oscillates juu na chini kwa\(25\) kila mwaka. Kama binadamu overfish, kuchukua idadi\(4\%\) ya watu kila mwaka, katika miaka ngapi ziwa kwanza kuwa wachache kuliko\(200\) whitefish?

Jibu

\(20.1\)miaka

44) Spring iliyounganishwa na dari ni vunjwa chini\(11\) cm kutoka usawa na kutolewa. Baada ya\(2\) sekunde, amplitude imepungua hadi\(6\) cm. Spring oscillates\(8\) mara kila pili. Pata wakati spring kwanza inakuja kati\(−0.1\) na\(0.1\) cm, kwa ufanisi wakati wa kupumzika.

45) Spring iliyounganishwa na dari ni vunjwa chini\(21\) cm kutoka usawa na kutolewa. Baada ya\(6\) sekunde, amplitude imepungua hadi\(4\) cm. Spring oscillates\(20\) mara kila pili. Pata wakati spring kwanza inakuja kati\(−0.1\) na\(0.1\) cm, kwa ufanisi wakati wa kupumzika.

Jibu

\(17.8\)sekunde

46) Maji mawili hutolewa kutoka dari na kutolewa kwa wakati mmoja. Spring kwanza, ambayo oscillates\(8\) mara kwa pili, awali vunjwa chini\(32\) cm kutoka usawa, na amplitude itapungua kwa\(50\%\) kila pili. Spring pili,\(18\) mara oscillating kwa pili, awali vunjwa chini\(15\) cm kutoka usawa na baada ya\(4\) sekunde ina amplitude ya\(2\) cm. Ni spring ipi inakuja kupumzika kwanza, na kwa wakati gani? Fikiria “kupumzika” kama amplitude chini ya\(0.1\) cm.

47) Chemchemi mbili hutolewa kutoka dari na kutolewa kwa wakati mmoja. Spring kwanza, ambayo oscillates\(14\) mara kwa pili, awali vunjwa chini\(2\) cm kutoka usawa, na amplitude itapungua kwa\(8\%\) kila pili. Spring pili,\(22\) mara oscillating kwa pili, awali vunjwa chini\(10\) cm kutoka usawa na baada ya\(3\) sekunde ina amplitude ya\(2\) cm. Ni spring ipi inakuja kupumzika kwanza, na kwa wakati gani? Fikiria “kupumzika” kama amplitude chini ya\(0.1\) cm.

Jibu

Spring 2 inakuja kupumzika kwanza baada ya\(8.0\) sekunde.

Upanuzi

48) Ndege inaruka\(1\) saa saa\(150\) mph\(22^∘\) mashariki mwa kaskazini, kisha inaendelea kuruka kwa\(1.5\) masaa kwa\(120\) mph, wakati huu kwa kuzaa\(112^∘\) mashariki ya kaskazini. Pata umbali wa jumla kutoka kwa mwanzo na angle ya moja kwa moja inazunguka kaskazini mwa mashariki.

49) ndege nzi\(2\) masaa katika\(200\) mph katika kuzaa ya\(60^∘\), kisha inaendelea kuruka kwa\(1.5\) saa kwa kasi sawa, wakati huu katika kuzaa ya\(150^∘\). Pata umbali kutoka kwa mwanzo na kuzaa kutoka mwanzo. (Kidokezo: kuzaa hupimwa kinyume chake kutoka kaskazini.)

Jibu

\(500\)maili, katika\(90^∘\)

Kwa mazoezi 50-52, pata kazi ya fomu\(y=ab^x \sin \left(\dfrac{π}{2}x \right)+c\) inayofaa data iliyotolewa.

50)

\(x\)	0	1	2	3
\(y\)	6	29	96	379

51)

\(x\)	0	1	2	3
\(y\)	6	34	150	746

Jibu

\(y=6(5)^x+4 \sin \left(\dfrac{π}{2}x \right)\)

52)

\(x\)	0	1	2	3
\(y\)	4	0	16	-40

Kwa mazoezi 53-54, tafuta kazi ya fomu\(y=ab^x \cos \left(\dfrac{π}{2}x \right)+c\) inayofaa data iliyotolewa.

53)

\(x\)	0	1	2	3
\(y\)	11	3	1	3

Jibu

\(y=8\left(\dfrac{1}{2} \right)^x \cos \left(\dfrac{π}{2}x \right)+3\)

54)

\(x\)	0	1	2	3
\(y\)	4	1	-11	1

Wachangiaji na Majina

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