7.R: Utambulisho wa Trigonometric na Ulinganisho (Mapitio)

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7.1: Kutatua Ulinganisho wa Trigonometric na Utambulisho

Kwa mazoezi ya 1-6, tafuta ufumbuzi wote hasa uliopo wakati$[0,2\pi )$.

1)$\csc ^2 t=3$

Jibu: $\sin^{-1}\left ( \dfrac{\sqrt{3}}{3} \right ), \pi -\sin^{-1}\left ( \dfrac{\sqrt{3}}{3} \right ), \pi +\sin^{-1}\left ( \dfrac{\sqrt{3}}{3} \right ), 2\pi -\sin^{-1}\left ( \dfrac{\sqrt{3}}{3} \right )$

2)$\cos ^2 x=\dfrac{1}{4}$

3)$2\sin \theta =-1$

Jibu: $\dfrac{7\pi }{6}, \dfrac{11\pi }{6}$

4)$\tan x \sin x+\sin(-x)=0$

5)$9\sin \omega -2=4\sin^2 \omega$

Jibu: $\sin^{-1}\left ( \dfrac{1}{4} \right ), \pi -\sin^{-1}\left ( \dfrac{1}{4} \right )$

6)$1-2\tan(\omega )=\tan^2(\omega )$

Kwa mazoezi 7-8, tumia utambulisho wa msingi ili kurahisisha maneno.

7)$\sec x \cos x+\cos x-\dfrac{1}{\sec x}$

Jibu: $1$

8)$\sin^3 x+\cos^2 x \sin x$

Kwa mazoezi 9-10, onyesha kama utambulisho uliopewa ni sawa.

9)$\sin^2 x+\sec^2 x -1=\dfrac{(1-\cos ^2 x)(1+\cos ^2 x)}{\cos ^2 x}$

Jibu: Ndio

10)$\tan^3 x \csc^2 x \cot^2 x \cos x \sin x=1$

7.2: Utambulisho wa Jumla na Tofauti

Kwa mazoezi 1-4, pata thamani halisi.

1)$\tan \left (\dfrac{7\pi }{12} \right )$

Jibu: $-2-\sqrt{3}$

2)$\cos \left (\dfrac{25\pi }{12} \right )$

3)$\sin(70^{\circ})\cos(25^{\circ})-\cos(70^{\circ})\sin(25^{\circ})$

Jibu: $\dfrac{\sqrt{2}}{2}$

4)$\cos(83^{\circ})\cos(23^{\circ})+\sin(83^{\circ})\sin(23^{\circ})$

Kwa mazoezi 5-6, kuthibitisha utambulisho.

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

Jibu: \ (\ kuanza {align*}
\ cos (4x) -\ cos (3x)\ cos x &=\ cos (2x+2x) -\ cos (x+2x)\ cos x\\
&=\ cos (2x)\ cos (2x) -\ dhambi (2x)\ 2x)\ cos x\\
&= (\ cos ^2 x-\ dhambi ^2 x) ^2-4\ cos ^2 x\ dhambi ^2 x-\ cos ^2 x (\ cos ^2
x-\ dhambi ^2 x) +\ dhambi x (2)\ dhambi x\ cos x\\
&= (\ cos ^2 x-\ dhambi ^2 x) ^2-4\ cos ^2 x\ dhambi ^2 x-\ cos ^2 x ^ 2 x (\ cos ^2 x-\ dhambi ^2 x\ cos ^2 x\\
cos ^ 2x\ dhambi ^ 2x+\ dhambi ^4-\ cos^2x\ dhambi ^ 2x-\ cos ^ 4x+\ cos ^ 2x\ dhambi ^ 2x+2\ dhambi ^ 2x\\ cos ^ 2x\\
&= \ dhambi ^ 4x-4\ cos^2x\ dhambi ^ 2x+\ cos ^ 2x\ dhambi ^ 2x\\
&=\ dhambi ^ 2x (\ dhambi ^ 2x+\ cos ^ 2x) -4\ cos ^ 2x\ dhambi ^ 2x\\
&=\ dhambi ^ 2 x
\ mwisho {align*}\)

6)$\cos(3x)-\cos^3x=-\cos x \sin^2x-\sin x \sin(2x)$

Kwa zoezi 7, kurahisisha maneno.

7)$\dfrac{\tan \left ( \tfrac{1}{2}x \right )+\tan \left ( \tfrac{1}{8}x \right )}{1-\tan \left ( \tfrac{1}{8}x \right )\tan \left ( \tfrac{1}{2}x \right )}$

Jibu: $\tan \left ( \dfrac{5}{8}x \right )$

Kwa mazoezi 8-9, pata thamani halisi.

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

9)$\tan \left ( \sin^{-1}(0)-\sin^{-1}\left ( \dfrac{1}{2} \right ) \right )$

Jibu: $\dfrac{\sqrt{3}}{3}$

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

Kwa mazoezi 1-4, pata thamani halisi.

1) Kupata$\sin (2\theta )$,$\cos (2\theta )$,na$\tan (2\theta )$ kupewa$\cos \theta =-\dfrac{1}{3}$ na$\theta $ ni katika kipindi$\left [\dfrac{\pi }{2} , \pi \right ]$.

2) Kupata$\sin (2\theta )$$\cos (2\theta )$,, na$\tan (2\theta )$ kupewa$\sec \theta =-\dfrac{5}{3}$ na$\theta $ ni katika kipindi$\left [\dfrac{\pi }{2} , \pi \right ]$.

Jibu: $-\dfrac{24}{25}, -\dfrac{7}{25}, \dfrac{24}{7}$

3)$\sin \left (\dfrac{7\pi }{8} \right )$

4)$\sec \left (\dfrac{3\pi }{8} \right )$

Jibu: $\sqrt{2(2+\sqrt{2})}$

Kwa mazoezi 5-6, tumia Kielelezo hapa chini ili kupata kiasi kilichohitajika.

$CNX_Precalc_Figure_07_07_201.jpg$

5)$\sin(2\beta ),\cos(2\beta ),\tan(2\beta ),\sin(2\alpha ),\cos(2\alpha ),\tan(2\alpha )$

6)$\sin \left (\frac{\beta }{2} \right ) ,\cos\left (\frac{\beta }{2} \right ),\tan\left (\frac{\beta }{2} \right ),\sin\left (\frac{\alpha }{2} \right ),\cos\left (\frac{\alpha }{2} \right ),\tan\left (\frac{\alpha }{2} \right )$

Jibu: $\dfrac{\sqrt{2}}{10},\dfrac{7\sqrt{2}}{10},\dfrac{1}{7},\dfrac{3}{5},\dfrac{4}{5},\dfrac{3}{4}$

Kwa mazoezi 7-8, kuthibitisha utambulisho.

7)$\dfrac{2\cos (2x)}{\sin (2x)}=\cot x-\tan x$

8)$\cot x \cos (2x) = -\sin (2x)+\cot x$

Jibu: $\begin{align*} \cot x \cos (2x) &= \cot x(1-2\sin ^2 x)\\ &= \cot x-\dfrac{\cos x}{\sin x}(2)\sin ^2 x\\ &= -2\sin x \cos \\ &= -\sin (2x)+\cot x \end{align*}$

Kwa mazoezi 9-10, fungua upya maneno bila mamlaka.

9)$\cos ^2 x \sin ^4 (2x)$

10)$\tan ^2 x \sin ^3 x$

Jibu: $\dfrac{10\sin x-5\sin (3x)+\sin (5x)}{8(\cos (2x)+1)}$

7.4: Jumla ya bidhaa na Bidhaa kwa-Jumla Formula

Kwa mazoezi 1-3, tathmini bidhaa kwa maneno yaliyotolewa kwa kutumia jumla au tofauti ya kazi mbili. Andika jibu halisi.

1)$\cos \left ( \dfrac{\pi }{3} \right )\sin \left ( \dfrac{\pi }{4} \right )$

2)$2\sin \left ( \dfrac{2\pi }{3} \right )\sin \left ( \dfrac{5\pi }{6} \right )$

Jibu: $\dfrac{\sqrt{3}}{2}$

3)$2\cos \left ( \dfrac{\pi }{5} \right )\cos \left ( \dfrac{\pi }{3} \right )$

Kwa mazoezi 4-5, tathmini jumla kwa kutumia formula ya bidhaa. Andika jibu halisi.

4)$\sin \left ( \dfrac{\pi }{12} \right )-\sin \left ( \dfrac{7\pi }{12} \right )$

Jibu: $-\dfrac{\sqrt{2}}{2}$

5)$\cos \left ( \dfrac{5\pi }{12} \right )+\cos \left ( \dfrac{7\pi }{12} \right )$

Kwa mazoezi 6-9, mabadiliko ya kazi kutoka kwa bidhaa hadi jumla au jumla kwa bidhaa.

6)$\sin(9x)\cos(3x)$

Jibu: $\dfrac{1}{2}(\sin(6x)+\sin(12x))$

7)$\cos(7x)\cos(12x)$

8)$\sin(11x)+\sin(2x)$

Jibu: $2\sin \left (\dfrac{13}{2}x \right )\cos \left (\dfrac{9}{2}x \right )$

9)$\cos(6x)+\cos(5x)$

7.5: Kutatua equations ya Trigonometric

Kwa mazoezi 1-2, pata ufumbuzi wote halisi kwa muda$[0,2\pi )$.

1)$\tan x+1=0$

Jibu: $\dfrac{3\pi }{4}, \dfrac{7\pi }{4}$

2)$2\sin(2x)+\sqrt{2}=0$

Kwa mazoezi 3-7, pata ufumbuzi wote halisi kwa muda$[0,2\pi )$.

3)$2\sin^2 x-\sin x=0$

Jibu: $0, \dfrac{\pi }{6}, \dfrac{5\pi }{6}, \pi $

4)$\cos^2 x-\cos x -1=0$

5)$2\sin^2 x+5\sin x +3=0$

Jibu: $\dfrac{3\pi }{2}$

6)$\cos x - 5\sin(2x)=0$

7)$\dfrac{1}{\sec ^2 x}+2+\sin^2 x+4\cos ^2 x=0$

Jibu: Hakuna ufumbuzi.

Kwa mazoezi 8-9, kurahisisha equation algebraically iwezekanavyo. Kisha tumia calculator ili kupata ufumbuzi kwa muda$[0,2\pi )$. Pande zote hadi sehemu nne za decimal.

8)$\sqrt{3}\cot ^2 x+\cot x=1$

9)$\csc ^2 x-3\csc x-4=0$

Jibu: $0.2527,2.8889,4.7124$

Kwa mazoezi 10-11, grafu kila upande wa equation ili kupata zero kwa muda$[0,2\pi )$.

10)$20\cos^2x+21\cos x+1=0$

11)$\sec^2x-2\sec x=15$

Jibu: $1.3694, 1.9106, 4.3726, 4.9137$

7.6: Mfano na Ulinganifu wa Trigonometric

Kwa mazoezi 1-3, graph pointi na kupata formula iwezekanavyo kwa maadili ya trigonometric katika meza iliyotolewa.

$x$	0	1	2	3	4	5
$y$	1	6	11	6	1	6

$x$	$y$
\ (x\) "> 0	\ (y\) ">-2
\ (x\) ">1	\ (y\) ">1
\ (x\) "> 2	\ (y\) ">-2
\ (x\) ">3	\ (y\) ">-5
\ (x\) ">4	\ (y\) ">-2
\ (x\) "> 5	\ (y\) ">1

Jibu: $3\sin \left ( \dfrac{x\pi }{2} \right )-2$

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

4) Mtu mwenye$6$ miguu ya jicho lake juu ya ardhi amesimama$3$ miguu mbali na msingi wa ngazi ya wima ya$15$ mguu. Ikiwa anaangalia juu ya ngazi, ni angani gani juu ya usawa anaangalia?

Jibu: $71.6^{\circ}$

5) Kutumia ngazi kutoka kwa zoezi la awali, ikiwa$6$ mfanyakazi wa ujenzi wa mguu amesimama juu ya ngazi anaangalia chini ya miguu ya mtu amesimama chini, ni angle gani kutoka usawa anaangalia?

Kwa mazoezi 6-7, jenga kazi ambazo zinaonyesha tabia iliyoelezwa.

6) Idadi ya lemmings inatofautiana na chini ya kila$500$ mwaka ya mwezi Machi. Kama wastani wa idadi ya kila mwaka ya lemmings ni$950$, kuandika kazi kwamba mifano ya idadi ya watu kwa heshima na$t$,mwezi.

Jibu: $P(t)=950-450\sin \left ( \dfrac{\pi }{6}t \right )$

7) Joto la kila siku katika jangwa linaweza kuwa kali sana. Ikiwa hali ya joto inatofautiana kutoka$90^{\circ}$$30^{\circ}$ F hadi F na wastani wa joto la kila siku hutokea saa 10 asubuhi, andika kazi ya kuimarisha tabia hii.

Kwa mazoezi 8-9, pata amplitude, frequency, na kipindi cha equations iliyotolewa.

8)$y=3\cos(x\pi )$

Jibu: Amplitude:$3$, kipindi:$2$, mzunguko:$\dfrac{1}{2}$ Hz

9)$y=-2\sin(16x\pi )$

Kwa mazoezi 10-11, mfano tabia iliyoelezwa na kupata maadili yaliyoombwa.

10) Aina ya uvamizi wa carp huletwa kwenye Ziwa Freshwater. Awali kuna$100$ carp katika ziwa na idadi ya watu inatofautiana na$20$ samaki msimu. Ikiwa kwa mwaka$5$, kuna$625$ carp, tafuta kazi inayofanana na idadi ya carp kwa heshima$t$,idadi ya miaka kuanzia sasa.

Jibu: $C(t)=20\sin (2\pi t)+100(1.4427)^t$

11) idadi ya samaki ya asili ya Ziwa Freshwater wastani$2500$ samaki, tofauti na$100$ samaki msimu. Kutokana na ushindani wa rasilimali kutoka kwa carp ya uvamizi, idadi ya samaki ya asili inatarajiwa kupungua kwa$5\%$ kila mwaka. Kupata kazi modeling idadi ya samaki asili kwa heshima na$t$,idadi ya miaka kuanzia sasa. Pia tambua miaka ngapi itachukua kwa carp ili kupata idadi ya samaki ya asili.

Mazoezi mtihani

Kwa mazoezi 1-2, kurahisisha maneno yaliyotolewa.

1)$\cos(-x)\sin x \cot x+\sin^2x$

Jibu: $1$

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

Kwa mazoezi 3-6, pata thamani halisi.

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

Jibu: $\dfrac{\sqrt{2}-\sqrt{6}}{4}$

4)$\tan \left ( \dfrac{3\pi }{8} \right )$

5)$\tan \left (\sin^{-1}\left (\dfrac{\sqrt{2}}{2} \right )+\tan^{-1}\sqrt{3} \right )$

Jibu: $-\sqrt{2}-\sqrt{3}$

6)$2\sin \left (\dfrac{\pi }{4} \right )\sin \left (\dfrac{\pi }{6} \right )$

Kwa mazoezi 7-16, pata ufumbuzi wote halisi wa equation juu$[0,2\pi )$.

7)$\cos^2x-\sin^2x-1=0$

Jibu: $0, \pi $

8)$\cos^2x=\cos x$

Jibu: $\sin^{-1}\left (\dfrac {1}{4}\left(\sqrt{13}-1\right) \right ), \pi - \sin^{-1}\left (\dfrac {1}{4}\left(\sqrt{13}-1\right) \right )$

9)$\cos (2x)+\sin ^2 x = 0$

10)$2\sin ^2 x - \sin x = 0$

Jibu: $0, \dfrac{\pi }{6}, \dfrac{5\pi }{6}, \pi$

11) Andika upya maneno kama bidhaa badala ya jumla:$\cos (2x)+\cos (-8x)$

12) Kupata ufumbuzi wote wa$\tan (x)-\sqrt{3}=0$.

Jibu: $\dfrac{\pi }{3}+k\pi$

13) Pata ufumbuzi wa$\sec ^2x -2\sec x=15$ kipindi cha$[0,2\pi )$ algebraically; kisha graph pande zote mbili za equation kuamua jibu.

14) Kupata$\sin (2\theta )$,$\cos (2\theta )$, na$\tan (2\theta )$ kupewa$\cot \theta =-\dfrac{3}{4}$ na$\theta $ ni juu ya muda$\left [ \dfrac{\pi }{2}, \pi \right ]$.

Jibu: $-\dfrac{24}{25}, -\dfrac{7}{25}, \dfrac{24}{7}$

15) Kupata$\sin \left (\dfrac{\theta }{2} \right )$,$\cos \left (\dfrac{\theta }{2} \right )$, na$\tan \left (\dfrac{\theta }{2} \right )$ kupewa$\cos \theta =-\dfrac{7}{25}$ na$\theta $ ni katika roboduara$\mathrm{IV}$.

16) Andika upya maneno$\sin ^4 x$ bila mamlaka zaidi kuliko$1$.

Jibu: $\dfrac{1}{8}(3+\cos (4x)-4\cos (2x))$

Kwa mazoezi 17-19, kuthibitisha utambulisho.

17)$\tan^3x-\tan x \sec^2x=\tan(-x)$

18)$\sin(3x)-\cos x \sin(2x)=\cos^2x \sin x-\sin^3x$

Jibu: $\begin{align*} \sin(3x)-\cos x \sin(2x) &= \\ \sin(x+2x)-\cos x(2\sin x \cos x) &= \\ \sin x \cos(2x)+\sin(2x)\cos x -2\sin x \cos ^2x &= \\ \sin x(\cos ^2x - \sin ^2x)+2\sin x \cos x \cos x - 2\sin x \cos ^2x &= \\ \sin x \cos ^2x - \sin ^3x +0 &= \\ \cos^2x \sin x - \sin ^3x &= \cos^2x \sin x-\sin^3x \end{align*}$

19)$\dfrac{\sin (2x)}{\sin x}-\dfrac{\cos (2x)}{\cos x}=\sec x$

20) Panda pointi na kupata kazi ya fomu$y=A\cos(Bx+C)+D$ inayofaa data iliyotolewa.

$x$	0	1	2	3	4	5
$y$	-2	2	-2	2	-2	2

Jibu: $y=2\cos(\pi x+\pi )$

21) Uhamisho$h(t)$ kwa sentimita ya wingi uliosimamishwa na chemchemi unatokana na kazi$h(t)=\dfrac{1}{4}\sin (120\pi t)$,ambapo$t$ hupimwa kwa sekunde. Pata amplitude, kipindi, na mzunguko wa makazi haya.

22) Mwanamke amesimama$300$ miguu mbali na jengo la$2000$ mguu. Ikiwa anaangalia juu ya jengo, ni angani gani juu ya usawa anaangalia? Mfanyakazi mwenye kuchoka anamtazama chini kutoka sakafu ^ya 15 ($1500$miguu juu yake). Ni pembe gani anamtazama chini? Pande zote hadi kumi ya karibu ya shahada.

Jibu: $81.5^{\circ}, 78.7^{\circ}$

23) Mifumo miwili ya sauti inachezwa kwenye chombo kinachosimamiwa na equation$n(t)=8\cos(20\pi t)\cos(1000\pi t)$. Je! Ni kipindi gani na mzunguko wa “haraka” na “polepole” oscillations? Amplitude ni nini?

24) Wastani wa theluji kila mwezi katika kijiji kidogo katika Himalaya ni$6$ inchi, na chini ya$1$ inchi hutokea Julai. Kujenga kazi kwamba mfano tabia hii. Wakati gani kuna zaidi ya$10$ inchi za theluji?

Jibu: $6+5\cos \left ( \dfrac{\pi }{6}(1-x) \right )$. Kuanzia Novemba 23 hadi Februari 6.

25) Spring masharti ya dari ni vunjwa chini$20$ cm. Baada ya$3$ sekunde, ambayo inakamilisha vipindi$6$ kamili, amplitude ni$15$ cm tu. Pata kazi ya kuimarisha nafasi ya$t$ sekunde za spring baada ya kutolewa. Je! Spring itapumzika wakati gani? Katika kesi hii, tumia$1$ cm amplitude kama kupumzika.

26) Viwango vya maji karibu na glacier kwa sasa$9$ miguu wastani, tofauti seasonally na$2$ inchi juu na chini ya wastani na kufikia hatua yao ya juu katika Januari. Kutokana na ongezeko la joto duniani, glacier imeanza kuyeyuka kwa kasi zaidi kuliko kawaida. Kila mwaka, viwango vya maji huongezeka kwa$3$ inchi za kutosha. Pata kazi ya kuimarisha kina cha$t$ miezi ya maji tangu sasa. Ikiwa docks ni$2$ miguu juu ya viwango vya sasa vya maji, wakati gani maji yatatokea kwanza juu ya docks?

Jibu: $D(t)=2\cos \left ( \dfrac{\pi }{6}t \right )+108+\dfrac{1}{4}t$,$93.5855$ miezi (au$7.8$ miaka) tangu sasa

Wachangiaji na Majina

Template:OpenStaxPreCalc

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