6.E: Kazi za mara kwa mara (Mazoezi)
- Page ID
- 181080
6.1: Grafu za Kazi za Sine na Cosine
Katika sura ya Kazi za Trigonometric, tulichunguza kazi za trigonometric kama kazi ya sine. Katika sehemu hii, tutatafsiri na kuunda grafu za kazi za sine na cosine
Maneno
1) Kwa nini kazi za sine na cosine zinaitwa kazi za mara kwa mara?
- Jibu
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Kazi za sine na cosine zina mali ambayo\(f(x+P)=f(x)\) kwa fulani\(P\). Hii ina maana kwamba maadili kazi kurudia kwa kila\(P\) vitengo juu ya\(x\) -axis.
2) Je, grafu ya\(y=\sin x\) kulinganisha na grafu ya\(y=\cos x\)? Eleza jinsi gani unaweza sambamba kutafsiri grafu ya\(y=\sin x\) kupata\(y=\cos x\).
3) Kwa equation\(A \cos(Bx+C)+D\)
- Jibu
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Thamani kamili ya mara kwa mara\(A\) (amplitude) huongeza kiwango cha jumla na mabadiliko ya mara kwa mara\(D\) (wima) hubadilisha grafu kwa wima.
4) Je mbalimbali ya kazi ya sine iliyotafsiriwa yanahusiana na equation\(y=A \sin(Bx+C)+D\)
5) Je, mduara wa kitengo unaweza kutumika kujenga grafu ya\(f(t)=\sin t\)?
- Jibu
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Katika hatua ambapo upande terminal ya\(t\) intersects kitengo mduara, unaweza kuamua kwamba\(\sin t\) sawa\(y\) -kuratibu ya uhakika.
Graphic
Kwa mazoezi yafuatayo, graph vipindi viwili kamili vya kila kazi na ueleze amplitude, kipindi, na midline. Hali ya kiwango cha juu na cha chini\(y\) -maadili na sambamba zao\(x\) -maadili kwa kipindi kimoja cha\(x>0\). Majibu ya pande zote kwa maeneo mawili ya decimal ikiwa ni lazima.
6)\(f(x)=2\sin x\)
7)\(f(x)=\dfrac{2}{3}\cos x\)
- Jibu
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ukubwa:\(\dfrac{2}{3}\)
kipindi:\(2\pi \); midline:\(y=0\); upeo:\(y=\dfrac{2}{3}\) hutokea\(x=0\); kiwango cha chini:\(y=-\dfrac{2}{3}\) hutokea\(x=\pi \); kwa kipindi kimoja, grafu huanza\(0\) na kuishia saa\(2\pi \).;
8)\(f(x)=-3\sin x\)
9)\(f(x)=4\sin x\)
- Jibu
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amplitude:\(4\); kipindi:\(2\pi \)
midline:\(y=0\); upeo\(y=4\) hutokea\(x=\dfrac{\pi }{2}\); kiwango cha chini:\(y=-4\) hutokea\(x=\dfrac{3\pi }{2}\); kipindi kimoja kamili hutokea kutoka\(x=0\) kwa\(x=2\pi\);
10)\(f(x)=2\cos x\)
11)\(f(x)=\cos (2x)\)
- Jibu
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amplitude:\(1\); kipindi:\(\pi\)
midline:\(y=0\); upeo:\(y=1\) hutokea\(x=\pi \); kiwango cha chini:\(y=-1\) hutokea\(x=\dfrac{\pi }{2}\); kipindi kimoja kamili kinawekwa kutoka\(x=0\) kwa\(x=\pi\);
12)\(f(x)=2 \sin \left(\dfrac{1}{2}x\right)\)
13)\(f(x)=4 \cos(\pi x)\)
- Jibu
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amplitude:\(4\); kipindi:\(2\); midline:\(y=0\)
upeo:\(y=4\) hutokea\(x=0\); kiwango cha chini:\(y=-4\) hutokea\(x=1\);
14)\(f(x)=3 \cos\left(\dfrac{6}{5}x\right)\)
15)\(y=3 \sin(8(x+4))+5\)
- Jibu
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amplitude:\(3\); kipindi:\(\dfrac{\pi}{4}\); midline:\(y=5\);
upeo:\(y=8\) hutokea\(x = -4+\frac{21\pi}{16} \approx 0.123\);
chini:\(y=2\) hutokea; mabadiliko ya
usawa:\(x = -4+\frac{23\pi}{16} \approx 0.516\); tafsiri ya wima\(-4\)\(5\); kipindi
kimoja hutokea kutoka\(x=-4+\frac{22\pi}{16} \approx 0.320\) kwa\(x=-4+\frac{26\pi}{16} \approx 1.105 \)
16)\(y=2 \sin(3x-21)+4\)
17)\(y=5 \sin(5x+20)-2\)
- Jibu
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amplitude:\(5\); kipindi:\(\dfrac{2\pi }{5}\); midline:\(y=-2\);
upeo:\(y=3\) hutokea\(x= -4+\frac{13\pi}{10} \approx 0.084\); kiwango
cha chini:\(y=-7\) hutokea; mabadiliko ya
awamu:\(x=-4+\frac{15\pi}{10} \approx 0.712\)\(-4\); tafsiri ya wima:\(-2\); kipindi
kimoja kamili kinaweza kuwa graphiced juu\(x=-4+\frac{7\pi}{5} \approx 0.398\) ya\(x=-4+\frac{9\pi}{5} \approx 1.655 \)
Kwa mazoezi yafuatayo, graph kipindi kimoja kamili cha kila kazi, kuanzia saa\(x=0\).
Kwa kila kazi, sema amplitude, kipindi, na midline.
Hali ya kiwango cha juu na cha chini\(y\) -maadili na sambamba zao\(x\) -maadili kwa kipindi kimoja cha\(x>0\).
Hali ya mabadiliko ya awamu na tafsiri ya wima, ikiwa inatumika.
Majibu ya pande zote kwa maeneo mawili ya decimal ikiwa ni lazima.
18)\(f(t)=2\sin \left(t-\dfrac{5\pi}{6} \right)\)
19)\(f(t)=-\cos \left(t+\dfrac{\pi}{3} \right)+1\)
- Jibu
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amplitude:\(1\); kipindi:\(2\pi \); midline:\(y=1\);
upeo:\(y=2\) hutokea\(t=\frac{2\pi}{3} \approx 2.094\); kiwango
cha chini:\(y=0\) hutokea\(t=\frac{2\pi}{3} \approx5.24\); mabadiliko ya
awamu:\(-\dfrac{\pi}{3}\); tafsiri ya wima:\(1\); kipindi
kimoja kamili ni kutoka\(t=\frac{2\pi}{3} \approx 2.094\) kwa\(t=\frac{8\pi}{3} \approx 8.378 \)
20)\(f(t)=4\cos \left(2\left (t+\dfrac{\pi}{4} \right ) \right)-3\)
21)\(f(t)=-\sin \left (\dfrac{1}{2}t+\dfrac{5\pi}{3} \right )\)
- Jibu
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amplitude:\(1\); kipindi:\(4\pi\); midline:\(y=0\);
upeo:\(y=1\) hutokea\(t=\frac{11\pi}{3} \approx 11.52\); kiwango
cha chini:\(y=-1\) hutokea\(t=\frac{5\pi}{3} \approx 5.24\); mabadiliko ya
awamu:\(-\dfrac{10\pi}{3}\); mabadiliko ya wima:\(0\); kipindi
kimoja kamili kinatoka \(t=\frac{2\pi}{3} \approx 2.094\)kwa\(t=\frac{14\pi}{3} \approx 14.661 \)
22)\(f(x)=4\sin \left (\dfrac{\pi}{2}(x-3) \right )+7\)
23) Kuamua amplitude, midline, kipindi, na equation kuwashirikisha kazi sine kwa grafu inavyoonekana katika Kielelezo hapa chini.
- Jibu
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23. amplitude:\(2\); midline:\(y=-3\) kipindi:\(4\); equation:\(f(x)=2\sin \left (\dfrac{\pi}{2}x \right )-3\)
24) Tambua amplitude, midline, kipindi, na equation inayohusisha kazi ya cosine kwa grafu iliyoonyeshwa kwenye Mchoro hapa chini.
25) Tambua amplitude, midline, kipindi, na equation inayohusisha kazi ya cosine kwa grafu iliyoonyeshwa kwenye Mchoro hapa chini.
- Jibu
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25. amplitude:\(2\); kipindi:\(5\); midline:\(y=3\) equation:\(f(x)=-2\cos \left (\dfrac{2\pi}{5}x \right )+3\)
26) Kuamua amplitude, midline, kipindi, na equation kuwashirikisha kazi sine kwa graph inavyoonekana katika Kielelezo hapa chini.
27) Tambua amplitude, midline, kipindi, na equation inayohusisha kazi ya cosine kwa grafu iliyoonyeshwa kwenye Mchoro hapa chini.
- Jibu
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27. amplitude:\(4\); kipindi:\(2\); midline:\(y=0\); equation:\(f(x)=-4\cos \left (\pi \left (x-\dfrac{\pi}{2} \right ) \right )\)
28) Kuamua amplitude, midline, kipindi, na equation kuwashirikisha kazi sine kwa graph inavyoonekana katika Kielelezo hapa chini.
29) Tambua amplitude, midline, kipindi, na equation inayohusisha kazi ya cosine kwa grafu iliyoonyeshwa kwenye Mchoro hapa chini.
- Jibu
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29. amplitude:\(2\); kipindi:\(2\); midline\(y=1\) equation:\(f(x)=2\cos \left (\pi x \right )+1\)
30) Kuamua amplitude, midline, kipindi, na equation kuwashirikisha kazi sine kwa graph inavyoonekana katika Kielelezo hapa chini.
Kialjebra
Kwa mazoezi yafuatayo, basi\(f(x)=\sin x \)
31) Juu\([0,2\pi )\)
32) Juu\([0,2\pi )\), tatua\(f(x)=\dfrac{1}{2}\).
- Jibu
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\(\dfrac{\pi }{6}\),\(\dfrac{5\pi }{6}\)
33) Tathmini\(f \left( \dfrac{\pi }{2} \right) \)
34) On\([0,2\pi)\),\(f(x)=\dfrac{\sqrt{2}}{2}\). Kupata maadili yote ya\(x\).
- Jibu
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\(\dfrac{\pi }{4}\),\(\dfrac{3\pi }{4}\)
35) Juu\([0,2\pi )\)
36) Juu\([0,2\pi )\)
- Jibu
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\(\dfrac{3\pi }{2}\)
37) Onyesha kwamba\(f(-x) = -f(x)\)
Kwa mazoezi yafuatayo, basi\(f(x)=\cos x\)
38) Juu\([0,2\pi )\)
- Jibu
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\(\dfrac{\pi }{2}\),\(\dfrac{3\pi }{2}\)
39) Juu\([0,2\pi )\)
40) Juu\([0,2\pi )\)
- Jibu
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\(\dfrac{\pi }{2}\),\(\dfrac{3\pi }{2}\)
41) Juu\([0,2\pi )\)
42) Juu\([0,2\pi )\)
- Jibu
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\(\dfrac{\pi }{6}\),\(\dfrac{11\pi }{6}\)
Teknolojia
43) Grafu\(h(x)=x+\sin x\) juu\([0,2\pi ]\)
44) Grafu\(h(x)=x+\sin x\) juu\([-100,100]\)
- Jibu
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Grafu inaonekana mstari. kazi linear kutawala sura ya grafu kwa maadili makubwa ya\(x\).
45) Grafu\(f(x)=x\sin x\) juu\([0,2\pi ]\) na verbalize jinsi graph inatofautiana kutoka grafu ya\(f(x)=x\sin x\).
46) Grafu\(f(x)=x\sin x\) kwenye dirisha\([-10,10]\) na kuelezea kile grafu inaonyesha.
- Jibu
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Grafu ni sawa na heshima na\(y\) -axis na hakuna amplitude kwa sababu kazi si mara kwa mara.
47) Grafu\(f(x)=\dfrac{\sin x}{x}\) kwenye dirisha\([-5\pi , 5\pi ]\) na kuelezea kile grafu inaonyesha.
Real-World Matumizi
48) Gurudumu la Ferris ni\(25\) 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. Kazi\(h(t)\) hutoa urefu wa mtu katika mita juu ya\(t\) dakika ya chini baada ya gurudumu kuanza kugeuka.
- Kupata amplitude, midline, na kipindi cha\(h(t)\).
- Pata fomu ya kazi ya urefu\(h(t)\).
- Jinsi ya juu ya ardhi ni mtu baada ya\(5\) dakika?
- Jibu
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- Amplitude:\(12.5\); kipindi:\(10\); midline:\(y=13.5\)
- \(h(t)=12.5\sin\left ( \dfrac{\pi}{5}(t-2.5) \right )+13.5\)
- \(26\)ft
6.2: Grafu za Kazi Zingine za Trigonometric
Sehemu hii inashughulikia graphing ya Tangent, Cosecant, Secant, na Cotangent curves.
Maneno
1) Eleza jinsi grafu ya kazi ya sine inaweza kutumika kwa grafu\(y=\csc x\).
Jibu
Tangu\(y=\csc x\) ni kazi ya kurudisha ya\(y=\sin x\)
2) Jinsi gani grafu ya\(y=\cos x\) kutumika kujenga grafu ya\(y=\sec x\)?
3) Eleza kwa nini kipindi cha\(y=\tan x\) ni sawa na\(\pi \).
Jibu
Majibu yatatofautiana. Kutumia mduara wa kitengo, mtu anaweza kuonyesha hilo\(y=\tan (x+\pi )=\tan x\).
4) Kwa nini hakuna intercepts kwenye grafu ya\(y=\csc x\)?
5) Kipindi cha\(y=\csc x\) kulinganisha na kipindi cha\(y=\sin x\)?
Jibu
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Kipindi hicho ni sawa:\(2\pi \)
Kialjebra
Kwa mazoezi 6-9, mechi kila kazi ya trigonometric na moja ya grafu zifuatazo.
6)\(f(x)=\tan x\)
7)\(f(x)=\sec x\)
- Jibu
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\(\mathrm{IV}\)
8)\(f(x)=\csc x\)
9)\(f(x)=\cot x\)
- Jibu
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\(\mathrm{III}\)
Kwa mazoezi 10-16, pata kipindi na mabadiliko ya usawa ya kila kazi.
10)\(f(x)=2\tan(4x-32)\)
11)\(h(x)=2\sec\left(\dfrac{\pi }{4}(x+1) \right)\)
- Jibu
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kipindi:\(8\); mabadiliko ya usawa:\(1\) kitengo cha kushoto
12)\(m(x)=6\csc\left(\dfrac{\pi }{3}x+\pi \right)\)
13) Kama\(\tan x=-1.5\)
- Jibu
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\(1.5\)
14) Kama\(\sec x=2\), kupata\(\sec (-x)\).
15) Kama\(\csc x=-5\), kupata\(\csc (-x)\).
- Jibu
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\(5\)
16) Kama\(x\sin x=2\), kupata\((-x)\sin (-x)\).
Kwa ajili ya mazoezi 17-18, kuandika upya kila kujieleza kama kwamba hoja\(x\) ni chanya.
17)\(\cot(-x)\cos(-x)+\sin(-x)\)
- Jibu
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\(-\cot x \cos x-\sin x\)
18)\(\cos(-x)+\tan(-x)\sin(-x)\)
Graphic
Kwa mazoezi 19-36, mchoro vipindi viwili vya grafu kwa kila kazi zifuatazo. Tambua sababu ya kuenea, kipindi, na asymptotes.
19)\(f(x)=2\tan(4x-32)\)
- Jibu
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sababu ya kuenea:\(2\); kipindi:\(\dfrac{\pi }{3}\)
asymptotes:\(x=\dfrac{1}{4}\left(\dfrac{\pi }{2}+\pi k \right)+8\),\(k\) wapi integer;
20)\(h(x)=2\sec\left(\dfrac{\pi }{4}(x+1) \right)\)
21)\(m(x)=6\csc\left(\dfrac{\pi }{3}x+\pi \right)\)
- Jibu
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sababu ya kuenea:\(6\); kipindi:\(6\); asymptotes:\(x=k\),\(k\) wapi integer
22)\(j(x)=\tan \left ( \dfrac{\pi }{2}x \right )\)
23)\(p(x)=\tan \left ( x-\dfrac{\pi }{2} \right )\)
- Jibu
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sababu ya kuenea:\(1\); kipindi:\(\pi \)
asymptotes:\(x=\pi k\),\(k\) wapi integer;
24)\(f(x)=4\tan (x)\)
25)\(f(x)=\tan \left ( x+\dfrac{\pi }{4} \right )\)
- Jibu
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Sababu ya kuenea:\(1\); kipindi:\(\pi \)
asymptotes:\(x=\dfrac{\pi}{4}+\pi k\),\(k\) wapi integer;
26)\(f(x)=\pi \tan(\pi x- \pi)-\pi\)
27)\(f(x)=2\csc (x)\)
- Jibu
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sababu ya kuenea:\(2\); kipindi:\(2\pi \)
asymptotes:\(x=\pi k\),\(k\) wapi integer;
28)\(f(x)=-\dfrac{1}{4}\csc(x)\)
29)\(f(x)=4\sec(3x)\)
- Jibu
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sababu ya kuenea:\(4\); kipindi:\(\dfrac{2\pi }{3}\)
asymptotes:\(x=\dfrac{\pi }{6}k\),\(k\) wapi integer isiyo ya kawaida;
30)\(f(x)=-3\cot(2x)\)
31)\(f(x)=7\sec(5x)\)
- Jibu
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sababu ya kuenea:\(7\); kipindi:\(\dfrac{2\pi }{5}\)
asymptotes:\(x=\dfrac{\pi }{10}k\),\(k\) wapi integer isiyo ya kawaida;
32)\(f(x)=\dfrac{9}{10}\csc(\pi x)\)
33)\(f(x)=2\csc \left(x+\dfrac{\pi }{4} \right)-1\)
- Jibu
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Sababu ya kuenea:\(2\); kipindi:\(2\pi \); asymptotes:\(x=-\dfrac{\pi}{4}+\pi k\),\(k\) wapi integer
34)\(f(x)=-\sec \left(x-\dfrac{\pi }{3} \right)-2\)
35)\(f(x)=\dfrac{7}{5}\csc \left(x-\dfrac{\pi }{4} \right)\)
- Jibu
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Sababu ya kuenea:\(\dfrac{7}{5}\); kipindi:\(2\pi \); asymptotes:\(x=\dfrac{\pi}{4}+\pi k\),\(k\) wapi integer
36)\(f(x)=5\left (\cot \left(x+\dfrac{\pi }{2} \right) -3 \right )\)
37) Curve tangent,\(A=1\)
- Jibu
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\(y=\tan\left(3\left(x-\dfrac{\pi}{4} \right) \right)+2\)
38) Curve tangent,\(A=-2\), kipindi cha\(\dfrac{\pi }{4}\); na mabadiliko ya awamu\((h, k)=\left (- \dfrac{\pi }{4},-2 \right )\)
Kwa mazoezi 39-45, pata equation kwa grafu ya kila kazi.
39)
- Jibu
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\(f(x)=\csc (2x)\)
40)
41)
- Jibu
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\(f(x)=\csc (4x)\)
42)
43)
- Jibu
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\(f(x)=2\csc x\)
44)
45)
- Jibu
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\(f(x)=\dfrac{1}{2}\tan (100\pi x)\)
Teknolojia
Kwa mazoezi 46-53, tumia calculator ya graphing kwa grafu vipindi viwili vya kazi iliyotolewa. Kumbuka: wengi calculators graphing hawana kifungo cosecant, kwa hiyo, unahitaji pembejeo\(\csc x\) kama\(\dfrac{1}{\sin x}\)
46)\(f(x)=| \csc (x) |\)
47)\(f(x)=| \cot (x) |\)
- Jibu
48)\(f(x)=2^{\csc (x)}\)
49)\(f(x)=\frac{\csc (x)}{\sec (x)}\)
- Jibu
50) Grafu\(f(x)=1+\sec^2(x)-\tan^2(x)\)
51)\(f(x)=\sec(0.001x)\)
- Jibu
52)\(f(x)=\cot(100 \pi x)\)
53)\(f(x)=\sin^2x +\cos^2x\)
- Jibu
Real-World Matumizi
54) Kazi\(f(x)=20\tan\left(\dfrac{\pi }{10}x\right)\) inaonyesha umbali katika harakati ya boriti ya mwanga kutoka gari la polisi kwenye ukuta kwa muda\(x\)
- Grafu juu ya muda\([0,5]\)
- Pata na kutafsiri sababu ya kuenea, kipindi, na asymptote.
- Tathmini\(f(10)\)\(f(2.5)\) na kujadili maadili ya kazi katika pembejeo hizo.
55) Amesimama kando ya ziwa, mvuvi hutazama mashua mbali mbali na kushoto kwake. Hebu\(x\)
- Je, ni uwanja wa busara kwa\(d(x)\) nini?
- Grafu\(d(x)\) kwenye uwanja huu.
- Kupata na kujadili maana ya asymptotes yoyote wima kwenye grafu ya\(d(x)\).
- Tumia na kutafsiri\(d\left ( -\dfrac{\pi }{3} \right )\)
Pande zote kwa nafasi ya pili ya decimal.. - Tumia na kutafsiri\(d\left ( \dfrac{\pi }{6} \right )\)
Pande zote kwa nafasi ya pili ya decimal.. - Umbali wa chini kati ya mvuvi na mashua ni nini? Hii inatokea lini?
- Jibu
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- \(\left ( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right )\)
- \(x=-\dfrac{\pi }{2}\)na\(x=\dfrac{\pi }{2}\)
umbali unakua bila kufungwa kama\(| x |\) mbinu\(\dfrac{\pi }{2}\) —yaani, katika pembe za kulia kwa mstari unaowakilisha kutokana kaskazini, mashua ingekuwa mbali sana, mvuvi hakuweza kuiona;; - \(3\); wakati\(x=-\dfrac{\pi }{3}\)
mashua ni\(3\) kilomita mbali;, - \(1.73\); wakati\(x=\dfrac{\pi }{6}\)
mashua ni karibu\(1.73\) kilomita;, - \(1.5\)km; wakati\(x=0\)
56) Laser rangefinder imefungwa kwenye comet inakaribia Dunia. Umbali\(g(x)\)
- Grafu\(g(x)\) juu ya muda\([0,35]\).
- Tathmini\(g(5)\) na kutafsiri habari.
- Umbali wa chini kati ya comet na Dunia ni nini? Hii inatokea lini? Ni mara kwa mara gani katika equation gani hii inafanana?
- Pata na kujadili maana ya asymptotes yoyote ya wima.
57) Kamera ya video inalenga roketi kwenye\(2\) maili ya uzinduzi wa pedi kutoka kamera. Pembe ya mwinuko kutoka ardhini hadi roketi baada ya\(x\) sekunde ni\(\dfrac{\pi }{120}x\).
- Andika kazi inayoonyesha urefu\(h(x)\)
katika maili, ya roketi juu ya ardhi baada ya\(x\) sekunde. Puuza curvature ya Dunia., - Grafu\(h(x)\) juu ya muda\((0,60)\).
- Tathmini na kutafsiri maadili\(h(0)\) na\(h(30)\).
- Nini kinatokea kwa maadili ya\(h(x)\) kama\(x\) mbinu\(60\) sekunde? Tafsiri maana ya hii kwa suala la tatizo.
- Jibu
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- \(h(x)=2\tan \left(\dfrac{\pi }{120}x \right)\)
- \(h(0)=0\)
baada ya\(0\) sekunde, roketi ni\(0\) mi juu ya ardhi;\(h(30)=2\): baada ya\(30\) sekunde, makombora ni\(2\) mi juu;: - Kama\(x\) inakaribia\(60\) sekunde, maadili ya\(h(x)\) kukua inazidi kubwa. Umbali wa roketi unaongezeka sana kiasi kwamba kamera haiwezi kuufuatilia tena.
6.3: Kazi za Trigonometric Inverse
Katika sehemu hii, tutazingatia kazi za trigonometric inverse. Kazi za trigonometric inverse “hupunguza” kile kazi ya awali ya trigonometric “inafanya,” kama ilivyo kwa kazi nyingine yoyote na inverse yake. Kwa maneno mengine, uwanja wa kazi ya inverse ni aina ya kazi ya awali, na kinyume chake.
Maneno
1) Kwa nini kazi\(f(x)=\sin^{-1}x\) na\(g(x)=\cos^{-1}x\) kuwa na safu tofauti?
- Jibu
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Kazi\(y=\sin x\) ni moja kwa moja\(\left [ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right ]\)
hivyo, muda huu ni aina mbalimbali ya kazi inverse ya\(y=\sin x\),\(f(x)=\sin^{-1}x\); Kazi\(y=\cos x\) ni moja kwa moja\([0,\pi ]\) hivyo, muda huu ni aina mbalimbali ya kazi inverse ya\(y=\cos x\),\(f(x)=\cos^{-1}x\);
2) Tangu kazi\(y=\cos x\) na\(y=\cos^{-1}x\) ni kazi inverse, kwa nini\(\cos^{-1}\left (\cos \left (-\dfrac{\pi }{6} \right ) \right )\)
3) Eleza maana ya\(\dfrac{\pi }{6}=\arcsin (0.5)\).
- Jibu
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\(\dfrac{\pi }{6}\)ni kipimo cha radian cha angle kati\(-\dfrac{\pi }{2}\) na\(\dfrac{\pi }{2}\) ambaye sine ni\(0.5\).
4) Wengi calculators hawana ufunguo wa kutathmini\(\sec ^{-1}(2)\)
5) Kwa nini lazima uwanja wa kazi sine,\(\sin x\)
- Jibu
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Ili kazi yoyote iwe na inverse, kazi lazima iwe moja kwa moja na inapaswa kupitisha mtihani wa mstari usio na usawa. Kazi ya kawaida ya sine sio moja kwa moja isipokuwa uwanja wake umezuiwa kwa namna fulani. Wanahisabati wamekubali kuzuia kazi sine kwa muda\(\left [ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right ]\) hivyo kuwa ni moja kwa moja na ana inverse.
6) Jadili kwa nini kauli hii si sahihi:\(\arccos(\cos x)=x\) kwa wote\(x\).
7) Kuamua kama kauli ifuatayo ni ya kweli au ya uongo na kuelezea jibu lako:\(\arccos(-x)=\pi - \arccos x\)
- Jibu
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Kweli. angle,\(\theta _1\) kwamba ni sawa\(\arccos(-x)\),\(x>0\), itakuwa pili quadrant angle na angle kumbukumbu\(\theta _2\),, ambapo\(\theta _2\) sawa\(\arccos x\),\(x>0\). Kwa kuwa\(\theta _2\) ni angle kumbukumbu kwa\(\theta _1\),\(\theta _2=\pi - \theta _1\) na\(\arccos(-x)=\pi - \arccos x-\)
Kialjebra
Kwa mazoezi 8-16, tathmini maneno.
8)\(\sin^{-1}\left(\dfrac{\sqrt{2}}{2}\right)\)
9)\(\sin^{-1}\left(-\dfrac{1}{2}\right)\)
- Jibu
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\(-\dfrac{\pi }{6}\)
10)\(\cos^{-1}\left(-\dfrac{1}{2}\right)\)
11)\(\cos^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)\)
- Jibu
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\(\dfrac{3\pi }{4}\)
12)\(\tan^{-1}(1)\)
13)\(\tan^{-1}(-\sqrt{3})\)
- Jibu
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\(-\dfrac{\pi }{3}\)
14)\(\tan^{-1}(-1)\)
15)\(\tan^{-1}(\sqrt{3})\)
- Jibu
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\(\dfrac{\pi }{3}\)
16)\(\tan^{-1}\left(\dfrac{-1}{\sqrt{3}}\right)\)
Kwa mazoezi 17-21, tumia calculator kutathmini kila kujieleza. Eleza majibu kwa karibu mia moja.
17)\(\cos^{-1}(-0.4)\)
- Jibu
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\(1.98\)
18)\(\arcsin (0.23)\)
19)\(\arccos \left(\dfrac{3}{5}\right)\)
- Jibu
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\(0.93\)
20)\(\cos^{-1}(-0.8)\)
21)\(\tan^{-1}(6)\)
- Jibu
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\(1.41\)
Kwa mazoezi 22-23, pata angle\(\theta\) katika pembetatu iliyotolewa. Majibu ya pande zote kwa karibu mia moja.
22)
23)
- Jibu
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\(0.56\)radiani
Kwa mazoezi 24-36, pata thamani halisi, ikiwa inawezekana, bila calculator. Ikiwa haiwezekani, eleza kwa nini.
24)\(\sin^{-1}(\cos(\pi))\)
25)\(\tan^{-1}(\sin(\pi))\)
- Jibu
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\(0\)
26)\(\cos^{-1}\left(\sin \left(\dfrac{\pi}{3} \right)\right)\)
27)\(\tan^{-1}\left(\sin \left(\dfrac{\pi}{3} \right)\right)\)
- Jibu
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\(0.71\)
28)\(\sin^{-1}\left(\cos \left(\dfrac{-\pi}{2} \right)\right)\)
29)\(\tan^{-1}\left(\sin \left(\dfrac{4\pi}{3} \right)\right)\)
- Jibu
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\(-0.71\)
30)\(\sin^{-1}\left(\sin \left(\dfrac{5\pi}{6} \right)\right)\)
31)\(\tan^{-1}\left(\sin \left(\dfrac{-5\pi}{2} \right)\right)\)
- Jibu
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\(-\dfrac{\pi}{4}\)
32)\(\cos \left(\sin^{-1} \left(\dfrac{4}{5} \right)\right)\)
33)\(\sin \left(\cos^{-1} \left(\dfrac{3}{5} \right)\right)\)
- Jibu
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\(0.8\)
34)\(\sin \left(\tan^{-1} \left(\dfrac{4}{3} \right)\right)\)
35)\(\cos \left(\tan^{-1} \left(\dfrac{12}{5} \right)\right)\)
- Jibu
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\(\dfrac{5}{13}\)
36)\(\cos \left(\sin^{-1} \left(\dfrac{1}{2} \right)\right)\)
Kwa mazoezi 37-41, pata thamani halisi ya kujieleza kwa\(x\) msaada wa pembetatu ya kumbukumbu.
37)\(\tan \left(\sin^{-1} (x-1)\right)\)
- Jibu
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\(\dfrac{x-1}{\sqrt{-x^2+2x}}\)
38)\(\sin \left(\sin^{-1} (1-x)\right)\)
39)\(\cos \left(\sin^{-1} \left(\dfrac{1}{x}\right)\right)\)
- Jibu
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\(\dfrac{\sqrt{x^2-1}}{x}\)
40)\(\cos \left(\tan^{-1} (3x-1)\right)\)
41)\(\tan \left(\sin^{-1} \left(x+\dfrac{1}{2}\right)\right)\)
- Jibu
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\(\dfrac{x+0.5}{\sqrt{-x^2-x+\tfrac{3}{4}}}\)
Upanuzi
Kwa zoezi 42, tathmini maneno bila kutumia calculator. Kutoa thamani halisi.
2)\(\dfrac{\sin^{-1}\left ( \tfrac{1}{2} \right )-\cos^{-1}\left ( \tfrac{\sqrt{2}}{2} \right )+\sin^{-1}\left ( \tfrac{\sqrt{3}}{2} \right )-\cos^{-1}(1)}{\cos^{-1}\left ( \tfrac{\sqrt{3}}{2} \right )-\sin^{-1}\left ( \tfrac{\sqrt{2}}{2} \right )+\cos^{-1}\left ( \tfrac{1}{2} \right )-\sin^{-1}(0)}\)
Kwa mazoezi 43-47, tafuta kazi ikiwa \(\sin t = \dfrac{x}{x+1}\)
43)\(\cos t\)
- Jibu
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\(\dfrac{\sqrt{2x+1}}{x+1}\)
44)\(\sec t\)
45)\(\cot t\)
- Jibu
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\(\dfrac{\sqrt{2x+1}}{x}\)
46)\(\cos \left(\sin^{-1} \left(\dfrac{x}{x+1}\right)\right)\)
47)\(\tan^{-1} \left(\dfrac{x}{\sqrt{2x+1}}\right)\)
- Jibu
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\(t\)
Graphic
48) Grafu\(y=\sin^{-1} x\) na ueleze kikoa na kazi mbalimbali.
49) Grafu\(y=\arccos x\)
- Jibu
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uwanja\([-1,1]\)
mbalimbali\([0,\pi ]\);
50) Graph mzunguko mmoja wa\(y=\tan^{-1} x\) na hali ya uwanja na aina mbalimbali ya kazi.
51) Kwa thamani\(x\) gani\(\sin x=\sin^{-1} x\)? Tumia calculator ya graphing ili takriban jibu.
- Jibu
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takriban\(x=0.00\)
52) Kwa thamani\(x\) gani\(\cos x=\cos^{-1} x\)? Tumia calculator ya graphing ili takriban jibu.
Real-World Matumizi
53) Tuseme ngazi\(13\) -foot ni leaning dhidi ya jengo, kufikia chini ya pili-na-oor dirisha\(12\) miguu juu ya ardhi. Ni angle gani, katika radians, ngazi hufanya na jengo?
Jibu
-
\(0.395\)radiani
54) Tuseme unaendesha\(0.6\) maili kwenye barabara ili umbali wa wima ubadilike kutoka\(0\) kwa\(150\) miguu. Je! Ni pembe gani ya mwinuko wa barabara?
55) Pembetatu ya isosceles ina pande mbili za urefu wa\(9\) inchi. Upande uliobaki una urefu wa\(8\) inchi. Kupata angle kwamba upande wa\(9\) inchi hufanya kwa upande\(8\) -inch.
- Jibu
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\(1.11\)radiani
56) Bila kutumia calculator, takriban thamani ya\(\arctan (10,000)\)
57) Truss kwa paa la nyumba hujengwa kutoka pembetatu mbili zinazofanana. Kila mmoja ana msingi wa\(12\) miguu na urefu wa\(4\) miguu. Pata kipimo cha angle ya papo hapo karibu na upande wa\(4\) mguu.
- Jibu
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\(1.25\)radiani
58) Mstari\(y=\dfrac{3}{5}x\) hupita kupitia asili katika\(x,y\) -ndege. Je! Ni kipimo gani cha angle ambayo mstari hufanya na\(x\) mhimili mzuri?
59) Mstari\(y=\dfrac{-3}{7}x\) hupita kupitia asili katika\(x,y\) -ndege. Je! Ni kipimo gani cha angle ambayo mstari hufanya na\(x\) mhimili hasi?
- Jibu
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\(0.405\)radiani
60) Ni asilimia gani ya daraja ambalo barabara inapaswa kuwa nayo ikiwa angle ya mwinuko wa barabara ni\(4\) digrii? (Daraja asilimia hufafanuliwa kama mabadiliko katika urefu wa barabara juu ya umbali\(100\) -foot usawa. Kwa mfano\(5\%\) daraja linamaanisha kwamba barabara inainuka\(5\) miguu kwa kila\(100\) miguu ya umbali usio na usawa.)
61) Ngazi ya\(20\) mguu hutegemea upande wa jengo ili mguu wa ngazi ni\(10\) miguu kutoka chini ya jengo hilo. Kama specifikationer wito kwa angle ya ngazi ya mwinuko kuwa kati\(35\) na\(45\) digrii, je, uwekaji wa ngazi hii kukidhi specifikationer usalama?
- Jibu
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Hapana. Pembe ngazi hufanya na usawa ni\(60\) digrii.
62) Tuseme ngazi ya\(15\) mguu hutegemea upande wa nyumba ili angle ya mwinuko wa ngazi ni\(42\) digrii. Je, ni mbali gani mguu wa ngazi kutoka upande wa nyumba?