Deck 6: Trigonometric Algebra and Geometry

Question 1

Suppose the figure below is part of the graph of the function 8sin(9x). What is the value of b?

A) 9.
B) 16
C) 8
D) 9

Answer 1

8

Question 2

Find the smallest positive number c such that the figure below is part of the graph of the function sin(x + 5c). Give the exact answer.

Answer 2

$\frac{2 \pi}{5}$

Question 3

Sketch the graphs of the functions -5sin(x) and sin(-5x) on the interval [- $\pi$ , $\pi$ ].

Answer 3

Question 4

What is the range of the function 3sin(6x)?

Question 5

What is the amplitude of the function 9sin(8x)?

Question 6

What is the period of the function 6sin(19x)?

A) $\frac{2 \pi}{19}$
B) $\frac{\pi}{19}$
C) $\frac{\pi}{38}$
D) 2 $\pi$

Question 7

What is the range of the function 4 + cos(4 - x)?

Question 8

What is the amplitude of the function 3 + cos(1 - x)?

Question 9

What is the period of the function 3 + cos(5 - x)?

Question 10

What is the range of the function $6 \cos \left(\frac{\pi}{2} x+\frac{2}{3} \pi\right)$ ?

Question 11

What is the amplitude of the function $2 \cos \left(\frac{\pi}{2} x+\frac{3}{4} \pi\right)$ ?

Question 12

What is the period of the function $8 \cos \left(\frac{\pi}{3} x+\frac{6}{7} \pi\right)$ ?

A) $\frac{6}{\pi}$
B) $\frac{\pi}{3}$
C) 3
D) 6

Question 13

By what fraction of the period of $9 \cos \left(\frac{\pi}{2} x\right)$ has the graph been shifted left to obtain the graph of $9 \cos \left(\frac{\pi}{2} x+\frac{4}{3} \pi\right)$ ?

Question 14

Find two distinct values for a so that acos(bx + c) + d, has amplitude 5.

Question 15

Find values for a and d, respectively, with a > 0, so that f(x) = acos(bx + c) + d has range [8, 14].

A) 4, 11
B) 3, 11
C) 4, 10
D) 3, 10

Question 16

Find values for a, d, c, and b, respectively, with a > 0, b > 0, and $0 \leq c \leq \pi$ , so that f(x) = acos(bx + c) + d has range [9, 17], f(0) = 15, and f has period 18. Give the exact answer.

Question 17

What is the range of the function 7sin²(4x)?

Question 18

What is the amplitude of the function 18cos²(9x)?

A) 18
B) 9
C) 81
D) 324

Question 19

What is the period of the function 5sin²(3x)? Give the exact answer.

Question 20

Sketch the graph of the function cos²(4x) on the interval [- $\pi$ , $\pi$ ].

Question 21

For $x=12^{\circ}$ and $y=18^{\circ}$ , evaluate each of the following. Round your answers to the nearest hundredth.

A) cos(x + y)
B) cos x + cos y

Question 22

For x = 5.6 radians and y = 2.6 radians, evaluate each of the following. Round your answers to the nearest hundredth.

A) sin(x - y)
B) sin x - sin y

Question 23

Find an exact expression for $\sin 292.5^{\circ}$ . The following information will be useful.
$\sin \left(22.5^{\circ}\right)=\frac{\sqrt{2-\sqrt{2}}}{2}$ , and $\cos \left(22.5^{\circ}\right)=\frac{\sqrt{2+\sqrt{2}}}{2}$

Question 24

Find an exact expression for $\sin 157.5^{\circ}$ . The following information will be useful:
$\sin \left(22.5^{\circ}\right)=\frac{\sqrt{2-\sqrt{2}}}{2}$ , and $\cos \left(22.5^{\circ}\right)=\frac{\sqrt{2+\sqrt{2}}}{2}$

A) $\frac{\sqrt{2+\sqrt{2}}}{2}$
B) $\frac{\sqrt{2-\sqrt{2}}}{2}$
C) $-\frac{\sqrt{2+\sqrt{2}}}{2}$
D) $-\frac{\sqrt{2-\sqrt{2}}}{2}$

Question 25

Find an exact expression for $\cos 42^{\circ}$ . The following information will be useful:
$\cos \left(18^{\circ}\right)=\sqrt{\frac{\sqrt{5}+5}{8}}$ and $\sin \left(18^{\circ}\right)=\frac{\sqrt{5}-1}{4}$

Question 26

Find an exact expression for $\cos 202.5^{\circ}$ . The following information will be useful:
$\sin \left(22.5^{\circ}\right)=\frac{\sqrt{2-\sqrt{2}}}{2}$ , and $\cos \left(22.5^{\circ}\right)=\frac{\sqrt{2+\sqrt{2}}}{2}$

A) $\frac{1}{2} \sqrt{2+\sqrt{2}}$
B) $\frac{1}{2} \sqrt{2-\sqrt{2}}$
C) $-\frac{1}{2} \sqrt{2-\sqrt{2}}$
D) $-\frac{1}{2} \sqrt{2+\sqrt{2}}$

Question 27

Evaluate cos(x + y) assuming that $-\frac{1}{2} \sqrt{2+\sqrt{2}}$ . Assume also that x is in the interval $\left(0, \frac{\pi}{2}\right)$ and y is in the interval $\left(\frac{\pi}{2}, \pi\right)$ . Give the exact answer.

Question 28

Evaluate sin(u + v) assuming that $\sin (u)=\frac{2}{3}$ and $\cos (v)=\frac{1}{2}$ . Assume also that u is in the interval $\left(0, \frac{\pi}{2}\right)$ and that v is in the interval $\left(-\frac{\pi}{2}, 0\right)$ . Give the exact answer.

Question 29

Evaluate cos(x - y) assuming that $\sin (y)=\frac{1}{14}$ and $\cos (x)=\frac{1}{16}$ . Assume also that x is in the interval $\left(0, \frac{\pi}{2}\right)$ and that y is in the interval $\left(\frac{\pi}{2}, \pi\right)$ . Give the exact answer.

Question 30

Evaluate sin(u - v) assuming that $\sin (u)=\frac{9}{10}$ and $\cos (\nu)=\frac{1}{\sqrt{5}}$ . Assume also that u is in the interval $\left(0, \frac{\pi}{2}\right)$ and that v is in the interval $\left(-\frac{\pi}{2}, 0\right)$ . Give the exact answer.

Question 31

Evaluate tan(x + y) assuming that $\sin (y)=\frac{1}{6}$ and $\cos (x)=\frac{1}{12}$ . Assume also that x is in the interval $\left(0, \frac{\pi}{2}\right)$ and that y is in the interval $\left(\frac{\pi}{2}, \pi\right)$ . Give the exact answer.

Question 32

Evaluate tan(u - v) assuming that $\sin (u)=\frac{11}{12}$ and $\cos (v)=\frac{1}{18}$ . Assume also that u is in the interval $\left(0, \frac{\pi}{2}\right)$ and that v is in the interval $\left(-\frac{\pi}{2}, 0\right)$ . Give the exact answer.

Question 33

Evaluate $\cos \left(\frac{\pi}{3}+\cos ^{-1}\left(\frac{2}{3}\right)\right)$ .

A) $\frac{1}{2}[2-\sqrt{15}]$
B) $\frac{1}{2}[2+\sqrt{15}]$
C) $\frac{1}{6}[2-\sqrt{15}]$
D) $\frac{1}{6}[2+\sqrt{15}]$

Question 34

Evaluate $\sin \left(\cos ^{-1} \frac{1}{4}+\tan ^{-1} 5\right)$ . Give the exact answer.

Question 35

Find a formula for $\tan \left(\theta-\frac{\pi}{3}\right)$ .

Question 36

Find a formula for: $\tan \left(\theta+\frac{\pi}{2}\right)$

A) $\cot \theta$
B) $-\cot \theta$
C) $\tan \theta$
D) $-\tan \theta$

Question 37

Find a formula for $\cos \left(\theta-\frac{\pi}{3}\right)$ .

Question 38

Find a formula for: $\cos \left(\theta+\frac{\pi}{2}\right)$

Question 39

Find a formula for $\sin \left(\theta-\frac{\pi}{3}\right)$ .

Question 40

Find a formula for $\sin \left(\theta+\frac{\pi}{2}\right)$

A) $\cos \theta$
B) $-\cos \theta$
C) $\sin \theta$
D) $-\sin \theta$

Question 41

For $\theta=26^{\circ}$ , evaluate each of the following; round your answers to the nearest hundredth.

A) cos 2 $\theta$
B) 2cos $\theta$

Question 42

For $\theta$ = -5 radians, evaluate each of the following; round your answers to the nearest hundredth.

A) sin 2 $\theta$
B) 2sin $\theta$

Question 43

For $\theta=64^{\circ}$ , evaluate each of the following; round your answers to the nearest hundredth.

A) $\sin \frac{\theta}{2}$
B) $\frac{\sin \theta}{2}$

Question 44

Given that $\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$ , find an exact expression for $\cos 288^{\circ}$ .

Question 45

Evaluate tan u assuming that u is in the interval $\left(0, \frac{\pi}{2}\right)$ $\cos u=\frac{1}{2}$ .

A) $-\sqrt{5}$
B) $\sqrt{5}$
C) $-\sqrt{3}$
D) $\sqrt{3}$

Question 46

Evaluate sin 2u assuming that u is in the interval $\left(0, \frac{\pi}{2}\right)$ and $\cos u=\frac{1}{15}$ . Give the exact answer.

Question 47

Evaluate $\cos \frac{\nu}{2}$ assuming that v is in the interval
$\left(0, \frac{\pi}{2}\right)$ and $\sin v=\frac{1}{7}$ . Give the exact answer.

Question 48

Evaluate $\tan \frac{u}{2}$ Assuming that u is in the interval
$\left(0, \frac{\pi}{2}\right)$ And $\cos u=\frac{1}{2}$ .

A) $\sqrt{\frac{1}{3}}$
B) $-\sqrt{\frac{1}{3}}$
C) $\sqrt{\frac{3}{1}}$
D) $-\sqrt{\frac{3}{1}}$

Question 49

Evaluate cos u assuming that u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ And $\sin u=\frac{1}{3}$ .

A) $\frac{\sqrt{8}}{3}$
B) $-\frac{\sqrt{8}}{3}$
C) $\frac{\sqrt{10}}{3}$
D) $-\frac{\sqrt{10}}{3}$

Question 50

Evaluate tan u assuming that u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ and $\sin u=\frac{1}{20}$ . Give the exact answer.

Question 51

Evaluate tan 2u assuming that u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ $\sin u=\frac{1}{7}$ . Give the exact answer.

Question 52

Evaluate $\tan \frac{u}{2}$ assuming that u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ and $\sin u=\frac{1}{2}$ . Give the exact answer.

Question 53

Evaluate $\cos \frac{u}{2}$ assuming that u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ and $\sin u=\frac{1}{2}$ . Give the exact answer.

Question 54

Evaluate cos u assuming that u is in the interval $\left(-\frac{\pi}{2}, 0\right)$ And $\tan u=-\frac{1}{16}$ .

A) $\frac{16 \sqrt{255}}{255}$
B) $-\frac{16 \sqrt{255}}{255}$
C) $\frac{16 \sqrt{257}}{257}$
D) $-\frac{16 \sqrt{257}}{257}$

Question 55

Evaluate sin 2u assuming that u is in the interval $\left(-\frac{\pi}{2}, 0\right)$ and $\tan u=-\frac{1}{2}$ . Give the exact answer.

Question 56

Evaluate $\sin \frac{u}{2}$ assuming that u is in the interval $\left(-\frac{\pi}{2}, 0\right)$ and $\tan u=-\frac{1}{8}$ . Give the exact answer.

Question 57

Evaluate $\tan \frac{u}{2}$ Assuming that u is in the interval $\left(-\frac{\pi}{2}, 0\right)$ And $\tan u=-\frac{1}{2}$ .

A) $2-\sqrt{3}$
B) $2+\sqrt{3}$
C) $2-\sqrt{5}$
D) $2+\sqrt{5}$

Question 58

Find an exact expression for $\sin \left(67.5^{\circ}\right)$ .

Question 59

Find an exact expression for $\sin \left(\frac{\pi}{8}\right)$ .

Question 60

Find an exact expression for $\tan \left(\frac{\pi}{12}\right)$ .

Question 61

-Given a = 5, B = $15^{\circ}$ , C = $20^{\circ}$ . Evaluate:

A) a
B) b
C) c
Round b and c to the nearest tenth.

Question 62

-Given a = 12, B = $90^{\circ}$ , C = $63^{\circ}$ . Evaluate b. Round b to the nearest tenth.

A) 23.6
B) 26.4
C) 24.6
D) 22.6

Question 63

-Given a = 10, $B=\frac{2 \pi}{11}$ , $C=\frac{3 \pi}{11}$ . Evaluate:

A) a
B) b
C) c
Round b and c to the nearest tenth.

Question 64

-Given a = 3, b = 5, c = 6. Evaluate:

A) A
B) B
C) C
Round your answers to the nearest tenth.

Question 65

-Given a = 8, b = 7, c = 5. Evaluate:

A) A
B) B
C) C
Round your answers to the nearest tenth.

Question 66

-Given a = 7, b = 5, c = 8. Evaluate:

A) A
B) B
C) C
Round your answers to the nearest tenth.

Question 67

-Given a = 14, $A=\frac{\pi}{7}$ , $C=\frac{4 \pi}{7}$ . Evaluate:
B , b , c
Round your answers for b and c to the nearest tenth.

A) a: $\frac{2 \pi}{7}$     b: 14.0     c: 31.5
B) a : $\frac{\pi}{7}$ ,     b: 25.2     c: 31.5
C) a: $\frac{2 \pi}{7}$     b: 25.2     c: 31.5
D) a: $\frac{\pi}{7}$     b: 14.0     c: 31.5

Question 68

-Given a = 3, b = 4, c = 5. Evaluate:

A) A
B) B
C) C
Round your answers to the nearest tenth.

Question 69

-Given a = 3, b = 4, C =

. Evaluate:

A) c
B) A
C) B
Round your answers to the nearest tenth.

Question 70

-Given a = 6, b = 7, C = 2 radians. Evaluate:

A) c
B) A
C) B
Angles should be given in radians rounded to the nearest hundredth.

Question 71

-Given a = 4, b = 3, C = 1. Evaluate:

A) c
B) A
C) B
Angles should be given in radians rounded to the nearest hundredth.

Question 72

-Given a = 7, b = 6, B = $10^{\circ}$ . Evaluate:

A) A where A < $90^{\circ}$
B) C
C) c
Round your answers to the nearest tenth.

Question 73

-Given a = 4, b = 3, C = $24^{\circ}$ . Evaluate A.

A) $36.2^{\circ}$
B) $44.1^{\circ}$
C) $68.1^{\circ}$
D) $44.6^{\circ}$

Question 74

-Given a = 9, b = 11, B = $15^{\circ}$ . Evaluate:

A) A where A < $90^{\circ}$
B) C
C) c
Round your answers to the nearest tenth.

Question 75

-Given a = 4, b = 3, B = $20^{\circ}$ . Evaluate:

A) A where A > $90^{\circ}$
B) C
C) c
Round your answers to the nearest tenth.

Question 76

-Given a = 9, b = 7, B = $30^{\circ}$ . Evaluate:

A) A where A > $90^{\circ}$
B) C
C) c
Round your answers to the nearest tenth.

Question 77

-Given a = 5, c = 8, B = $20^{\circ}$ . Evaluate b.

A) 3.2
B) 4.7
C) 2.7
D) 3.7

Question 78

-Given a = 11, c = 8, B = $20^{\circ}$ . Evaluate:

A) b
B) A
C) C
Round your answers to the nearest hundredth.

Question 79

-Given a = 11, c = 13, B = $90^{\circ}$ . Evaluate:

A) b
B) A
C) C
Round your answers to the nearest hundredth.

Question 80

-Given a = 12, c = 10, B = $75^{\circ}$ . Evaluate C.

A) $45.7^{\circ}$
B) $44.7^{\circ}$
C) $43.7^{\circ}$
D) $46.7^{\circ}$