Deck 5: Trigonometric Functions

Question 1

For $\theta=4^{\circ}$ evaluate each of the following. Round your answers to four decimal places.

A) $\cos ^{2} \theta$
B) $\cos \left(\theta^{2}\right)$

Answer 1

Part A: 0.9951
Part B: 0.9613

Question 2

For $\theta=-7^{\circ}$ evaluate each of the following. Round your answers to four decimal places.

A) $\sin ^{2} \theta$
B) $\sin \left(\theta^{2}\right)$

Answer 2

Part A: 0.0149
Part B: 0.7547

Question 3

Given that $\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2}$ , find an exact expression for $\sin \left(\frac{\pi}{12}\right)$ .

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

Answer 3

$\frac{\sqrt{2-\sqrt{3}}}{2}$

Question 4

Given that $\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2}$ , find an exact expression for $\tan \left(\frac{\pi}{12}\right)$ .

Question 5

Given that $\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2}$ , find an exact expression for $\cos \left(\frac{13 \pi}{12}\right)$ .

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

Question 6

Given that $\sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$ , find an exact expression for $\sec \left(\frac{7 \pi}{8}\right)$ .

Question 7

Given that $\sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$ , find an exact expression for $\tan \left(\frac{25 \pi}{8}\right)$ .

Question 8

Given that $\cos \frac{\pi}{8}=\frac{\sqrt{2+\sqrt{2}}}{2}$ , find an exact expression for $\sin \left(\frac{39 \pi}{8}\right)$ .

Question 9

Given that $\sin \frac{\pi}{12}=\frac{\sqrt{2-\sqrt{3}}}{2}$ , find an exact expression for $\tan \left(\frac{47 \pi}{12}\right)$ .

Question 10

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ , with tan(u) = -3. Find an exact expression for tan(-u).

Question 11

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ , with tan(u) = -3. Find an exact expression for cos(-u).

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

Question 12

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ , with tan(u) = -9. Find an exact expression for sin(u).

Question 13

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ , with tan(u) = -3. Find an exact expression for tan(3 $\pi$ - u).

Question 14

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$
, with tan(u) = -2. Find an exact expression for sin(6 $\pi$ + u).

Question 15

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ , with tan(u) = -6. Find an exact expression for tan(u - 9 $\pi$ ).

A) $\frac{1}{6}$
B) 6
C) -6
D) $-\frac{1}{6}$

Question 16

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ , with tan(u) = -3. Find an exact expression for $\sin \left(\frac{\pi}{2}-u\right)$ .

Question 17

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ , with tan(u) = -6. Find an exact expression for $\tan \left(\frac{\pi}{2}-u\right)$ .

Question 18

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ , with tan(u) = -1. Find an exact expression for $\cos \left(\frac{\pi}{2}-u\right)$ .

Question 19

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ , with tan(u) = -5. Find an exact expression for sin(8 $\pi$ + u).

A) $\frac{\sqrt{26}}{26}$
B) $-\frac{\sqrt{26}}{26}$
C) $\frac{5 \sqrt{26}}{26}$
D) $-\frac{5 \sqrt{26}}{26}$

Question 20

Suppose u is in the interval $\left(\frac{\pi}{2}, \pi\right)$ , with tan(u) = -3. Find an exact expression for $\sin \left(\frac{\pi}{2}+u\right)$ .

Question 21

-Suppose a = 8 and b = 13. Evaluate c. Give the exact answer.

Question 22

-Suppose a = 9 and b = 11. Evaluate sin(u). Give the exact answer.

Question 23

-Suppose a = 7 and b = 8. Evaluate cos(v). Give the exact answer.

Question 24

-Suppose a = 6 and b = 7. Evaluate tan(v).

A) $\frac{6}{7}$
B) $-\frac{6}{7}$
C) $\frac{7}{6}$
D) $-\frac{7}{6}$

Question 25

-Suppose b = 6 and c = 7. Evaluate cos(u). Give the exact answer.

Question 26

-Suppose b = 7 and c = 8. Evaluate tan(u). Give the exact answer.

Question 27

-Suppose b = 8 and c = 11. Evaluate sin(v).

A) $\frac{\sqrt{17}}{8}$
B) $\frac{\sqrt{57}}{8}$
C) $\frac{\sqrt{17}}{11}$
D) $\frac{\sqrt{57}}{11}$

Question 28

-Suppose a = 3 and $u=18^{\circ}$ . Evaluate b. Give the exact answer.

Question 29

-Suppose a = 2 and $u=27^{\circ}$ . Evaluate cos(v). Give the exact answer.

Question 30

-Suppose a = 9 and $u=34^{\circ}$ . Evaluate tan(v).

A) $\tan \left(34^{\circ}\right)$
B) $\tan \left(19^{\circ}\right)$
C) $\cot \left(34^{\circ}\right)$
D) $\cot \left(19^{\circ}\right)$

Question 31

-Suppose c = 8 and u = 1 radian. Evaluate b. Give the exact answer.

Question 32

-Suppose c = 4 and u = 1 radian. Evaluate sin(v). Give the exact answer.

Question 33

-Suppose c = 6 and $\cos (u)=\frac{1}{7}$ . Evaluate a. Give the exact answer.

Question 34

-Suppose c = 5 and $\cos (u)=\frac{1}{6}$ . Evaluate sin(u).

A) $-\frac{\sqrt{37}}{6}$
B) $\frac{\sqrt{35}}{6}$
C) $-\frac{\sqrt{35}}{6}$
D) $\frac{\sqrt{37}}{6}$

Question 35

-Suppose c = 17 and $\cos (u)=\frac{1}{18}$ . Evaluate cos(v). Give the exact answer.

Question 36

-Suppose c = 15 and $\cos (u)=\frac{1}{16}$ . Evaluate tan(v). Give the exact answer.

Question 37

-Suppose b = 15 and $\sin (u)=\frac{1}{16}$ . Evaluate c. Give the exact answer.

Question 38

-Suppose b = 5 and $\sin (u)=\frac{1}{6}$ . Evaluate sin(v). Give the exact answer.

Question 39

-Suppose b = 11 and $\sin (u)=\frac{1}{12}$ . Evaluate cos(v).

A) $-\frac{\sqrt{145}}{12}$
B) $\frac{1}{12}$
C) $-\frac{\sqrt{143}}{12}$
D) $\frac{\sqrt{145}}{12}$

Question 40

Suppose that a 29-foot ladder is leaning against a wall, making a $69^{\circ}$ angle with the ground (as measured from a perpendicular line from the base of the ladder to the wall). How many feet up the wall is the end of the ladder, to the nearest tenth of a foot?

Question 41

Find the perimeter of a right triangle that has hypotenuse of length 7 mm and an 18° angle. Round the answer to two decimal places.

Question 42

Find the four smallest positive numbers $\theta$ such that tan(2 $\theta$ ) = 1. Give the exact answers in radians.

Question 43

Suppose $0<\theta<\frac{\pi}{2}$ and $\cos (\theta)=\frac{3}{4}$ . Evaluate:

A) sin( $\theta$ )
B) tan( $\theta$ )
Give the exact answers.

Question 44

Suppose $\frac{\pi}{2}<\theta<\pi$ and $\sin (\theta)=\frac{1}{2}$ . Evaluate:

A) cos( $\theta$ )
B) tan( $\theta$ )
Give the exact answers.

Question 45

Suppose $-\frac{\pi}{2}<\theta<0$ and $\cos (\theta)=\frac{15}{16}$ . Evaluate:

A) sin( $\theta$ )
B) tan( $\theta$ )
Give the exact answers.

Question 46

Suppose $-\frac{\pi}{2}<\theta<0$ and $\sin (\theta)=-\frac{3}{4}$ . Evaluate:

A) cos( $\theta$ )
B) tan( $\theta$ )
Give the exact answers.

Question 47

Suppose $0<\theta<\frac{\pi}{2}$ and $\tan (\theta)=\frac{5}{6}$ . Evaluate:

A) sin((\theta\))
B) cos( $\theta$ )
Give the exact answers.

Question 48

Suppose $-\frac{\pi}{2}<\theta<0$ and $\tan (\theta)=-\frac{5}{6}$ . Evaluate:

A) sin( $\theta$ )
B) cos( $\theta$ )
Give the exact answers.

Question 49

Suppose $\frac{\pi}{2}<\theta<\pi$ and $\tan (\theta)=-\frac{6}{7}$ . Evaluate:

A) sin( $\theta$ )
B) cos( $\theta$ )
Give the exact answers.

Question 50

Given that $\cos \left(15^{\circ}\right)=\frac{\sqrt{2+\sqrt{3}}}{2}$ , find an exact expression for $\cos 165^{\circ}$ .

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

Question 51

Given that $\cos \left(15^{\circ}\right)=\frac{\sqrt{2+\sqrt{3}}}{2}$ , find an exact expression for $\sin 285^{\circ}$ .

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

Question 52

Given that $\cos \left(15^{\circ}\right)=\frac{\sqrt{2+\sqrt{3}}}{2}$ , find an exact expression for $\cot 15^{\circ}$ :

Question 53

Given that $\sin \left(22.5^{\circ}\right)=\frac{\sqrt{2-\sqrt{2}}}{2}$ , find an exact expression for $\tan 22.5^{\circ}$ :

Question 54

Given that $\sin \left(22.5^{\circ}\right)=\frac{\sqrt{2-\sqrt{2}}}{2}$ , find an exact expression for $\csc 22.5^{\circ}$ :

Question 55

Suppose u is in the interval $\left(0, \frac{\pi}{2}\right)$ , with tan(u) = 4. Find an exact expression for cot(u).

A) -4
B) 4
C) $-\frac{1}{4}$
D) $\frac{1}{4}$

Question 56

Suppose u is in the interval $\left(0, \frac{\pi}{2}\right)$ , with tan(u) = 9. Find an exact expression for sec(u).

Question 57

Suppose u is in the interval $\left(0, \frac{\pi}{2}\right)$ , with tan(u) = 4. Find an exact expression for csc(u).

A) $\frac{4}{\sqrt{17}}$
B) $-\frac{4}{\sqrt{17}}$
C) $\frac{\sqrt{17}}{4}$
D) $-\frac{\sqrt{17}}{4}$

Question 58

Suppose u is in the interval $\left(0, \frac{\pi}{2}\right)$ , with tan(u) = 4. Find an exact expression for sin(u).

Question 59

Suppose u is in the interval $\left(0, \frac{\pi}{2}\right)$ , with tan(u) = 3. Find an exact expression for cos(u).

Question 60

Suppose u is in the interval $\left(0, \frac{\pi}{2}\right)$ , with cot(u) = 5. Find an exact expression for sec(u).

A) $\frac{5}{\sqrt{26}}$
B) $-\frac{5}{\sqrt{26}}$
C) $\frac{\sqrt{26}}{5}$
D) $-\frac{\sqrt{26}}{5}$

Question 61

Suppose u is in the interval $\left(0, \frac{\pi}{2}\right)$
, with cot(u) = 9. Find an exact expression for sin(u).

Question 62

Find the equation of the line in the xy-plane that goes through the origin and makes an angle of 0.5 radians with the positive horizontal axis. Round the slope to three decimal places.

Question 63

Find $\theta$ such that tan( $\theta$ ) = $\sqrt{3}$ for $0<\theta<\pi$ . Give the exact angle in radians.

Question 64

Give the exact values for the following quantities.
(i) cos(13 $\pi$ ) , (ii) sin(13 $\pi$ )

A) (i) -1, (ii) 1
B) (i) 1, (ii) 0
C) (i) -1, (ii) 0
D) (i) 1, (ii) -1

Question 65

Give the exact values for the following quantities.

A) $\cos \left(\frac{3 \pi}{4}\right)$
B) $\sin \left(\frac{3 \pi}{4}\right)$

Question 66

Give the exact values for the following quantities.

A) $\cos \left(\frac{2 \pi}{3}\right)$
B) $\sin \left(\frac{2 \pi}{3}\right)$

Question 67

Give the exact values for the following quantities.

A) $\cos \left(\frac{37 \pi}{4}\right)$
B) $\sin \left(\frac{37 \pi}{4}\right)$

Question 68

Give the exact values for the following quantities.

A) $\cos \left(\frac{7 \pi}{3}\right)$
B) $\sin \left(\frac{7 \pi}{3}\right)$

Question 69

Give the exact values for the following quantities.
(i) $\cos \left(\frac{25 \pi}{6}\right)$ , (ii) $\sin \left(\frac{25 \pi}{6}\right)$

A) (i) $-\frac{\sqrt{3}}{2}$ , (ii) $\frac{1}{2}$
B) (i) $\frac{\sqrt{3}}{2}$ , (ii) $-\frac{1}{2}$
C) (i) $-\frac{\sqrt{3}}{2}$ , (ii) $-\frac{1}{2}$
D) (i) $\frac{\sqrt{3}}{2}$ , (ii) $\frac{1}{2}$

Question 70

Give the exact values for the following quantities.

A) $\cos \left(\frac{47 \pi}{6}\right)$
B) $\sin \left(\frac{47 \pi}{6}\right)$

Question 71

Find the smallest number $\theta$ larger than 16 $\theta$ such that cos $\theta$ = 0. Give the exact answer in radians.

Question 72

Find the smallest number $\theta$ larger than 11 $\pi$ such that $\cos \theta=-\frac{1}{2}$

A) $\frac{32 \pi}{3}$
B) $\frac{19 \pi}{3}$
C) $\frac{34 \pi}{3}$
D) $\frac{17 \pi}{3}$

Question 73

Find the four smallest positive numbers $\theta$ such that $\sin (17 \theta)=0$ . Give the exact answers.

Question 74

Find the four smallest positive numbers $\theta$ such that $\cos (4 \theta)=0$ . Give the exact answers.

Question 75

Find the four smallest positive numbers $\theta$ such that $\sin (19 \theta)=\frac{1}{2}$ . Give the exact answers.

Question 76

Find the four smallest positive numbers $\theta$ such that $\cos (7 \theta)=-\frac{1}{2}$ . Give the exact answers.

Question 77

Suppose $0<\theta<\pi$ and $\cos \theta=\frac{4}{9}$ . Evaluate sin $\theta$ . Give the exact answer.

Question 78

Suppose $\frac{\pi}{2}<\theta<\pi$ And $\sin \theta=\frac{8}{9}$ . Evaluate cos $\theta$ .

A) $\frac{\sqrt{17}}{9}$
B) $-\frac{\sqrt{17}}{9}$
C) $-\frac{\sqrt{15}}{9}$
D) $\frac{\sqrt{15}}{9}$

Question 79

Suppose $\frac{3 \pi}{2}<\theta<2 \pi$ and $\cos \theta=\frac{8}{9}$ . Evaluate sin $\theta$ . Give the exact answer.

Question 80

Suppose $\frac{\pi}{2}<\theta<\frac{3 \pi}{2}$ and $\sin \theta=\frac{1}{2}$ . Evaluate cos $\theta$ . Give the exact answer.