Deck 8: Polar Coordinates and Parametric Equations

Question 1

Write the sum as a product.

Answer 1

Question 2

Solve the equation in the interval
.

Answer 2

General solution:
.In the interval
:

Question 3

Find the exact value.

Answer 3

Question 4

Find $\sin 2 x$ , given that $\sin x = \frac { 3 } { 5 }$ and $x$ is in quadrant II.

Question 5

Find the exact value.

Question 6

Find the exact value of
given that
, and x and y are in quadrant I.

Question 7

Solve the equation in the interval
.

Question 8

Find the exact value.

Question 9

Verify the identity.

Question 10

Prove
.

Question 11

Solve the equation in the interval
.

Question 12

Use a double-angle formula or half-angle formula to simplify
.

Question 13

Make the indicated trigonometric substitution in the given algebraic expression and simplify.Assume that

Question 14

Find the exact value.

Question 15

Find the exact value.

Question 16

Simplify the trigonometric expression.

Question 17

Verify the identity.

Question 18

Write
in terms of
and
then simplify.

Question 19

Write the product as a sum.

Question 20

Solve the equation in the interval
.

Question 21

Find the exact value.

Question 22

Verify the identity.

Question 23

Find the exact value.

Question 24

Solve the equation in the interval
.

Question 25

Find the exact value. $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$

Question 26

Find the exact value.

A)
B)
C)
D)
E) none of these

Question 27

Find the exact value.

Question 28

Let
,
. Simplify the expression.

Question 29

Solve the equation in the interval
.

Question 30

Simplify.

Question 31

Find the exact value.

Question 32

Write
in terms of
and
then simplify.

Question 33

Solve the given equation.

Question 34

Solve the given equation.

Question 35

Find the exact value of the expression.

Question 36

Verify the identity.

Question 37

Prove the identity.

Question 38

Find the exact value.

Question 39

Solve the given equation.

Question 40

Solve the equation in the interval
.

Question 41

$( \sin x + \cos x ) ^ { 2 } = 1 + \sin 2 x$

Question 42

Solve the given equation.

Question 43

Write the product as a sum.

Question 44

Solve the equation in the interval $[ 0,2 \pi )$ . $\sin x = \cos 2 x$

A) $\frac { \pi } { 6 } , \frac { \pi } { 2 }$

B) $\frac { \pi } { 4 } , \frac { \pi } { 2 }$

C) $\frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$

D) $\frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$

E) none

Question 45

Write in terms of
And
Then simplify

A)
B)
C)
D)
E)

Question 46

Find all solutions of $2 \sin x - \sqrt { 2 } = 0$

A) $x = \frac { \pi } { 4 }$ and $x = \frac { 3 \pi } { 4 }$

B) $x = \frac { \pi } { 6 } + 2 k \pi$ and $x = \frac { \pi } { 2 } + 2 k \pi , k \in \text { integer }$

C) $x = \frac { \pi } { 3 } + 2 k \pi$ and $x = \frac { \pi } { 2 } + 2 k \pi , k \in \text { integer }$

D) $x = \frac { \pi } { 2 } \text { and } x = \frac { 3 \pi } { 2 }$

E) $x = \frac { \pi } { 4 } + 2 k \pi$ and $x = \frac { 3 \pi } { 4 } + 2 k \pi , k \in \text { integer }$

Question 47

Solve the equation in the interval $[ 0,2 \pi )$ . $2 \cos ^ { 2 } x + \sin x = 1$

A) $\frac { \pi } { 6 } , \frac { \pi } { 2 }$

B) $\frac { \pi } { 6 } , \frac { \pi } { 3 }$

C) $\frac { \pi } { 2 } , \frac { 7 \pi } { 6 }$

D) $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$

E) none

Question 48

Find the exact value of the expression.

Question 49

$\cos \left( x + \frac { \pi } { 2 } \right) + \sin \left( x + \frac { \pi } { 2 } \right) = \cos x - \sin x$

Question 50

Find the exact value of
given that
, and x and y are in quadrant I.

Question 51

Let $x = 2 \tan \theta$ , $0 \leq \theta < \frac { \pi } { 2 }$ ) Simplify the expression. $\frac { 1 } { x ^ { 2 } \sqrt { 4 + x ^ { 2 } } }$

A) $8 \tan ^ { 3 } \theta$

B) $\frac { \cos ^ { 3 } \theta } { 8 \sin ^ { 2 } \theta }$

C) $\sec ^ { 3 } \theta$

D) $\frac { \cos ^ { 2 } \theta } { 4 \sin \theta }$

E) $\frac { 8 \cos \theta } { \sin ^ { 2 } \theta }$

Question 52

$\cos ( x + y ) = \cos x + \cos y$

Question 53

Find $\sin 2 x$ , given that $\sin x = 3 / 5$ and $x$ is in quadrant II.

Question 54

Find all solutions of
.

Question 55

Solve the given equation.

Question 56

Write the sum as a product.

Question 57

Solve the equation in the interval $[ 0,2 \pi )$ . $\tan x \cos x - \cos x = 0$

A) $x = \frac { \pi } { 4 }$ , $x = \frac { 3 \pi } { 4 }$ , $x = \frac { \pi } { 2 }$ , $x = \frac { 3 \pi } { 2 }$

B) $x = 0$ , $x = \frac { \pi } { 2 }$ , $x = \frac { \pi } { 4 }$ , $x = \frac { 5 \pi } { 4 }$ , $x = \frac { 3 \pi } { 2 }$

C) $x = 0$ , $x = \frac { \pi } { 2 }$ , $x = \frac { 3 \pi } { 4 }$

D) $x = \frac { \pi } { 2 }$ , $x = \frac { 3 \pi } { 4 }$ , $x = \frac { 5 \pi } { 4 }$

E) $x = 0$ , $x = \frac { \pi } { 2 }$

Question 58

Find
, given that
and
is in quadrant II.

Question 59

Simplify.

A)
B)
C)
D)
E)

Question 60

Find the exact value.

Question 61

Find the exact value. $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$

A) $\frac { 3 } { 10 }$

B) $\frac { 2 \sqrt { 3 } } { 5 }$

C) $\frac { 3 - 4 \sqrt { 3 } } { 10 }$

D) $\frac { 3 - \sqrt { 2 } } { 10 }$

E) $\frac { 3 - 4 \sqrt { 3 } } { 5 }$

Question 62

Write $\sec x - \cos x$$\tan x$ in terms of $\sin x$ and $\cos x$ then simplify

A) $1$
B) $\sin ^ { 2 } x$
C) $\sin x$
D) $\cos x$
E) $- \cos ^ { 2 } x$

Question 63

Find the exact value of given that
, and x and y are in quadrant I

A)
B)
C)
D)
E)

Question 64

Find $\tan 2 x$ , given that $\sin x = \frac { 3 } { 5 }$ and $x$ is in quadrant II

A) $- 24$

B) $- \frac { 24 } { 7 }$

C) $\frac { 4 \sqrt { 6 } } { 7 }$

D) $\frac { \sqrt { 6 } } { 7 }$

E) $- \frac { 4 } { 7 }$

Question 65

Find the exact value.

A)
B)
C)
D)
E) none

Question 66

Find the exact value.

A)
B)
C)
D)
E) none

Question 67

Find the exact value.

A)
B)
C)
D)
E)

Question 68

$\cos \left( x + \frac { \pi } { 2 } \right) + \sin \left( x + \frac { \pi } { 2 } \right) = \cos x - 2 \sin x$

Question 69

Solve the given equation.

A)
B)
C)
D)
E) no solution

Question 70

$\sin ( x - y ) = \sin x - \sin y$

Question 71

Use a graphing device to find the solution of $\sin 4 x = x$ , correct to two decimal places

A) $0$ , $\pm 0.62$

B) $\pm 0.25$

C) $- 1$ , $\pm 0.62$

D) $\pm 1$

E) $0 , \pm 0.12$

Question 72

Write the sum as a product.

A)
B)
C)
D)
E)

Question 73

Determine which of the following is a solution of the equation

A)
B)
C)
D)
E) none of these

Question 74

Find the exact value.

A)
B)
C)
D)
E)

Question 75

Find the exact value.

A)
B)
C)
D)
E) none

Question 76

Use a double-angle formula or half-angle formula to simplify

A)
B)
C)
D)
E)

Question 77

Write the product as a sum.

A)
B)
C)
D)
E) none of these

Question 78

Use a double-angle formula or half-angle formula to simplify $\frac { \sin 48 ^ { \circ } } { 1 + \cos 48 ^ { \circ } }$

A) $\tan 48 ^ { \circ }$
B) $\tan 24 ^ { \circ }$
C) $\csc 24 ^ { \circ }$
D) $\cot 48 ^ { \circ }$
E) $1 + \sin 24 ^ { \circ }$

Question 79

$\cos x + \sin x \tan x = \csc x$

Question 80

Find the exact value.

A)
B)
C)
D)
E)