Deck 6: Trigonometric Identities, Inverses, and Equations

Question 1

Starting with the ratio identity given, use substitution and fundamental identities to write a new identity belonging to the ratio family.

A)
B)
C)
D)

Answer 1

Question 2

Write csc x entirely in terms of tan x.

A)
B)
C)
D)

Answer 2

Question 3

Use algebra and fundamental identities to rewrite the given expression to create a new identity relationship.
8 sec x tan x

Answer 3

8 sec x tan x = (Answers may vary.)

Question 4

Verify that the equation is an identity.

Question 5

Verify that the equation is an identity.

Question 6

Find the exact value of the expression using a sum or difference identity. Some simplifications may involve using symmetry and the formula for negatives. cos 165°

A)
B)
C)
D)

Question 7

Write cot x entirely in terms of cos x.

Question 8

Verify the equation is an identity using multiplication and fundamental identities.
cos x (sec x - cos x) =

Question 9

Show that the equation is not an identity.

Question 10

For the function f( $\theta$ ) and the quadrant in which $\theta$ terminates, state the value of the other five trig functions. with $\theta$ in QIV.

Question 11

Verify the equation is an identity using special products and fundamental identities.
(1 + cos x)[1 - cos(-x)] = sin² x

Question 12

For the function and the quadrant QI in which $\theta$ terminates, state the value of sin $\theta$ .

A)
B)
C)
D)

Question 13

Verify that the equation is an identity.

Question 14

Use algebra and fundamental identities to rewrite the given expression to create a new identity relationship. csc x + cot x

A) csc x + cot x =
B) csc x + cot x =
C) csc x + cot x = tan x
D) csc x + cot x = 1 + sin² x

Question 15

Rewrite as a single expression. cos(6 $\theta$ )cos(2 $\theta$ ) + sin(6 $\theta$ )sin(2 $\theta$ )

A) cos(8 $\theta$ )
B) cos(4 $\theta$ )
C) sin(8 $\theta$ )
D) sin(4 $\theta$ )

Question 16

Show that the equation is not an identity.

Question 17

Verify the equation is an identity using factoring and fundamental identities.

Question 18

Verify the equation is an identity by using fundamental identities and to combine terms.

Question 19

Verify that the equation is an identity.

Question 20

Verify the equation is an identity using special products and fundamental identities.

Question 21

Find the exact value.

A)
B)
C)
D)

Question 22

Rewrite as a single expression.

Question 23

Use the following to answer questions :
$\alpha$ and $\beta$ are acute angles with tan( $\alpha$ ) = and sec( $\beta$ ) = .

-Find cos( $\alpha$ - $\beta$ ).

A)
B)
C)
D)

Question 24

Use the following to answer questions :
$\alpha$ and $\beta$ are acute angles with tan( $\alpha$ ) = and sec( $\beta$ ) = .

-Find tan( $\alpha$ - $\beta$ ).

A)
B)
C)
D)

Question 25

Use a cofunction identity to write an equivalent expression.
cos 13°

Question 26

Verify the identity.

Question 27

Use an angle reduction formula to find the exact value. sin 2940°

A)
B)
C)
D)

Question 28

Rewrite as a single expression. sin(7 $\theta$ )cos(3 $\theta$ ) + cos(7 $\theta$ )sin(3 $\theta$ )

A) cos(10 $\theta$ )
B) cos(4 $\theta$ )
C) sin(10 $\theta$ )
D) sin(4 $\theta$ )

Question 29

Use the following to answer questions :
$\alpha$ and $\beta$ are acute angles with tan( $\alpha$ ) = and sec( $\beta$ ) = .

-Find sin( $\alpha$ - $\beta$ ).

A)
B)
C)
D)

Question 30

Find the exact value. sin 190° cos 55° - cos 190° sin 55°

A)
B)
C)
D)

Question 31

Rewrite as a single expression.

A)
B)
C)
D)

Question 32

Use the following to answer questions :
$\alpha$ and $\beta$ are acute angles with tan( $\alpha$ ) = and sec( $\beta$ ) = .

-Find tan( $\alpha$ + $\beta$ ).

A)
B)
C)
D)

Question 33

Use the following to answer questions :
$\alpha$ and $\beta$ are acute angles with tan( $\alpha$ ) = and sec( $\beta$ ) = .

-Find cos( $\alpha$ + $\beta$ ).

A)
B)
C)
D)

Question 34

Use a cofunction identity to write an equivalent expression.

A)
B)
C)
D)

Question 35

Find the exact value using a sum or difference identity. Some simplifications may involve using symmetry and the formulas for negatives.
sin(-15°)

Question 36

Find the exact value using a sum or difference identity. Some simplifications may involve using symmetry and the formulas for negatives.

A)
B)
C)
D)

Question 37

Rewrite as a single expression. cos(10 $\theta$ )cos(6 $\theta$ ) - sin(10 $\theta$ )sin(6 $\theta$ )

A) cos(16 $\theta$ )
B) cos(4 $\theta$ )
C) sin(16 $\theta$ )
D) sin(4 $\theta$ )

Question 38

Find the exact value.

A)
B)
C)
D)

Question 39

Use the following to answer questions :
$\alpha$ and $\beta$ are acute angles with tan( $\alpha$ ) = and sec( $\beta$ ) = .

-Find sin( $\alpha$ + $\beta$ ).

A)
B)
C)
D)

Question 40

Rewrite as a single expression. sin(6 $\theta$ )cos(3 $\theta$ ) - cos(6 $\theta$ )sin(3 $\theta$ )

A) cos(9 $\theta$ )
B) cos(3 $\theta$ )
C) sin(9 $\theta$ )
D) sin(3 $\theta$ )

Question 41

Find exact values for ;

Question 42

Use a half-angle identity to rewrite the expression as a single, non-radical function.

Question 43

Find the exact value using a product-to-sum identity.

Question 44

Use the following to answer questions :
tan $\theta$ = ; $\theta$ in QII.

-Find the exact value of sin(2 $\theta$ ).

A)
B)
C)
D)

Question 45

Use a double-angle identity to find the exact value. cos 105° sin 105°

A)
B)
C)
D)

Question 46

Find the exact values for sin $\theta$ , cos $\theta$ , and tan $\theta$ .

Question 47

Rewrite in terms of an expression containing only cosines to the power 1.
sin² x cos⁴ x

Question 48

Rewrite the product as a sum using a product-to-sum identity. sin(-10 $\alpha$ )sin(4 $\alpha$ )

A)
B)
C)
D)

Question 49

Rewrite the sum as a product using a sum-to-product identity.

Question 50

Find the exact value using a sum-to-product identity.

A)
B) -1
C)
D)

Question 51

Use a double-angle identity to find the exact value.

A) 0
B) 1
C) -1
D)

Question 52

Rewrite the product as a sum using a product-to-sum identity. cos(10 $\alpha$ )sin(-7 $\alpha$ )

A)
B)
C)
D)

Question 53

Use the following to answer questions :
tan $\theta$ = ; $\theta$ in QII.

-Find the exact value of tan(2 $\theta$ ).

A)
B)
C)
D)

Question 54

Rewrite the sum as a product using a sum-to-product identity.

A) 2cos(3 $\alpha$ )sin(2 $\alpha$ )
B) 2sin(3 $\alpha$ )cos(2 $\alpha$ )
C) 2cos(3 $\alpha$ )cos(2 $\alpha$ )
D) 2sin(3 $\alpha$ )sin(2 $\alpha$ )

Question 55

Use the following to answer questions :

-Use a half-angle identity to find the exact value of cos $\theta$ .

A)
B)
C)
D)

Question 56

Rewrite the product as a sum using a product-to-sum identity.
2sin(2073\9\pi\)t)cos(813 $\pi$ t)

Question 57

Use the following to answer questions :
tan $\theta$ = ; $\theta$ in QII.

-Find the exact value of cos(2 $\theta$ ).

A)
B)
C)
D)

Question 58

Use the following to answer questions :

-Use a half-angle identity to find the exact value of tan $\theta$ .

A)
B)
C)
D)

Question 59

Use a double-angle identity to find the exact value.

Question 60

Use the following to answer questions :

-Use a half-angle identity to find the exact value of sin $\theta$ .

A)
B)
C)
D)

Question 61

Evaluate without the aid of calculators or tables. Answer in radians.
cos^-1 1

Question 62

Simplify without using a calculator.

Question 63

Evaluate without using a calculator. Answer in radians.

A)
B)
C)
D)

Question 64

Verify the identity.

Question 65

Evaluate without the aid of calculators or tables, keeping the domain and range of the function in mind. Answer in radians.

A) 0
B)
C)
D)

Question 66

Use the following to answer questions :
The graph of y = csc(x) is shown below. Draw the horizontal line y = -1.5.
State the quadrant of the principal root.

A) QI
B) QII
C) QIII
D) QIV

Question 67

Evaluate. Answer in exact form.

Question 68

Evaluate without using a calculator. Answer in radians.

Question 69

Evaluate. Answer in exact form.

Question 70

Evaluate without the aid of calculators or tables. Answer in radians.

A)
B)
C)
D)

Question 71

Evaluate using a calculator. Answer to the nearest tenth of a degree.
sec^-1 5.593

Question 72

Evaluate using a calculator. Answer in radians to the nearest ten-thousandth, in degrees to the nearest tenth.

Question 73

Evaluate using a calculator, keeping the domain and range of the function in mind. Answer in radians to the nearest ten-thousandth and in degrees to the nearest tenth.

Question 74

Evaluate without the aid of calculators or tables. Answer in radians.

A)
B)
C)
D)

Question 75

The range of a projectile is modeled by the function , where v is the initial velocity and $\theta$ is the angle at which the object is initially propelled. The maximum range is achieved when $\theta$ = 45°. Use exact values to compute how many feet short of maximum the projectile falls if $\theta$ = 67.5° and v = 104 ft/sec.

A) feet
B) feet
C) feet
D) feet

Question 76

Use the following to answer questions :
The graph of y = csc(x) is shown below. Draw the horizontal line y = -1.5.

-State the number of roots in [0, 2 $\pi$ ].

A) 4
B) 3
C) 2
D) 1

Question 77

Evaluate using a calculator, keeping in mind the domain and range of the function. Answer in radians to the nearest ten-thousandth and in degrees to the nearest tenth.

Question 78

Evaluate without the aid of calculators or tables, keeping the domain and range of the function in mind. Answer in radians.

Question 79

Evaluate. Answer in exact form.

A)
B)
C)
D)

Question 80

Evaluate. Answer in exact form.