Free Printable Unit 9 simplifying trig identities

1.

What is csc(x) equivalent to?

a)

$\frac{1}{\sin x}$

b)

$\frac{1}{\tan x}$

c)

sin(x)

d)

$\frac{1}{\cos x}$

2.

Please select the correct solution

a)

csc² x

b)

cot² x

c)

sin² x

d)

sec² x

3.

Please select the correct solution
cos²x + sin ²x=

a)

1

b)

csc²x + sec²x

c)

sin²x

d)

1 - sin²x

4.

Please select the correct solution

a)

csc x

b)

sec x

c)

cot x

d)

tan x

5.

Simplify

a)

sin²θ

b)

cosθ

c)

tanθ

d)

1- sin²θ

6.

Simplify

a)

-1

b)

sin θ

c)

csc θ

d)

1

7.

Simplify

a)

secθ

b)

cos²θ

c)

sin²θ

d)

$\frac{\sin^2x}{\cos^2x}$

8.

Simplify: (secx-1)(secx+1)

Hint: you will need to FOIL first

a)

tan²x

b)

sinx

c)

cot²x

d)

cosx

9.

Simplify: tanxcotx-cos²x

a)

tanx

b)

cotx

c)

sin²x

d)

cos²x

10.

Is this a pythagorean identity?
tan²x + 1 = secx

a)

Yes

b)

No

11.

Simplify this to a basic trigonometry function:
tan(x)csc(x)

a)

tanx

b)

secx

c)

cotx

d)

cscx

12.

a)

csc²θ

b)

sec²θ

c)

cscθ

d)

1

13.

a)

$\frac{1}{\cos\ x}$

b)

1

c)

cot x

d)

-1

14.

Verify the following:

a)

cos²x

b)

sin²x

c)

cot²x

d)

tan²x

15.

Simplify.

a)

sinθ

b)

cos²θ

c)

sin²θ

d)

1- sin²θ

16.

Simplify.

a)

sin x

b)

cos x

c)

tan x

d)

1

17.

5) tanxcotx-cos²x

a)

tanx

b)

cotx

c)

sin²x

d)

cos²x

18.

Which of the following is not a pythagorean identity?

a)

cot²x + 1 = csc²x

b)

csc²x + 1 = cot²x

c)

cot²x - csc²x = -1

d)

csc²x - cot²x = 1

19.

sin x =

a)

cos x

b)

1/cosx

c)

1/secx

d)

1/cscx

20.

cos x =

a)

sin x

b)

sin²x-1

c)

1/sec x

d)

1/csc x

21.

tan x =

a)

sinx/cos x

b)

cosx/sinx

c)

1/cotx

d)

sec²x

22.

sec x =

a)

1/cosx

b)

1/sinx

c)

1/cscx

d)

1/tanx

23.

csc x =

a)

1/cosx

b)

1/sinx

c)

cot²x-1

d)

1/secx

24.

cotx =

a)

sinx/cosx

b)

cosx/sinx

c)

1/tanx

d)

1/cotx

25.

sin²x + cos²x =

a)

1

b)

1/sinx

c)

sec²x

d)

csc²x

26.

1 - sin²x =

a)

cos²x

b)

cos²x+1

c)

csc²x

d)

tan²x

27.

tan²x + 1 =

a)

csc²x

b)

sec²x

c)

sec²x - 1

d)

cscx

28.

1 + cot²x =

a)

sec²x

b)

sec²x - 1

c)

tan²x

d)

csc²x

29.

1 - cos²x =

a)

sec²x

b)

csc²x

c)

sin²x

d)

sec²x - 1

30.

sec²x =

a)

sinx + cosx

b)

1 + csc²x

c)

1 + tan²x

d)

1 - sin²x

31.

csc²x - 1 =

a)

cot²x

b)

tan²x

c)

cot²x + 1

d)

sin²x + cos²x

32.

sec²x - 1 =

a)

cot²x

b)

csc²x

c)

1 + tan²x

d)

tan²x

33.

1/cosx =

a)

secx

b)

cscx

c)

tanx

d)

sinx

34.

1/tanx =

a)

tanx

b)

sinx/cosx

c)

cotx

d)

sin²x

35.

cos²x =

a)

1 - sec²x

b)

1 - sin²x

c)

sin²x

d)

1

36.

1/sinx

a)

tanx

b)

secx

c)

cscx

d)

cosx

37.

Solve the following for tan²θ:
1 + tan²θ = sec²θ

a)

tan²θ = sec²θ + 1

b)

tan²θ = sec²θ - 1

c)

tanθ = secθ - 1

d)

tan²θ = opp/adj

38.

Solve the following for cos²θ:
sin²θ + cos²θ = 1

a)

cos²θ = 1 - sin²θ

b)

cosθ = 1 + sin²θ

c)

cosθ = 1 - sinθ

d)

cos²θ = Adj/hyp

39.

Rewrite tan(x) in terms of sin(x) and cos(x).

a)

tan(x) = cos(x)/sin(x)

b)

tan(x) = 1/cot(x)

c)

tan(x) = sin(x)/cos(x)

d)

tan(x) = opp/adj

40.

What is csc(x) equivalent to?

a)

1/sin(x)

b)

1/tan(x)

c)

sin(x)

d)

1/cos(x)

41.

Please select the correct solution

a)

csc² x

b)

cot² x

c)

sin² x

d)

sec² x

42.

Please select the correct solution cos²x + sin ²x=

a)

1

b)

csc²x + sec²x

c)

sin²x

d)

1 - sin²x

43.

Simplify

$\tan x\csc x\cos x$

a)

1/cos x

b)

1

c)

cot x

d)

-1

44.

Simplify

$\sin\theta\left(\csc\theta-\sin\theta\right)$

a)

secθ

b)

cos²θ

c)

sin²θ

d)

sin²θ/cos²θ

45.

Simplify $\tan x\left(\cot x\ +\ \csc x\right)$

a)

1 + sec x

b)

1 + csc x

c)

sec x - 1

d)

tan² x

46.

What happens when you multiply two reciprocal functions?

a)

You get a pythagorean identity

b)

It equals 1

c)

You get a quotient identity

d)

It equals 0

47.

Simplify $\left(\csc x+1\right)\left(\csc x-1\right)$

a)

$\csc^2x+1$

b)

$\tan^2x$

c)

$\cot^2x$

d)

$2\csc x$

48.

$1-\sin^2x\ =$

a)

$\sin^2x$

b)

$\cos^2x$

c)

$\sec^2x$

d)

$\csc^2x$

49.

a)

sec(u)

b)

cos(u)

c)

csc(u)

d)

sin(u)

e)

cot(u)

50.

a)

csc(u)

b)

cos(u)

c)

tan(u)

d)

sin(u)

e)

cot(u)

51.

cos(90 - u)

a)

sec(u)

b)

csc(u)

c)

tan(u)

d)

sin(u)

e)

cot(u)

52.

sin(90 - u)

a)

sec(u)

b)

cos(u)

c)

tan(u)

d)

csc(u)

e)

cot(u)

53.

a)

sec(u)

b)

cos(u)

c)

tan(u)

d)

sin(u)

e)

cot(u)

54.

a)

sec(u)

b)

cos(u)

c)

tan(u)

d)

sin(u)

e)

csc(u)

55.

cos(-u)

a)

cos(u)

b)

-cos(u)

c)

sin(u)

d)

-sec(u)

e)

sec(u)

56.

sec(-u)

a)

cos(u)

b)

-cos(u)

c)

sin(u)

d)

-sec(u)

e)

sec(u)

57.

sin(-u)

a)

cos(u)

b)

-sin(u)

c)

sin(u)

d)

-csc(u)

e)

csc(u)

58.

csc(-u)

a)

cos(u)

b)

-sin(u)

c)

sin(u)

d)

-csc(u)

e)

csc(u)

59.

tan(-u)

a)

tan(u)

b)

-cot(u)

c)

sin(u)

d)

-tan(u)

e)

cot(u)

60.

cot(-u)

a)

tan(u)

b)

-cot(u)

c)

sin(u)

d)

-tan(u)

e)

cot(u)

61.

Select all that are Pythagorean Identities.

a)

sin²(u) + cos²(u) = 1

b)

sin²(u) - cos²(u) = 1

c)

tan²(u) + 1 = sec²(u)

d)

cot²(u) + csc²(u) = 1

e)

cot²(u) + 1 = csc²(u)

62.

Simplify: $\secθ\sinθ$

a)

cos θ

b)

csc θ

c)

tan θ

d)

cot θ

63.

Simplify: $\cos\theta\tan\theta$

a)

sin θ

b)

cos θ

c)

tan θ

d)

csc θ

64.

Simplify: $(\secθ-1)(\secθ+1)$

a)

sin ² θ

b)

cos ² θ

c)

tan ² θ

d)

sec ² θ

65.

Simplify: $\left(\cos\ θ\right)\left(\sec\ θ-\cosθ\right)$

a)

$\sin^2θ$

b)

$\cos^2θ$

c)

$\csc^2\theta$

d)

$\sec^2\theta$

66.

Simplify: $\tan x\ \cot x$

a)

$\sin\ x$

b)

$\cos\ x$

c)

$-1$

d)

$1$

67.

Simplify: $\frac{\sec^2x}{\sec^2x-1}$

a)

$\sin^2x$

b)

$\csc^2x$

c)

$\cos^2x$

d)

$\sec^2x$

68.

Simplify: $\sec x-\tan x\sin x$

a)

$\sin x$

b)

$\cos x$

c)

$\csc x$

d)

$\sec x$

69.

Simplify: $\frac{1-\sin^2x}{\sec x}$

a)

$\sin^2x$

b)

$\cos^2x$

c)

$\sin^3x$

d)

$\cos^3x$

70.

Simplify: $\frac{\cos^2\left(-x\right)\csc x}{\cot x}$

a)

$-\sin x$

b)

$-\cos x$

c)

$\sin x$

d)

$\cos x$

71.

Simplify: $\frac{\sin^2x+\cos^2x+\cot^2x}{1+\tan^2x}$

a)

$\cot^2x$

b)

$\tan^2x$

c)

$\csc^2x$

d)

$\sec^2x$

72.

Which of the following is NOT a pythagorean identity?

a)

sin²x + cos²x = 1

b)

1 - cos²x = sin²x

c)

1 - sin²x = cos² x

d)

sin²x - cos²x = -1

73.

Which of the following is NOT a pythagorean identity?

a)

1 + tan²x = sec²x

b)

sec²x - 1= tan²x

c)

tan²x - sec²x = -1

d)

tan²x - 1 = -sec²x

74.

5) tanxcotx-cos²x

a)

tanx

b)

cotx

c)

sin²x

d)

cos²x

75.

The Pythagorean Theorem ONLY works on which triangle?

a)

obtuse

b)

scalene

c)

isosceles

d)

right

76.

a)

1/cos x

b)

1

c)

cot x

d)

-1

77.

Simplify

a)

-1

b)

sin θ

c)

csc θ

d)

1

78.

Simplify

a)

secθ

b)

cos²θ

c)

sin²θ

d)

sin²θ/cos²θ

79.

Please select the correct solution

a)

csc² x

b)

cot² x

c)

sin² x

d)

sec² x

80.

Simplify

a)

sin²θ

b)

cosθ

c)

tanθ

d)

1- sin²θ

81.

Simplify: tanxcotx-cos²x

a)

tanx

b)

cotx

c)

sin²x

d)

cos²x

82.

Simplify this to a basic trigonometry function:
tan(x)csc(x)

a)

tanx

b)

secx

c)

cotx

d)

cscx

83.

If sinθ = 3 / 5 and θ is in Q II, find tan θ.

a)

3 / 4

b)

-3 / 4

c)

4 / 3

d)

- 4 / 3

84.

Simplify (Hint: you will need to FOIL first)

a)

sinθ

b)

cot²θ

c)

tan²θ

d)

cos²θ

85.

Given $\tan x=\frac{9}{12}$ and the angle is in quadrant I. What is the value of secx?

a)

$\sec x=\frac{5}{3}$

b)

$\sec x=\frac{4}{5}$

c)

$\sec x=\frac{5}{4}$

d)

$\sec=\frac{4}{3}$

86.

Simplify the expression $\cot x\sin^2x\csc x$

a)

sinx

b)

cosx

c)

tanx

d)

cscx

87.

Simplify the expression $\csc^2x\left(1-\sec^2x\right)$

a)

$-\csc^2x$

b)

$\cos x$

c)

$-\sec^2x$

d)

$\tan x$

88.

$\cos\left(\theta\right)=$

a)

$-\cos\left(\theta\right)$

b)

$\cos\left(-\theta\right)$

c)

$-\cos\left(-\theta\right)$

89.

$\sin\left(\theta\right)=$

a)

$-\sin\left(\theta\right)$

b)

$\sin\left(-\theta\right)$

c)

$-\sin\left(-\theta\right)$

90.

$\tan\left(\theta\right)=$

a)

$-\tan\left(\theta\right)$

b)

$\tan\left(-\theta\right)$

c)

$-\tan\left(-\theta\right)$

91.

$\cos\left(-\theta\right)=$

a)

$-\cos\left(\theta\right)$

b)

$\cos\left(\theta\right)$

c)

$-\cos\left(-\theta\right)$

92.

$\sin\left(-\theta\right)=$

a)

$-\sin\left(\theta\right)$

b)

$\sin\left(\theta\right)$

c)

$-\sin\left(-\theta\right)$

93.

$\tan\left(-\theta\right)=$

a)

$-\tan\left(\theta\right)$

b)

$\tan\left(\theta\right)$

c)

$-\tan\left(-\theta\right)$

94.

$\cos\left(\frac{\pi}{2}+\theta\right)=$

a)

$\sin\left(\theta\right)$

b)

$-\sin\left(\theta\right)$

95.

$\sin\left(\frac{\pi}{2}+\theta\right)=$

a)

$\cos\left(\theta\right)$

b)

$-\cos\left(\theta\right)$

96.

$\cos\left(\pi+\theta\right)=$

a)

$\cos\left(\theta\right)$

b)

$-\cos\left(\theta\right)$

97.

$\cos\left(\pi-\theta\right)=$

a)

$\cos\left(\theta\right)$

b)

$-\cos\left(\theta\right)$

98.

$\sin\left(\pi-\theta\right)=$

a)

$\sin\left(\theta\right)$

b)

$-\sin\left(\theta\right)$

99.

$\sin\left(\pi+\theta\right)=$

a)

$\sin\left(\theta\right)$

b)

$-\sin\left(\theta\right)$

100.

$\tan\left(\pi+\theta\right)=$

a)

$\tan\left(\theta\right)$

b)

$-\tan\left(\theta\right)$

101.

$\tan\left(\pi-\theta\right)=$

a)

$\tan\left(\theta\right)$

b)

$-\tan\left(\theta\right)$

102.

cos²θ(1+cot²θ)

a)

sinθ

b)

sin²θ

c)

cotθ

d)

cot²θ

103.

Verify the following.
tanB (cotB + tanB) = sec²B

a)

1+ tan²B

b)

sec²B

c)

cot²B

d)

tan²B

104.

a)

sin²θ

b)

cosθ

c)

tanθ

d)

1- sin²θ

105.

a)

-1

b)

sin θ

c)

csc θ

d)

1

106.

a)

csc²θ

b)

sec²θ

c)

cscθ

d)

1

107.

a)

sinθ

b)

cot²θ

c)

tan²θ

d)

cos²θ

108.

a)

secθ

b)

cos²θ

c)

sin²θ

d)

sin²θ/cos²θ

109.

a)

csc s

b)

tan s + 1

c)

cot s

d)

cot s + 1

110.

a)

csc²θ

b)

csc θ sec θ

c)

cot θ tan θ

d)

1/sinθ

111.

a)

1 + sec s

b)

1 + csc s

c)

sec s - 1

d)

tan² s

112.

a)

1/cos x

b)

1

c)

cot x

d)

-1

113.

a)

sinθ

b)

cos²θ

c)

sin²θ

d)

1- sin²θ

114.

a)

cos²x

b)

sin²x

c)

cot²x

d)

tan²x

115.

a)

csc x

b)

sec x

c)

1/sec x

d)

cos x

116.

a)

sin x

b)

cos x

c)

tan x

d)

1

117.

What is csc(x) equivalent to?

a)

1/sin(x)

b)

1/tan(x)

c)

sin(x)

d)

1/cos(x)

118.

Simplify:
(secx-1)(secx+1).
Hint: you will need to FOIL first

a)

tan²x

b)

sinx

c)

cot²x

d)

cosx

119.

Simplify: tanxcotx-cos²x

a)

tanx

b)

cotx

c)

sin²x

d)

cos²x

120.

What is csc(x) equivalent to?

a)

1/sin(x)

b)

1/tan(x)

c)

sin(x)

d)

1/cos(x)

121.

cot²θ=

a)

sec²θ - 1

b)

1 - cos²θ

c)

csc²θ - 1

d)

1 - sin²θ

122.

Sin means

a)

opposite / hypotenuse

b)

adjacent / hypotenuse

c)

opposite / adjacent

d)

adjacent / opposite

123.

True or False, cos(x)/tan(x) = cot(x)cos(x)

a)

True

b)

False

124.

True or False, tan(x)cot(x) = 1?

a)

True

b)

False

125.

True or False, (cot(x)) / (sec(x)) = sin(x)?

a)

True

b)

False

126.

True or False, cos(x)/cot(x) = sin(x)?

a)

True

b)

False

127.

What is csc(x) equivalent to?

a)

1/sin(x)

b)

1/tan(x)

c)

sin(x)

d)

1/cos(x)

128.

Please select the correct solution

a)

csc² x

b)

cot² x

c)

sin² x

d)

sec² x

129.

Please select the correct solution
cos²x + sin ²x=

a)

1

b)

csc²x + sec²x

c)

sin²x

d)

1 - sin²x

130.

Please select the correct solution

a)

csc x

b)

sec x

c)

cot x

d)

tan x

131.

Is this a pythagorean identity?
tan²x + 1 = secx

a)

Yes

b)

No

132.

What pair of trigonometric functions are both reciprocals and cofunctions?

a)

tan θ and cot θ

b)

sec θ and csc θ

c)

sin θ and cos θ

133.

What is the value of sin210^ocos30^o - cos210^osin30^ois

a)

0

b)

-1

c)

1

d)

sin240^o

134.

a)

sinθ

b)

cos²θ

c)

sin²θ

d)

1- sin²θ

135.

From the given list, select all equivalent forms of the Pythagorean Identity

\cos^2\theta+\sin^2\theta=1

a)

$1+\tan^2\theta=\sec^2\theta$

b)

$1+\cot^2\theta=\sec^2\theta$

c)

$1+\tan^2\theta=\csc^2\theta$

d)

$1+\cot^2\theta=\csc^2\theta$

136.

Select all correct even/odd identities from the given list.

a)

$\sin\left(-x\right)=\sin\left(x\right)$

b)

$\cos\left(-x\right)=\cos\left(x\right)$

c)

$\tan\left(-x\right)=-\tan\left(x\right)$

d)

$\sin\left(-x\right)=-\sin\left(x\right)$

e)

$\cos\left(-x\right)=-\cos\left(x\right)$

137.

From the given list, select all equivalent forms of the Pythagorean Identity

\cos^2\theta+\sin^2\theta=1

a)

$\sin^2\theta=\cos^2\theta-1$

b)

$\cos^2\theta=1-\sin^2\theta$

c)

$\cos^2\theta=1+\sin^2\theta$

d)

$\sin^2\theta=1-\cos^2\theta$

138.

What is the best first step for proving $\frac{1}{1+\cos x}=\frac{1-\cos x}{\sin^2x}$ ?

a)

Multiply the numerator and denominator by the conjugate $\left(1-\cos x\right)$

b)

Square the numerator and denominator

c)

Use a Pythagorean Identity for the denominator to change the $\left(1+\cos x\right)$ to $\left(\sin^2x\ +\cos^2x+\cos x\right)$

d)

Split up the denominator

139.

Which of the following is the best first step to verify the identity

\cos^4x-\sin^4x=2\cos^2x-1

a)

Split up the $\left(\sin^4x\right)$ as $\left(\sin^2x\right)\left(\sin^2x\right)$ to set up a Pythagorean Identity substitution

b)

Split up the $\left(\cos^4x\right)$ as $\left(\cos^2x\right)\left(\cos^2x\right)$ to set up a Pythagorean Identity substitution

c)

Factor the left side using the difference of squares

d)

Use a Pythagorean Identity on the right side to change $\left(2\cos^2x-1\right)$ to $\left(2\left(1-\sin^2x\right)-1\right)$

140.

Find the exact value of

\cos105\degree

a)

$\frac{\sqrt{2}+\sqrt{6}}{4}$

b)

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

c)

$\frac{\sqrt{6}-\sqrt{2}}{4}$

d)

$\frac{\sqrt{2}-\sqrt{6}}{4}$

141.

If $\cos A=\frac{3}{5}$ where $0\degree\le A\le90\degree$ and $\sin B=\frac{5}{13}$ where $90\degree\le B\le180\degree$ find $\cos\left(A+B\right)$ .

(a)

142.

Write the following as a single trigonometric expression, then evaluate the expression.

\sin37\degree\cos113\degree+\cos37\degree\sin113\degree

a)

$\sin150\degree=\frac{\sqrt{3}}{2}$

b)

$\sin150\degree=\frac{1}{2}$

c)

$\sin\left(-76\degree\right)\approx-0.9703$

d)

$\sin\left(-76\degree\right)\approx0.2419$

143.

Use Sum or Difference Identities to find the exact value of each expression.
cos(75°)

a)

1/4

b)

√6/4 - √2 / 4

c)

√6 + √2 / 4

d)

-√6/4 - √2 / 4

144.

Write the following expression as the sine, cosine, or tangent of an angle.
cos(175)cos(55)+sin(175)sin(55)

a)

sin(120)

b)

cos(120)

c)

cos(230)

d)

sin(230)

145.

Which of the following is NOT
a solution to
sin θ = √(3) / 2 ?

a)

π / 3

b)

2π / 3

c)

5π / 3

d)

7π / 3

146.

a)

A

b)

B

c)

C

d)

D

147.

Please choose the correct sum and difference formula:
cos (α-β) =

a)

sin α cos β + cos α sin β

b)

sin α cos β - cos α sin β

c)

cos α cos β - sin α sin β

d)

cos α cos β + sin α sin β

148.

Find the exact value of the expression.
cos(20)cos(40) - sin(20)sin(40)

a)

1/4

b)

1/2

c)

√(3)/2

d)

√(3)

149.

sin 105º

a)

√3/2

b)

-1/4(√2 + √6)

c)

1/4 (√2 + √6)

d)

2-√3

150.

a)

A

b)

B

c)

C

d)

D

151.

a)

A

b)

B

c)

C

d)

D

152.

a)

A

b)

B

c)

C

d)

D

153.

If $\sin A=\frac{4}{5},\tan B=\frac{5}{12},$ and A and B are first quadrant angles, what is the value of $\sin\left(A+B\right)$ ?

a)

$\frac{63}{65}$

b)

$-\frac{33}{65}$

c)

$\frac{33}{65}$

d)

$-\frac{63}{65}$

154.

Find the exact value of the expression.
$\sin\left(\frac{5\pi}{12}\right)\cos\left(\frac{\pi}{4}\right)-\cos\left(\frac{5\pi}{12}\right)\sin\left(\frac{\pi}{4}\right)$

a)

$\frac{1}{2}$

b)

$-\frac{1}{2}$

c)

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

d)

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

155.

Find the exact value of $\sin2x$ if $\sin x=\frac{12}{13}$ and x is in the first quadrant.

a)

$\frac{120}{169}$

b)

$\frac{25}{169}$

c)

$\frac{60}{169}$

d)

$\frac{5}{13}$

156.

Use a sum or difference formula to find an exact value of $\sin\left(\frac{7\pi}{12}\right)$

a)

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

b)

$\frac{-\sqrt{2}-\sqrt{6}}{4}$

c)

$\frac{\sqrt{2}+\sqrt{6}}{4}$

d)

$2-\sqrt{3}$

157.

If cos θ = 4/5 and $\frac{3\pi}{2}<\theta<2\pi$

Find sin 2θ

a)

$-\frac{1}{5}$

b)

$\frac{24}{25}$

c)

$-\frac{24}{25}$

d)

$-\frac{25}{24}$

158.

If $\sin A=\frac{3}{5},\sin B=\frac{2}{3},$ $and\ \angle A\ and\ \angle B$ are in the first quadrant, what is the value of $\cos\left(A-B\right)?$

a)

$-\frac{2}{3}$

b)

$\frac{4\sqrt{5}-6}{15}$

c)

$\frac{4\sqrt{5}+2}{5}$

d)

$\frac{4\sqrt{5}+6}{15}$

159.

If $\sin\theta=\frac{\sqrt{5}}{3}$ and $\frac{\pi}{2}<\theta<\pi$ , then $\cos2\theta$ equals

a)

$\frac{1}{3}$

b)

$-\frac{1}{3}$

c)

$\frac{1}{9}$

d)

$-\frac{1}{9}$

160.

Use sum or difference angles identity to find the exact value for $\sin\ \left(\frac{-\pi}{12}\right)$

a)

$\frac{\sqrt{6}+\sqrt{2}}{4}$

b)

$\frac{\sqrt{6}-\sqrt{2}}{4}$

c)

$\frac{\sqrt{2}-\sqrt{6}}{4}$

d)

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

161.

Find the exact value of $\sin2\theta$ if $\sin\theta=\frac{12}{13}$ and $\theta$ is in the first quadrant.

a)

120/169

b)

25/169

c)

60/169

d)

5/13

162.

If cos θ = 4/5 and $\frac{3\pi}{2}<\theta<2\pi$ , find sin 2θ

a)

-1/5

b)

24/25

c)

-24/25

d)

-25/24

163.

If A and B are first quadrant angles with sinA=15/17 and cosB=4/5, find cos(A - B).

a)

3/8

b)

8/17

c)

-13/85

d)

77/85

164.

Find the exact value of the expression.
cos(20)cos(40) - sin(20)sin(40)

a)

1/4

b)

1/2

c)

√(3)/2

d)

√(3)

165.

Find the exact value of the expression.
sin(5π/12)cos(π/4) - cos(5π/12)sin(π/4)

a)

1/2

b)

-1/2

c)

-√(2)/2

d)

√(3)/2

166.

Find the exact value.
cos (265 - 25) =

a)

53/12

b)

-√(3)/2

c)

-1/2

d)

√(3)/2

167.

Find the exact value.
cos (5π/18 - π/9)

a)

√(3)/2

b)

1/4

c)

1/2

d)

1

168.

Write the following expression as the sine, cosine, or tangent of an angle.
cos(175)cos(55)+sin(175)sin(55)

a)

sin(120)

b)

cos(120)

c)

cos(230)

d)

sin(230)

169.

Find the exact value of the expression.
cos(20)cos(40) - sin(20)sin(40)

a)

1/4

b)

1/2

c)

√(3)/2

d)

√(3)

170.

Find the exact value.
cos (265 - 25) =

a)

53/12

b)

-√(3)/2

c)

-1/2

d)

√(3)/2

171.

Given sin α = -3/5, cos α = -4/5 and cos β = (2√5)/5, sin β = (-√5)/5
Please find the sum or difference:
cos (α+β)=

a)

(-11√5)/25

b)

(-2√5)/25

c)

(√2-√6)/4

d)

2

172.

Given sin α = -3/5, cos α = -4/5 and cos β = (2√5)/5, sin β = (-√5)/5
Please find the sum or difference:
sin (α-β)=

a)

0

b)

(-10√5)/25

c)

(√2-√6)/4

d)

(√8)/4

173.

Simplify
sin(x+π)

a)

-1

b)

csc(x)

c)

sin(x)

d)

-sin(x)

174.

Use Sum or Difference Identities to find the exact value of
sin(π/12)

a)

(-√(2)-√(6))/2

b)

(√(6)-√(2))/4

c)

(√(2)+√(3))/4

d)

(-√(6)+√(2))/4

175.

Use Sum or Difference Identities to find the exact value of each expression.
cos(75°)

a)

1/4

b)

( √(6) - √(2) ) / 4

c)

(√(6) + √(2)) / 4

d)

(-√(6) - √(2)) / 4

176.

solve
cos(π/3)cos(2π/3)-sin(π/3)cos(2π/3)

a)

1

b)

(-1+√(3)) / 4

c)

1/2

d)

-1/2

177.

Find the exact value.
cos (5π/18 - π/9)

a)

√(3)/2

b)

1/4

c)

1/2

d)

1

178.

Write the following expression as the sine, cosine, or tangent of an angle.
cos(175)cos(55)+sin(175)sin(55)

a)

sin(120)

b)

cos(120)

c)

cos(230)

d)

sin(230)

179.

Find the exact value of the expression.
sin(5π/12)cos(π/4) - cos(5π/12)sin(π/4)

a)

1/2

b)

-1/2

c)

-√(2)/2

d)

√(3)/2

180.

Write the following expression as the sine, cosine, or tangent of an angle.
cos(175)cos(55)+sin(175)sin(55)

a)

sin(120)

b)

cos(120)

c)

cos(230)

d)

sin(230)

181.

Which of the following is NOT
a solution to
sin θ = √(3) / 2 ?

a)

π / 3

b)

2π / 3

c)

5π / 3

d)

7π / 3

182.

Please choose the correct sum and difference formula:
cos (α-β) =

a)

sin α cos β + cos α sin β

b)

sin α cos β - cos α sin β

c)

cos α cos β - sin α sin β

d)

cos α cos β + sin α sin β

183.

Find the exact value of the expression.
cos(20)cos(40) - sin(20)sin(40)

a)

1/4

b)

1/2

c)

√(3)/2

d)

√(3)

184.

sin 105º

a)

√3/2

b)

-1/4(√2 + √6)

c)

1/4 (√2 + √6)

d)

2-√3

185.

Which of the following is the same as $\cos\ \left(A+B\right)$

a)

$\sin A\cos\ B\ +\ \cos\ A\ \sin\ B$

b)

$\sin\ A\ \cos\ B\ -\ \cos\ A\ \sin\ B$

c)

$\cos\ A\ \cos\ B\ +\ \sin\ A\ \sin\ B$

d)

$\cos\ A\ \cos\ B\ -\ \sin\ A\ \sin\ B$

186.

Which of the following is equivalent to $\tan\ \left(A-B\right)$

a)

$\tan\ A\ -\ \tan\ B$

b)

$\frac{\tan\ A\ -\tan\ B}{1+\ \tan A\tan B}$

c)

$\frac{\tan\ A\ +\tan\ B}{1-\ \tan A\tan B}$

d)

$\frac{\sin\ A}{\cos\ B}$

187.

Expand $\sin\ \left(33^{o\ }+42^o\right)$

a)

$\sin\ 75^{o\ }$

b)

$\sin\ 33^o\cos42^o+\cos33^o\sin42^o$

c)

$\sin\ 33^o\cos42^o-\cos33^o\sin42^o$

d)

$\cos33^o\cos42^o+\sin33^o\sin42^o$

188.

Expand $\cos\ \left(\frac{\pi}{5}+\frac{\pi}{6}\right)$

a)

$\cos\frac{\pi}{5}\cos\frac{\pi}{6}-\sin\frac{\pi}{5}\sin\frac{\pi}{6}\$

b)

$\cos\frac{\pi}{5}\cos\frac{\pi}{6}+\sin\frac{\pi}{5}\sin\frac{\pi}{6}\$

c)

$\cos\ \frac{2\pi}{11}$

d)

$\cos\frac{\pi}{5}\sin\frac{\pi}{6}-\cos\frac{\pi}{5}\sin\frac{\pi}{6}\$

189.

$\cos75^o\cos15^o-\sin75^{o\ }\sin15^o$ is equivalent to

a)

$\sin\ 90^{o\ }$

b)

$\sin\ 60^{o\ }$

c)

$\cos\ 90^{o\ }$

d)

$\cos\ 60^{o\ }$

190.

$\frac{\tan45^o+\tan30^o}{1-\tan45^o\tan30^o}$ is equivalent to

a)

$\tan75^o$

b)

$\tan\ 15^o$

c)

$\frac{\sin45^o}{\cos30^o}$

d)

$\tan90^{o\ }$

191.

Use sum or difference angles identity to find the exact value for $\cos105^o$

a)

$\frac{\sqrt{6}+\sqrt{2}}{4}$

b)

$\frac{\sqrt{6}-\sqrt{2}}{4}$

c)

$\frac{-\sqrt{6}-\sqrt{2}}{4}$

d)

$\frac{\sqrt{2}-\sqrt{6}}{4}$

192.

Use sum or difference angles identity to find the exact value for $\sin\ \left(-15^o\right)$

a)

$\frac{\sqrt{6}+\sqrt{2}}{4}$

b)

$\frac{\sqrt{6}-\sqrt{2}}{4}$

c)

$\frac{\sqrt{2}-\sqrt{6}}{4}$

d)

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

193.

Given $\sin x=\frac{3}{5}and\ \sin y=\frac{2}{3},$ where $x\ and\ y$ are both in first quadrant. $Evaluate\ \sin\left(x+y\right)$

a)

$\frac{3\sqrt{5}+8}{15}$

b)

$\frac{4\sqrt{5}+6}{15}$

c)

$\frac{2\sqrt{5}+12}{15}$

d)

$\frac{4\sqrt{5}+2}{5}$

194.

If $\sin A=\frac{3}{5},\sin B=\frac{2}{3},$ $and\ \angle A\ and\ \angle B$ are acute angles, what is the value of $\cos\left(A-B\right)?$

a)

$-\frac{2}{3}$

b)

$\frac{4\sqrt{5}-6}{15}$

c)

$\frac{4\sqrt{5}+2}{5}$

d)

$\frac{4\sqrt{5}+6}{15}$

195.

If $\cos A=\frac{1}{3}$ , then the positive value of $\tan\ \frac{A}{2}$ ?

a)

$\sqrt{2}$

b)

$\sqrt{3}$

c)

$\frac{\sqrt{3}}{3}$

d)

$\frac{\sqrt{2}}{2}$

196.

Factor: sec²x − secx − 2

a)

(sec x)(secx − 2)

b)

(secx − 2)(secx − 1)

c)

(secx − 2)(secx + 1)

d)

(secx + 2)(secx − 1)

197.

csc²x = 2 is equivalent to sin²x = ½

a)

True

b)

False

198.

On the domain [0, 2π), solve this equation 0 = sinx(sinx − 1)

a)

0, π

b)

0, π, ½π

c)

½π

d)

0, 90, 270

199.

Factor 0 = sin²x − sinx

a)

0 = sinx(sinx)

b)

0 = sinx(1 − sinx)

c)

0 = cosx(sinx − 1)

d)

0 = sinx(sinx − 1)

200.

Choose a good way to start solving this equation

tanx sin²x = 2 tanx

a)

divide tanx from both sides

b)

factor out tanx

c)

subtract 2 tanx from the left and then factor tanx out

d)

cancel sin²x out

201.

Why doesn't 2cosx − 3 = 0 have solutions?

a)

cosx is never bigger than one

b)

cosx is never equal to a fraction

c)

Actually, this equation does have a solution, x = π

d)

This equation will have a solution tomorrow.

202.

Find an expression equivalent to the following:

$\frac{\sec\theta\tan\theta}{\sin\theta}$

a)

$\sec^2\theta$

b)

$\cot\theta$

c)

$\tan^2\theta$

d)

$\cos^2\theta$

203.

If $\sec\theta=-\frac{5}{4}$ and $180^o<\theta<270^o$ , find $\tan\theta$ .

a)

$-\frac{3}{5}$

b)

$-\frac{4}{5}$

c)

$\frac{3}{4}$

d)

$\frac{3}{5}$

204.

Simplify the following:

$\frac{\tan^2\theta+1}{\tan^2\theta}$

a)

$\csc^2\theta$

b)

-1

c)

$\tan^2\theta$

d)

1

205.

Simplify the following:

$\frac{\tan x}{\sin x}+\frac{1}{\cos x}$

a)

$2\ \tan^2x$

b)

$2\ \cos x$

c)

$2\cos x-1$

d)

$2\ \sec x$

206.

Use a sum or difference identity to find the exact value of $\sin105^o$ .

a)

$\frac{-\sqrt{2}-\sqrt{6}}{4}$

b)

$\frac{\sqrt{6}-\sqrt{2}}{4}$

c)

$\frac{\sqrt{6}+\sqrt{2}}{4}$

d)

$\frac{\sqrt{2}-\sqrt{6}}{4}$

207.

Find the value of $\tan\left(\alpha-\beta\right)$ if $\cos a=\frac{4}{5}$ , $\sin\beta=-\frac{5}{13}$ , $270^o<\alpha<360^o$ , and $270^o<\beta<360^o$ .

a)

$\frac{16}{63}$

b)

$-\frac{16}{63}$

c)

$-\frac{56}{33}$

d)

$\frac{56}{33}$

208.

Which expression is equivalent to the following:

$\cos\left(\pi+\theta\right)$

a)

$-\cos\theta$

b)

$\cos\theta$

c)

$-\sin\theta$

d)

$\sin\theta$

209.

Which expression is NOT equivalent to the following:

$\cos2\theta$

a)

$2\cos^2\theta-1$

b)

$1-2\sin^2\theta$

c)

$\cos^2\theta+\sin^2\theta$

d)

$\cos^2\theta-\sin^2\theta$

210.

If $\sin\theta=0.6$ and $90^o<\theta<180^o$ , find the exact value of $\sin2\theta$ .

a)

-0.6

b)

-0.96

c)

0.96

d)

0.28

211.

If $\cos\theta=-\frac{3}{5}$ and $\theta$ has its terminal side in Quadrant II, find the exact value of $\tan2\theta$ .

a)

$\frac{24}{25}$

b)

$\frac{7}{25}$

c)

$\frac{24}{7}$

d)

$-\frac{7}{25}$

212.

Use a half-angle identity to find the exact value of $\cos75^o$ .

a)

$\frac{1}{2}\sqrt{2+\sqrt{3}}$

b)

$\frac{1}{2}\sqrt{2-\sqrt{3}}$

c)

$\frac{1}{2}\sqrt{2+\sqrt{2}}$

d)

$-\frac{1}{2}\sqrt{1+\sqrt{3}}$

213.

Solve $2\cos x-1=0$ for $0\le x<2\pi$ .

a)

$\frac{\pi}{6};\frac{5\pi}{6}$

b)

$\frac{\pi}{3};\ \frac{5\pi}{3}$

c)

$\frac{\pi}{3};\ \frac{2\pi}{3}$

d)

$\frac{7\pi}{6};\ \frac{11\pi}{6}$

214.

Solve $2\sin^2x-\sin x=0$ for principal values of x.

a)

$60^o;\ 120^o$

b)

$0^o;\ 150^o$

c)

$0^o;\ 30^o$

d)

$60^o$

215.

Solve $2\sin x+\sqrt{3}=0$ for $0\le x<2\pi$ .

a)

$\frac{4\pi}{3};\frac{5\pi}{3}$

b)

$\frac{2\pi}{3};\frac{4\pi}{3}$

c)

$\frac{7\pi}{6};\frac{11\pi}{6}$

d)

$\frac{5\pi}{6};\frac{7\pi}{6}$

216.

Find a numerical value of one trigonometric function of x if the following is true:

$\sec x\cot x=4$

a)

$\csc x=\frac{1}{4}$

b)

$\sec x=\frac{1}{4}$

c)

$\sec x=4$

d)

$\csc x=4$

217.

Given sin⊖ = $\frac{-1}{3}$ and cos⊖ = $\frac{2\sqrt{2}}{3}$ , find the exact value for tan⊖.

a)

$-2\sqrt{2}$

b)

$\frac{\sqrt{2}}{4}$

c)

$-3$

d)

$\frac{-\sqrt{2}}{4}$

218.

Given cos ⊖ = $\frac{\sqrt{3}}{5}$ and tan⊖ < 0 , find the exact value for sin⊖ using pythagorean identities.

a)

$\frac{5\sqrt{3}}{3}$

b)

$\frac{\sqrt{22}}{5}$

c)

$\frac{22}{25}$

d)

$\frac{-\sqrt{22}}{5}$

219.

The expression $\sin^2x+\cos^2x-b^2$ is equal to

a)

1

b)

$b^2$

c)

$\left(1+b\right)\left(1-b\right)$

d)

$\sin x\cos x-b$

220.

Pythagorean: $\sin^2x-1=?$

a)

$-\cos^2x$

b)

$\cos^2x$

c)

$\cos^2x-1$

d)

$1-\cos^2x$

221.

Given sin⊖ = 2/3 and tan⊖ <0, find the exact value for tan⊖

a)

3/2

b)

(3√5)/5

c)

(-2√5)/5

d)

-√5/2

222.

Given sin⊖ = 2/3 and tan⊖ <0, find the exact value for cos⊖

a)

3/2

b)

-√5/3

c)

√5/3

d)

-2/3

223.

csc(x)sin(x) + cot²(x)

a)

A. tan²(x)

b)

B. sec²(x)

c)

C. csc²(x)

d)

D. -csc²(x)

224.

Select the MOST SIMPLIFIED answer for the problem:

$\frac{1-\cos^2x}{\sin^2x}$

a)

$\sin^2x$

b)

$\cos^2x$

c)

1

d)

$\frac{\sin^2x}{\sin^2x}$

225.

Select the MOST SIMPLIFIED answer for the problem:

$\frac{\cot^2x}{1-\sin^2x}$

a)

1

b)

$\frac{\cot^2x}{\cos^2x}$

c)

$\frac{1}{\sin^2x}$

d)

$\csc^2x$

226.

$3\left(\csc^2x-\cot^2x\right)$

a)

0

b)

1

c)

2

d)

3

227.

Select the MOST SIMPLIFIED answer for the problem:

$\frac{\sec^2x-1}{\sin^2x}$

a)

$\frac{1}{\cos^2x}$

b)

$\sec x$

c)

$\sec^2x$

d)

$\frac{1}{\cos x}$

228.

Select the MOST SIMPLIFIED answer for the problem:

$\frac{\cot^2\theta}{1+\cot^2\theta}$

a)

$\cos\theta$

b)

$\cos^2\theta$

c)

1

d)

$\left(\frac{\cos^2\theta}{\sin^2\theta}\right)\left(\frac{\sin^2\theta}{1}\right)$

229.

Select the MOST SIMPLIFIED answer for the problem:

$\frac{1-\sin^2x}{1-\cos^2x}$

a)

$\frac{\cos x}{\sin x}$

b)

$\frac{\cos^2x}{\sin^2x}$

c)

$\tan^2x$

d)

$\cot^2x$

230.

$\frac{\cos^2x+\sin^2x}{\sec x}$

a)

$\frac{1}{\sec x}$

b)

$\sin x$

c)

$\cos x$

231.

Select the MOST SIMPLIFIED answer for the problem:

$\sin^2x+\sin x+\cos^2x-1$

a)

$\sin x$

b)

2

c)

0

232.

Solve on the Interval [0,2π)

tan(x)+1=2

a)

0 and π

b)

3π/4 and 7π/4

c)

π/4 and 5π/4

d)

3π/4 and 5π/4

233.

Solve

cos²θ = ½

on θ∈[0, 2π)

a)

θ = π /4, 7π /4

b)

θ = π /4, 3π /4

c)

θ = 3π /4, 5π /4

d)

θ = π/4, 3π/4, 5π/4, 7π/4

234.

a)

No solution

b)

π/3 +2πn, 5π/3 +2πn

c)

π/6 +2πn, 11π/6 +2πn

d)

2π/3 +2πn, 4π/3 +2πn

235.

a)

A

b)

B

c)

C

d)

D

236.

Find the solution over the interval [0, 2π)

a)

0, π, π/6, 5π/6,

b)

0, π/6, 7π/6

c)

π, 5π/6, 11π/6

d)

π, π/6, 5π/6, 7π/6, 11π/6

237.

Which of these is equivalent to 2cos²x − 3cosx = 0 ?

a)

-cos²x = 0

b)

cosx(2cosx + 3) = 0

c)

cosx(2cosx − 3) = 0

d)

cos x = ⅔

238.

Choose a good way to start solving this equation

tanx sin²x = 2 tanx

a)

divide tanx from both sides

b)

factor out tanx

c)

subtract 2 tanx from the left and then factor tanx out

d)

cancel sin²x out

239.

Simplify: $\frac{\cos^2\left(-x\right)\csc x}{\cot x}$

a)

$-\sin x$

b)

$-\cos x$

c)

$\sin x$

d)

$\cos x$

Unit 9 simplifying trig identities

Total questions: 239

Worksheet time: 10hrs 11mins

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