Table of contents

0. Review of College Algebra(0)
1. Measuring Angles(0)
2. Trigonometric Functions on Right Triangles(0)
3. Unit Circle(0)
4. Graphing Trigonometric Functions(0)
5. Inverse Trigonometric Functions and Basic Trigonometric Equations(0)
6. Trigonometric Identities and More Equations(0)
7. Non-Right Triangles(0)
8. Vectors(0)
9. Polar Equations(0)
10. Parametric Equations(0)
11. Graphing Complex Numbers(0)

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent - Online Tutor, Practice Problems & Exam Prep

Inverse Sine, Cosine, & Tangent Practice Problems

147 problems

1PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression without the help of the calculator.

sin(sin^-1 0.7)

2PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression without the help of the calculator.

sin^-1 (sin 2π/3)

3PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression without the help of the calculator.

tan (tan^-1 175)

4PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression without the help of the calculator.

tan^-1 [tan(-π/3)]

5PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression without the help of the calculator.

tan^-1 (tan 3π/4)

6PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression without the help of the calculator.

sin^-1 (sin 2π)

7PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression without the help of the calculator.

sin(sin^-1 2π)

8PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

sin^{- 1}((√3)/2)

9PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

sin^{- 1}(- (√3)/2)

10PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

sin^{- 1}(- (√2)/2)

11PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

cos^{- 1}((√2)/2)

12PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

cos^{- 1}(- 1/2)

13PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

tan^{- 1}(1)

14PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

tan^{- 1}(- 1)

15PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

cot^{- 1}(1)

16PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

cot^{- 1}(- 1)

17PRACTICE PROBLEM

For the following trigonometric expression, evaluate and approximate the answer (in radians) to four significant figures.

sin^{- 1}(0.8)

18PRACTICE PROBLEM

For the following trigonometric expression, evaluate and approximate the answer (in radians) to four significant figures.

sin^{- 1}(- 0.87)

19PRACTICE PROBLEM

For the following trigonometric expression, evaluate and approximate the answer (in radians) to three significant figures.

cos^{- 1}(4/9)

20PRACTICE PROBLEM

For the following trigonometric expression, evaluate and approximate the answer (in radians) to three significant figures.

cos^{- 1}((√3)/8)

21PRACTICE PROBLEM

For the following trigonometric expression, evaluate and approximate the answer (in radians) to three significant figures.

tan^{- 1}(- 54)

22PRACTICE PROBLEM

For the following trigonometric expression, evaluate and approximate the answer (in radians) to three significant figures.

tan^{- 1}(- √289)

23PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression using the properties of the inverse functions.

cot(cot^-1 7π)

24PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression using the properties of the inverse functions.

sec(sec^-1 5π)

25PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression using the properties of the inverse functions.

cot^-1 (cot 5π/6)

26PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

csc^{- 1}(- √2)

27PRACTICE PROBLEM

For the following trigonometric expression, evaluate and give the answer as an exact value in radians.

sec^{- 1}(- 2)

28PRACTICE PROBLEM

Transform the following expression into an algebraic expression. Use a right triangle in writing the algebraic expression. Assume that the inverse trigonometric function is defined for its argument and assume that x > 0.

tan (sin⁻¹ 2x)

sin (tan⁻¹ 4x)

sin (cos⁻¹ (5/x))

csc (sin⁻¹ (3/x))

sec (sin⁻¹ (4/√(x² + 16)))

33PRACTICE PROBLEM

csc (tan⁻¹ 3x)

34PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression by sketching.

cos (sin^-1 (35/37))

35PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression by sketching.

tan (cos^-1 (12/37))

36PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression by sketching.

$\tan\left[\sin^{-1}\left(-\frac{20}{29}\right)\right\rbrack$

37PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression by sketching.

sin (cos⁻¹ (√3/2))

38PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression by sketching.

tan [cos^-1 (-2/5)]

39PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression by sketching.

cos [tan^-1 (-6/7)]

40PRACTICE PROBLEM

Determine the exact value of the following trigonometric expression by sketching.

cot (csc^-1 6)

41PRACTICE PROBLEM

Perform transformation on the given graph of y = sin⁻¹ x to graph the following function. Then, identify the domain and range. Express the domain and range in interval notation.

f(x) = sin⁻¹ x + 3π/2

42PRACTICE PROBLEM

Perform transformation on the given graph of y = cos⁻¹ x to graph the following function. Then, identify the domain and range. Express the domain and range in interval notation.

f(x) = cos⁻¹ (x + 8)

43PRACTICE PROBLEM

Perform transformation on the given graph of y = tan⁻¹ x to graph the following function. Then, identify the domain and range. Express the domain and range in interval notation.

f(x) = -5 tan⁻¹ x

44PRACTICE PROBLEM

Perform transformation on the given graph of y = cos⁻¹ x to graph the following function. Then, identify the domain and range. Express the domain and range in interval notation.

f(x) = cos⁻¹ (x/5)

45PRACTICE PROBLEM

Without using a calculator, determine the exact value of the expression:

sin⁻¹ (-1/2)

46PRACTICE PROBLEM

Without using a calculator, determine the exact value of the expression:

cos⁻¹ (-(√3)/2)

47PRACTICE PROBLEM

Without using a calculator, determine the exact value of the expression:

sec⁻¹ (1)

48PRACTICE PROBLEM

Without using a calculator, determine the exact value of the expression:

sin (cos⁻¹ ((√3)/2))

49PRACTICE PROBLEM

Without using a calculator, determine the exact value of the expression:

tan (cos⁻¹ ((√2)/2))

50PRACTICE PROBLEM

Without using a calculator, determine the exact value of the expression:

sec(tan⁻¹ √3)

51PRACTICE PROBLEM

Without using a calculator, determine the exact value of the expression:

cos (sin⁻¹ (63/65))

52PRACTICE PROBLEM

Without using a calculator, determine the exact value of the expression:

tan (sin⁻¹ (-21/29))

53PRACTICE PROBLEM

Without using a calculator, determine the exact value of the expression:

cos⁻¹ (cos (2π/7))

54PRACTICE PROBLEM

Without using a calculator, determine the exact value of the expression:

cos⁻¹ (sin (4π/3))

55PRACTICE PROBLEM

csc (cos⁻¹ (7/x))

56PRACTICE PROBLEM

Identify whether the given statement is true or false.

The cotangent and cosecant trigonometric functions are undefined for the same values of θ.

57PRACTICE PROBLEM

Solve the following equation for x if x lies in the interval $\left[0,\pi\right\rbrack$ .

$y=13\cos x$

58PRACTICE PROBLEM

Solve the following equation for x if x lies in the interval $\left(-\frac{\pi}{10},\frac{\pi}{10}\right)$ .

$y=7\tan5x$

59PRACTICE PROBLEM

Solve the following equation for x if x lies in the interval $\left[0,6\pi\right\rbrack$ .

$y=12\cos\left(\frac{x}{6}\right)$

60PRACTICE PROBLEM

Solve the following equation for x if x lies in the interval $\left[0,\frac{\pi}{8}\right\rbrack$ .

$y=-4\cos8x$

61PRACTICE PROBLEM

Solve the following equation for x if x lies in the interval $\left[-\frac{\pi}{2},\right.\left.\frac{\pi}{2}\right\rbrack$ .

$y=\sin x-7$

62PRACTICE PROBLEM

Solve the following equation for x if x lies in the interval $\left[-\frac{\pi}{2},\right.\left.\frac{\pi}{2}\right\rbrack$ .

$y=-8+3\sin x$

63PRACTICE PROBLEM

Solve the following equation for x if x lies in the interval $\left(0,\right.\left.\frac{\pi}{5}\right)$ .

$y=\frac16\cot5x$

64PRACTICE PROBLEM

Solve the following equation for x if x lies in the interval $\left[-5,\pi-5\right\rbrack$ .

$y=\cos\left(x+5\right)$

65PRACTICE PROBLEM

Solve the following equation for x if x lies in the interval $\left[0,\frac{\pi}{8}\right)\cup\left(\frac{\pi}{8},\right.\left.\frac{\pi}{4}\right\rbrack$ .

$y=\sqrt5+7\sec4x$

66PRACTICE PROBLEM

Identify the equation which has a solution equal to $\frac{\pi}{3}$ .

67PRACTICE PROBLEM

Identify the equation which has a solution equal to $\frac{2\pi}{3}$ .

68PRACTICE PROBLEM

Identify the equation which has a solution equal to $\frac{5\pi}{6}$ .

69PRACTICE PROBLEM

Determine the exact solution for the given equation.

$-6\sin^{-1}x=\pi$

70PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\sin^{-1}\left(\frac{\sqrt2}{2}\right)$

71PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\sin^{-1}\left(-\frac{\sqrt2}{2}\right)$

72PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\cos^{-1}\left(-\frac12\right)$

73PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\tan^{-1}\left(\sqrt3\right)$

74PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\tan^{-1}\left(\frac{\sqrt3}{3}\right)$

75PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\sin^{-1}\left(-\frac12\right)$

76PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\cos^{-1}\left(-\frac{\sqrt2}{2}\right)$

77PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\sin^{-1}\sqrt5$

78PRACTICE PROBLEM

Find the exact solution of the equation.

$\frac32\cos^{-1}\frac{x}{2}=\pi$

79PRACTICE PROBLEM

Find the exact solution of the equation.

$4\sin^{-1}x=3\pi$

80PRACTICE PROBLEM

Find the exact solution of the equation.

$\cos^{-1}x=\sin^{-1}\frac{12}{37}$

81PRACTICE PROBLEM

Find the exact solution of the equation.

$\tan^{-1}x=\cot^{-1}\frac{17}{3}$

82PRACTICE PROBLEM

Find the exact solution of the equation.

$\arcsin x = \arctan \frac{13}{84}$

83PRACTICE PROBLEM

Find the exact solution of the equation.

$7\arccos\left(\frac{x}{5}-\frac{\pi}{5}\right)=7\pi$

84PRACTICE PROBLEM

Find the exact solution of the equation.

$\cos^{- 1} x + \tan^{- 1} \sqrt{3} = \frac{π}{3}$

85PRACTICE PROBLEM

Find the exact solution of the equation.

$\arcsin x - \frac{1}{2} \arccos \frac{\sqrt{3}}{2} = \frac{π}{4}$

86PRACTICE PROBLEM

Find the exact solution of the equation.

$\sin^{- 1} x - 8 \tan^{- 1} \frac{\sqrt{3}}{3} = π$

87PRACTICE PROBLEM

Find the solution of the equation on the interval [0, 4] using a calculator. Round the answer to four decimal places.

$\left(\arctan x\right)^3-3x+5=0$

88PRACTICE PROBLEM

Find the exact solution of the equation.

$\arccos x+\arctan\sqrt3=\frac{7\pi}{12}$

89PRACTICE PROBLEM

Write the point that lies on the graph of $y=\sin^{-1}x$ if the point $\left(\frac{\pi}{6},\frac12\right)$ lies on the graph of $y=\sin x$ .

90PRACTICE PROBLEM

Find the exact solution of the equation.

$\arccos2x-\arcsin x=\frac{7\pi}{6}$

91PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\cot^{-1}\left(-\sqrt3\right)$

92PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\csc^{-1}\left(-\frac{2\sqrt3}{3}\right)$

93PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\sec^{-1}2$

94PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\sec^{-1}\left(\sqrt2\right)$

95PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\csc^{-1}\left(\frac{\sqrt3}{2}\right)$

96PRACTICE PROBLEM

Find the value of the given inverse trigonometric function using a calculator. Write your answer in radians.

$y=\sin^{-1}\left(0.564763238\right)$

97PRACTICE PROBLEM

Find the value of the given inverse trigonometric function using a calculator. Write your answer in radians.

$y=\cos^{-1}(-0.87563524)$

98PRACTICE PROBLEM

Find the value of the given inverse trigonometric function using a calculator. Write your answer in radians.

$y=\tan^{-1}(1.5679827)$

99PRACTICE PROBLEM

Find the value of the given inverse trigonometric function using a calculator. Write your answer in radians.

$y=\cot^{-1}(-0.26746463)$

100PRACTICE PROBLEM

Find the value of the given inverse trigonometric function using a calculator. Write your answer in radians.

$y=\sec^{-1}(-1.8962734)$

101PRACTICE PROBLEM

Determine the angle measure $\theta$ without using a calculator. Express the answer in degrees.

$\theta=\arctan\left(\frac{\sqrt3}{3}\right)$

102PRACTICE PROBLEM

Determine the angle measure $\theta$ without using a calculator. Express the answer in degrees.

$\theta=\sin^{-1}\left(-\frac12\right)$

103PRACTICE PROBLEM

Determine the angle measure $\theta$ without using a calculator. Express the answer in degrees.

$\theta=\cos^{-1}\left(-\frac{\sqrt3}{2}\right)$

104PRACTICE PROBLEM

Determine the angle measure $\theta$ without using a calculator. Express the answer in degrees.

$\theta=arc\cot\left(-1\right)$

105PRACTICE PROBLEM

Determine the angle measure $\theta$ without using a calculator. Express the answer in degrees.

$\theta=\csc^{-1}\left(-\frac{2\sqrt3}{3}\right)$

106PRACTICE PROBLEM

Determine the angle measure $\theta$ without using a calculator. Express the answer in degrees.

$\theta=\arcsin\left(7\right)$

107PRACTICE PROBLEM

Estimate the value of $θ$ by using a calculator. Express your answer in degrees rounded to the nearest thousandth.

$θ = \arcsin (- 0.26938721)$

108PRACTICE PROBLEM

Estimate the value of $θ$ by using a calculator. Express your answer in degrees rounded to the nearest thousandth.

$θ = \cos^{- 1} (- 0.59375518)$

109PRACTICE PROBLEM

Estimate the value of $θ$ by using a calculator. Express your answer in degrees rounded to the nearest thousandth.

$theta equals a r c cosecant 2.93880461$

110PRACTICE PROBLEM

Estimate the value of $\theta$ by using a calculator. Express your answer in degrees rounded to the nearest thousandth.

$\theta=arc\cot\left(-0.59367721\right)$

111PRACTICE PROBLEM

Estimate the value of $θ$ by using a calculator. Express your answer in degrees rounded to the nearest thousandth.

$θ = \arctan (- 5.3674925)$

112PRACTICE PROBLEM

Transform the following expression into an algebraic expression in $x$ . Assume that the inverse trigonometric function is defined for its argument and assume that $x>0$ .

$\sin\left(\cos^{-1}2x\right)$

113PRACTICE PROBLEM

Transform the following expression into an algebraic expression in $x$ . Assume that the inverse trigonometric function is defined for its argument and assume that $x>0$ .

$\cos\left(\sin^{-1}4x\right)$

114PRACTICE PROBLEM

Transform the following expression into an algebraic expression in $x$ . Assume that the inverse trigonometric function is defined for its argument and assume that $x>0$ .

$\sin\left(2\sec^{-1}\frac{x}{4}\right)$

115PRACTICE PROBLEM

Transform the following expression into an algebraic expression in $x$ . Assume that the inverse trigonometric function is defined for its argument and assume that $x>0$ .

$\tan\left(2\sin^{-1}\frac{x}{\sqrt{x^2}+6}\right)$

116PRACTICE PROBLEM

Transform the following expression into an algebraic expression in $x$ . Assume that the inverse trigonometric function is defined for its argument and assume that $x>0$ .

$\sec\left(\cot^{-1}\frac{\sqrt{25-x^2}}{x}\right)$

117PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\sin^{-1}0+\sin^{-1}\left(-1\right)$

118PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\tan^{-1}\left(-\frac{\sqrt3}{3}\right)+\tan^{-1}\left(\sqrt3\right)$

119PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\cos^{-1}\left(-\frac12\right)+\cos^{-1}\left(-1\right)$

120PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\sec^{-1}\left(-1\right)+\sec^{-1}\left(2\right)$

121PRACTICE PROBLEM

Determine the exact value of the given inverse trigonometric function without using a calculator.

$y=\cot^{-1}\left(-\frac{\sqrt3}{3}\right)+\cot^{-1}\left(-\sqrt3\right)$

122PRACTICE PROBLEM

Determine the angle measure $\theta$ without using a calculator. Express the answer in degrees.

$\theta=\csc^{-1}\left(-2\right)+\csc^{-1}\left(1\right)$

123PRACTICE PROBLEM

Determine the angle measure θ. Express the answer in degrees.

$\theta=\tan^{-1}1.8484838$

124PRACTICE PROBLEM

Find the value of the given inverse trigonometric function using a calculator. Write your answer in degrees.

$\theta=\cos^{-1}0.76463521$

125PRACTICE PROBLEM

Find the value of the given inverse trigonometric function using a calculator. Write your answer in degrees.

$\theta=\sec^{-1}5.68737372$

126PRACTICE PROBLEM

Determine the exact value of the expression without using a calculator.

$\tan\left(\arccos\frac{3}{11}\right)$

127PRACTICE PROBLEM

Determine the exact value of the expression without using a calculator.

$\cos\left(\tan^{-1}\left(-5\right)\right)$

128PRACTICE PROBLEM

Determine the exact value of the expression without using a calculator.

$\sin\left(2\tan^{-1}\frac{35}{12}\right)$

129PRACTICE PROBLEM

Determine the exact value of the expression without using a calculator.

$\cos\left(2\arctan\frac{84}{13}\right)$

130PRACTICE PROBLEM

Determine the exact value of the expression without using a calculator.

$\sin\left(2\cos^{-1}\frac{3}{11}\right)$

131PRACTICE PROBLEM

Determine the exact value of the expression without using a calculator.

$\sec\left(\sec^{-1}14\right)$

132PRACTICE PROBLEM

Determine the exact value of the expression without using a calculator.

$\cos\left(\tan^{-1}\frac{9}{40}-\tan^{-1}\frac{20}{21}\right)$

133PRACTICE PROBLEM

Determine the exact value of the expression without using a calculator.

$\sin\left(\sin^{-1}\frac25+\tan^{-1}\left(-4\right)\right)$

134PRACTICE PROBLEM

Find the exact solution of the equation.

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

135PRACTICE PROBLEM

Find the exact solution of the equation.

$\sin^{-1}x+\csc^{-1}\frac{1}{x}=\frac{\pi}{3}$

136PRACTICE PROBLEM

Transform the following expression into an algebraic expression in $x$ . Assume that the inverse trigonometric function is defined for its argument and assume that $x>0$ .