5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Use a calculator to approximate each value in decimal degrees.

θ = arccos (-0.39876459)

105

Use a calculator to approximate each value in decimal degrees.

θ = csc⁻¹ 1.9422833

110

Use a calculator to approximate each value in decimal degrees.

θ = cot⁻¹ (-0.60724226)

117

Solve each equation for x.

4/3 arctan x/2 = π

105

Solve each equation for x.

arccos x + arctan 1 = 11π/12

104

Use a calculator to approximate each value in decimal degrees.

θ = tan⁻¹ (-7.7828641)

98

Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.)

y = arcsin 0.92837781

112

Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.)

y = cos⁻¹ (―0.32647891)

113

Solve each equation for x.

y = 1/2 tan (3x + 2), for x in [-2/3 - π/6, -2/3 + π/6]

112

Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.)

y = arctan 1.1111111

119

Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.)

y = cot⁻¹ (―0.92170128)

99

Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.)

y = sec⁻¹ (―1.2871684)

107

Solve each equation for x, where x is restricted to the given interval.

y = 5 cos x , for x in [0, π]

94

Find the exact value of each real number y. Do not use a calculator.

y = tan⁻¹ (―√3)

115

Graph each inverse circular function by hand.

y = arcsec [(1/2)x]

111

Evaluate each expression without using a calculator.

tan (arccos 3/4)

110

Evaluate each expression without using a calculator.

cos (tan⁻¹ (-2))

115

Evaluate each expression without using a calculator.

sin (2 tan⁻¹ (12/5))

101

Evaluate each expression without using a calculator.

cos (2 arctan (4/3))

103

Evaluate each expression without using a calculator.

sin (2 cos⁻¹ (1/5))

117

Evaluate each expression without using a calculator.

sec (sec⁻¹ 2)

118

Evaluate each expression without using a calculator.

cos (tan⁻¹ (5/12) - tan⁻¹ (3/4))

155

Evaluate each expression without using a calculator.

sin (sin⁻¹ 1/2 + tan⁻¹ (-3))

113

Solve each equation for x, where x is restricted to the given interval.

y = 3 tan 2x , for x in [―π/4, π/4]

101

Find the exact value of each real number y. Do not use a calculator.

y = cos⁻¹ (―√2/2)

134

Use a calculator to find each value. Give answers as real numbers.

cos (tan⁻¹ 0.5)

100

Use a calculator to find each value. Give answers as real numbers.

tan (arcsin 0.12251014)

106

Write each trigonometric expression as an algebraic expression in u, for u > 0.

sin (arccos u)

148

Write each trigonometric expression as an algebraic expression in u, for u > 0.

cos (arcsin u)

111

Write each trigonometric expression as an algebraic expression in u, for u > 0.

sin (2 sec⁻¹ u/2)

156

In Exercises 1–26, find the exact value of each expression. _ cos⁻¹ √3/2

162

In Exercises 1–26, find the exact value of each expression. _ cos⁻¹ (- √2/2)

In Exercises 1–26, find the exact value of each expression. _ tan⁻¹ √3/3

171

In Exercises 1–26, find the exact value of each expression. _ tan⁻¹ (−√3)

184

In Exercises 1–26, find the exact value of each expression. _ cot⁻¹ √3

In Exercises 1–26, find the exact value of each expression. _ cot⁻¹ (−√3)

182

In Exercises 1–26, find the exact value of each expression. _ csc⁻¹ (− 2√3/3)

247

In Exercises 1–26, find the exact value of each expression. _ sec⁻¹ (−√2)

184

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. sin⁻¹ 0.3

172

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. sin⁻¹ (-0.32)

170

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. cos⁻¹ 3/8

182

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ sin⁻¹ (− √3/2)

171

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. cos⁻¹ (− 1/2)

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. _ cos⁻¹ √5/7

237

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. tan⁻¹ (−20)

199

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sec⁻¹ (−1)

170

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. ___ tan⁻¹ (−√473)

180

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ cos(sin⁻¹ √2/2)

185

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin(sin⁻¹ 0.9)

146

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan[sin⁻¹ (− 1/2)]

182

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ csc(tan⁻¹ √3/3)

188

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin 5π/6)

194

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin(cos⁻¹ 3/5)

181

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan (tan⁻¹ 125)

180

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan [cos⁻¹ (− 4/5)]

202

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan⁻¹ [tan(− π/6)]

195

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(sin π/3)

192

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan⁻¹ (tan 2π/3)

179

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(cos 2π/3)

183

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin π)

199

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin(sin⁻¹ π)

196

In Exercises 52–53, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. sec(sin⁻¹ 1/x)

141

In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. cot(cot⁻¹ 9π)

189

In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. sec(sec⁻¹ 7π)

174

In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. cot⁻¹ (cot 3π/4)

200

In Exercises 63–82, use a sketch to find the exact value of each expression. cos (sin⁻¹ 4/5)

165

In Exercises 63–82, use a sketch to find the exact value of each expression. tan (cos⁻¹ 5/13)

183

In Exercises 63–82, use a sketch to find the exact value of each expression. tan [sin⁻¹ (− 3/5)]

190

In Exercises 63–82, use a sketch to find the exact value of each expression. _ sin (cos⁻¹ √2/2)

309

In Exercises 63–82, use a sketch to find the exact value of each expression. tan [cos⁻¹ (− 1/3)]

235

In Exercises 63–82, use a sketch to find the exact value of each expression. cos [tan⁻¹ (− 2/3)]

162

In Exercises 63–82, use a sketch to find the exact value of each expression. cot (csc⁻¹ 8)

In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. tan (cos⁻¹ x)

In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. sin (tan⁻¹ x)

310

In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. cos (sin⁻¹ 1/x)

188

In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. sec (cos⁻¹ 1/x)

In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. ___ sec (sin⁻¹ x/√x²+4)

172

In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. csc (cot⁻¹ x)

226

The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. f(x) = sin⁻¹ x + π/2

133

The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. f(x) = cos⁻¹ (x + 1)

151

The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. h(x) = −2 tan⁻¹ x

142

The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. f(x) = cos⁻¹ x/2

132