5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Use a calculator to approximate each value in decimal degrees.

θ = arccos (-0.39876459)

219

Use a calculator to approximate each value in decimal degrees.

θ = csc⁻¹ 1.9422833

215

Use a calculator to approximate each value in decimal degrees.

θ = cot⁻¹ (-0.60724226)

235

Solve each equation for x.

4/3 arctan x/2 = π

213

Solve each equation for x.

arccos x + arctan 1 = 11π/12

210

Use a calculator to approximate each value in decimal degrees.

θ = tan⁻¹ (-7.7828641)

194

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

y = arcsin 0.92837781

211

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

y = cos⁻¹ (―0.32647891)

211

Solve each equation for x.

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

203

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

y = arctan 1.1111111

216

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

y = cot⁻¹ (―0.92170128)

193

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

y = sec⁻¹ (―1.2871684)

205

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

y = 5 cos x , for x in [0, π]

213

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

y = tan⁻¹ (―√3)

211

Graph each inverse circular function by hand.

y = arcsec [(1/2)x]

217

Evaluate each expression without using a calculator.

tan (arccos 3/4)

231

Evaluate each expression without using a calculator.

sin (2 tan⁻¹ (12/5))

197

Evaluate each expression without using a calculator.

sin (2 cos⁻¹ (1/5))

230

Evaluate each expression without using a calculator.

sec (sec⁻¹ 2)

225

Evaluate each expression without using a calculator.

cos (tan⁻¹ (5/12) - tan⁻¹ (3/4))

270

Evaluate each expression without using a calculator.

sin (sin⁻¹ 1/2 + tan⁻¹ (-3))

205

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

y = 3 tan 2x , for x in [―π/4, π/4]

190

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

y = cos⁻¹ (―√2/2)

224

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

cos (tan⁻¹ 0.5)

214

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

tan (arcsin 0.12251014)

218

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

sin (arccos u)

241

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

cos (arcsin u)

199

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

sin (2 sec⁻¹ u/2)

286

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

258

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

275

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

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

287

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

271

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

289

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

353

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

274

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

268

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

289

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

281

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

270

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

277

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

348

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

303

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

262

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

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

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

287

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

299

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

272

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

308

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

286

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

296

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

260

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

269

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

295

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

309

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)

235

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)

282

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)

270

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)

304

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

255

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

297

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

459

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

260

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

278

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)

455

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)

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)

327

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

236

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)

246

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

241

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

212