5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Use a calculator to approximate each value in decimal degrees.

θ = arccos (-0.39876459)

Use a calculator to approximate each value in decimal degrees.

θ = csc⁻¹ 1.9422833

Use a calculator to approximate each value in decimal degrees.

θ = cot⁻¹ (-0.60724226)

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Solve each equation for x.

4/3 arctan x/2 = π

Solve each equation for x.

arccos x + arctan 1 = 11π/12

Use a calculator to approximate each value in decimal degrees.

θ = tan⁻¹ (-7.7828641)

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

y = arcsin 0.92837781

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

y = cos⁻¹ (―0.32647891)

Solve each equation for x.

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

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

y = arctan 1.1111111

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

y = cot⁻¹ (―0.92170128)

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

y = sec⁻¹ (―1.2871684)

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

y = 5 cos x , for x in [0, π]

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

y = tan⁻¹ (―√3)

Graph each inverse circular function by hand.

y = arcsec [(1/2)x]

Evaluate each expression without using a calculator.

tan (arccos 3/4)

Evaluate each expression without using a calculator.

cos (tan⁻¹ (-2))

Evaluate each expression without using a calculator.

sin (2 tan⁻¹ (12/5))

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Evaluate each expression without using a calculator.

cos (2 arctan (4/3))

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Evaluate each expression without using a calculator.

sin (2 cos⁻¹ (1/5))

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Evaluate each expression without using a calculator.

sec (sec⁻¹ 2)

Evaluate each expression without using a calculator.

cos (tan⁻¹ (5/12) - tan⁻¹ (3/4))

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Evaluate each expression without using a calculator.

sin (sin⁻¹ 1/2 + tan⁻¹ (-3))

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

y = 3 tan 2x , for x in [―π/4, π/4]

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

y = cos⁻¹ (―√2/2)

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

cos (tan⁻¹ 0.5)

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

tan (arcsin 0.12251014)

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

sin (arccos u)

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

cos (arcsin u)

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

sin (2 sec⁻¹ u/2)

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

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

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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)

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

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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. _ csc⁻¹ (− 2√3/3)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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)

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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)

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)

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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)

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

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

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

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

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

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

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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)

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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)

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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)

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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)

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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)

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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

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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)

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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

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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

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