5. Inverse Trigonometric Functions and Basic Trigonometric Equations

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

y = cos⁻¹ (―0.32647891)

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

y = sin⁻¹ √3

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

y = sin⁻¹ 0

Use a calculator to approximate each value in decimal degrees.

θ = arcsec 3.4723155

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

y = cos⁻¹ (―√2/2)

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)

Solve each equation for exact solutions.

4/3 cos⁻¹ x/4 = π

Use a calculator to approximate each value in decimal degrees.

θ = sin⁻¹ (-0.13349122)

Which one of the following equations has solution 0?

a. arctan 1 = x

b. arccos 0 = x

c. arcsin 0 = x

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)

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

The point (π/4, 1) lies on the graph of y = tan x. Therefore, the point _______ lies on the graph of y = tan⁻¹ x.

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

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

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

y = arcsin 0.92837781