5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Solve each equation for exact solutions.

-4 arcsin x = π

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)

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

y = ― 2 cos 5x , for x in [0, π/5]

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

y = cos (x + 3) , for x in [―3, π―3]

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

y = tan⁻¹ (―√3)

Evaluate each expression without using a calculator.

cos (arccos (-1))

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

y = csc⁻¹ (―2)

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

y = √2 + 3 sec 2x, for x in [0, π/4) ⋃ (π/4, π/2]

Find the exact value of each expression. sin⁻¹ (- 1/2)

Evaluate each expression without using a calculator.

tan (arcsin (3/5) + arccos (5/7))

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)

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

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

y = arctan 1.1111111

Solve each equation for exact solutions.

sin⁻¹ x - 4 tan⁻¹ (-1) = 2π

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

y = 6 cos x/4 , for x in [0, 4π]