5. Inverse Trigonometric Functions and Basic Trigonometric Equations

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)

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

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

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 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)

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)

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

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

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

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. sin (tan⁻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)