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.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)

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)

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 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)