5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Solve each equation for x.

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

Which one of the following equations has solution 3π/4

a. arctan 1 = x

b. arcsin √2/2 = x

c. arccos (―√2 /2) = x

Evaluate the expression.

$\(\sin\)^{-1}1$

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

y = sin x ―2 , for x in [―π/2. π/2]

Decide whether each statement is true or false. If false, explain why.

The tangent and secant functions are undefined for the same values.

Solve each equation for exact solutions.

cos⁻¹ x + tan⁻¹ x = π/2

Evaluate each expression without using a calculator.

arccos (cos (3π/4))

Evaluate the expression.

$\(\tan\)^{-1}\(\left\)(-\(\frac{\sqrt3}{3}\]\right\))$

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

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

y = ―4 + 2 sin x , for x in [―π/2. π/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. sin (tan⁻¹ x)

Solve each equation for exact solutions.

arccos x + 2 arcsin √3/2 = π

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)

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

Find the degree measure of θ if it exists. Do not use a calculator.

θ = arccos (-1/2)