Solve each equation for exact solutions.
sin⁻¹ x - 4 tan⁻¹ (-1) = 2π

Verified step by step guidance

1. Recognize that \( \sin^{-1}(x) \) represents the angle whose sine is \( x \), and \( \tan^{-1}(-1) \) represents the angle whose tangent is \( -1 \).
2. Recall that \( \tan^{-1}(-1) \) is an angle in the fourth quadrant where the tangent is \( -1 \), which is \( -\frac{\pi}{4} \).
3. Substitute \( \tan^{-1}(-1) = -\frac{\pi}{4} \) into the equation: \( \sin^{-1}(x) - 4(-\frac{\pi}{4}) = 2\pi \).
4. Simplify the equation: \( \sin^{-1}(x) + \pi = 2\pi \).
5. Solve for \( \sin^{-1}(x) \) by isolating it: \( \sin^{-1}(x) = 2\pi - \pi \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹(x) and tan⁻¹(x), are used to find angles when given a ratio. For example, sin⁻¹(x) gives the angle whose sine is x, while tan⁻¹(x) gives the angle whose tangent is x. Understanding how to manipulate these functions is crucial for solving equations involving them.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, angle sum and difference identities, and double angle formulas. These identities can simplify complex equations and help in finding exact solutions.

Periodic Nature of Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For instance, the sine and tangent functions have periods of 2π and π, respectively. This periodicity is essential when solving equations, as it allows for the identification of all possible solutions within specified intervals.