Solve each equation for exact solutions.
4/3 cos⁻¹ x/4 = π

Verified step by step guidance

1. Start by isolating the inverse cosine expression. Multiply both sides of the equation by \( \frac{3}{4} \) to get \( \cos^{-1}\left( \frac{x}{4} \right) = \frac{3}{4} \pi \).
2. Recall that \( \cos^{-1}(y) = \theta \) means \( \cos(\theta) = y \). So rewrite the equation as \( \cos\left( \frac{3}{4} \pi \right) = \frac{x}{4} \).
3. Evaluate \( \cos\left( \frac{3}{4} \pi \right) \) using the unit circle or known cosine values for special angles. Remember that \( \frac{3}{4} \pi = 135^\circ \) and cosine is negative in the second quadrant.
4. Set \( \frac{x}{4} \) equal to the value found in step 3, then solve for \( x \) by multiplying both sides by 4.
5. Consider the domain of \( \cos^{-1} \), which is \( [-1,1] \), to verify that the solution for \( x \) is valid. If necessary, check for any additional solutions within the domain.

Key Concepts

Inverse Cosine Function (cos⁻¹ or arccos)

The inverse cosine function returns the angle whose cosine is a given number. Its output range is from 0 to π radians. Understanding this function is essential to isolate the variable inside the cosine and solve for exact angle values.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and then applying inverse functions to find the angle. It also requires considering the domain and range of the inverse function to determine all possible solutions.

Manipulating Algebraic Expressions

Algebraic manipulation is necessary to isolate the variable inside the inverse cosine function. This includes multiplying or dividing both sides of the equation and simplifying fractions to express the variable clearly for substitution or evaluation.