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 function. Multiply both sides of the equation by \( \frac{3}{4} \) to get \( \cos^{-1} \left( \frac{x}{4} \right) = \frac{3\pi}{4} \).
2. Apply the cosine function to both sides to eliminate the inverse cosine, resulting in \( \frac{x}{4} = \cos \left( \frac{3\pi}{4} \right) \).
3. Recall that \( \cos \left( \frac{3\pi}{4} \right) \) is a known value. Since \( \frac{3\pi}{4} \) is in the second quadrant, where cosine is negative, \( \cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2} \).
4. Substitute \( -\frac{\sqrt{2}}{2} \) for \( \cos \left( \frac{3\pi}{4} \right) \) in the equation, giving \( \frac{x}{4} = -\frac{\sqrt{2}}{2} \).
5. Solve for \( x \) by multiplying both sides by 4, resulting in \( x = -2\sqrt{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹ (arccosine), are used to find the angle whose cosine is a given value. In this context, cos⁻¹(x/4) represents the angle whose cosine equals x/4. Understanding how to manipulate these functions is essential for solving equations involving them.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the variable and finding all possible angles that satisfy the equation. In this case, the equation includes a coefficient (4/3) and a constant (π), which requires careful algebraic manipulation to isolate x and determine its exact values.

Understanding Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. In this problem, π radians corresponds to 180 degrees. Recognizing the relationship between radians and degrees is crucial for interpreting the solutions correctly and ensuring they fall within the appropriate range for the cosine function.