Solve each equation for exact solutions.
arccos x + 2 arcsin √3/2 = π

Verified step by step guidance

1. Recognize that the equation is \(\arccos x + 2 \arcsin \frac{\sqrt{3}}{2} = \pi\). Our goal is to solve for \(x\).
2. Evaluate the known inverse trigonometric value: find \(\arcsin \frac{\sqrt{3}}{2}\). Recall that \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\), so \(\arcsin \frac{\sqrt{3}}{2} = \frac{\pi}{3}\).
3. Substitute this value back into the equation: \(\arccos x + 2 \times \frac{\pi}{3} = \pi\), which simplifies to \(\arccos x + \frac{2\pi}{3} = \pi\).
4. Isolate \(\arccos x\) by subtracting \(\frac{2\pi}{3}\) from both sides: \(\arccos x = \pi - \frac{2\pi}{3} = \frac{\pi}{3}\).
5. Use the definition of \(\arccos\) to solve for \(x\): since \(\arccos x = \frac{\pi}{3}\), then \(x = \cos \frac{\pi}{3}\). Recall that \(\cos \frac{\pi}{3} = \frac{1}{2}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccos and arcsin, return the angle whose trigonometric ratio equals a given value. For example, arccos x gives the angle whose cosine is x, and arcsin y gives the angle whose sine is y. Understanding their ranges and outputs is essential for solving equations involving these functions.

Exact Values of Trigonometric Functions

Certain trigonometric values correspond to well-known angles and can be expressed exactly using fractions of π and square roots. For instance, sin(π/3) = √3/2. Recognizing these exact values helps simplify expressions and solve equations without approximations.

Trigonometric Equation Solving Techniques

Solving trigonometric equations often involves isolating inverse functions, using identities, and applying domain restrictions. In this problem, combining inverse cosine and sine functions and using their properties allows finding exact solutions for x within the valid domain.