Solve each equation for exact solutions.
-4 arcsin x = π

Verified step by step guidance

1. Start with the given equation: \(-4 \arcsin x = \pi\).
2. Isolate \(\arcsin x\) by dividing both sides of the equation by \(-4\): \(\arcsin x = \frac{\pi}{-4} = -\frac{\pi}{4}\).
3. Recall that \(\arcsin x\) is the inverse sine function, which means \(x = \sin(\arcsin x)\), so \(x = \sin\left(-\frac{\pi}{4}\right)\).
4. Use the property of sine for negative angles: \(\sin(-\theta) = -\sin(\theta)\), so \(x = -\sin\left(\frac{\pi}{4}\right)\).
5. Evaluate \(\sin\left(\frac{\pi}{4}\right)\) using known exact values, then write the exact value for \(x\).

Key Concepts

Inverse Sine Function (arcsin)

The inverse sine function, arcsin, returns the angle whose sine is a given number. Its domain is limited to [-1, 1], and its range is [-π/2, π/2]. Understanding arcsin is essential to solve equations involving it by isolating the variable inside the function.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and then applying inverse functions to find the angle. It is important to consider the domain and range restrictions of the inverse functions to find all valid solutions.

Exact Values of π in Trigonometry

Trigonometric equations often involve multiples of π, representing angles in radians. Recognizing and manipulating these exact values helps in expressing solutions precisely, rather than as decimal approximations.