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

Verified step by step guidance

1. Step 1: Start by isolating the arcsin function. Divide both sides of the equation by -4 to get \( \arcsin(x) = -\frac{\pi}{4} \).
2. Step 2: Recall that \( \arcsin(x) \) is the inverse function of \( \sin(x) \), which means \( x = \sin(-\frac{\pi}{4}) \).
3. Step 3: Use the property of sine that \( \sin(-\theta) = -\sin(\theta) \). Therefore, \( \sin(-\frac{\pi}{4}) = -\sin(\frac{\pi}{4}) \).
4. Step 4: Calculate \( \sin(\frac{\pi}{4}) \), which is a well-known angle in trigonometry. \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).
5. Step 5: Substitute back to find \( x \). Thus, \( x = -\frac{\sqrt{2}}{2} \).

Key Concepts

Arcsine Function

The arcsine function, denoted as arcsin or sin⁻¹, is the inverse of the sine function. It takes a value from the range [-1, 1] and returns an angle in the range [-π/2, π/2]. Understanding this function is crucial for solving equations involving arcsin, as it allows us to find the angle whose sine is a given value.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable that satisfy the equation. This often requires using inverse trigonometric functions, identities, and understanding the periodic nature of trigonometric functions. In this case, we need to isolate x and determine its value based on the given equation involving arcsin.

Exact Solutions

Exact solutions in trigonometry refer to the precise values of angles or lengths, often expressed in terms of π or radicals, rather than decimal approximations. When solving equations like -4 arcsin x = π, finding the exact solution means determining the exact value of x that satisfies the equation without resorting to numerical methods.