# Chapter 7 - Trigonometric Identities and Equations - 7.7 Equations Involving Inverse Trigonometric Functions - 7.7 Exercises: 10

$x=2 \displaystyle \arcsin\frac{y}{3}$

#### Work Step by Step

The goal is to isolate x. Divide with 3... $\displaystyle \frac{y}{3}=\sin\frac{x}{2}$ $\qquad$ ...use the definition of arcsin $\displaystyle \frac{x}{2}=\arcsin\frac{y}{3}$ ($\displaystyle \frac{x}{2}$ is the number from $[-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}]$ such that $\displaystyle \sin\frac{y}{3}=\frac{x}{2}$) (...If $\displaystyle \frac{x}{2}\in(-\frac{\pi}{2}, \displaystyle \frac{\pi}{2})$, then $x\in(-\pi, \pi)$...) ... isolate the x by multiplying with 2 ... $x=2 \displaystyle \arcsin\frac{y}{3}$

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