Problem 120 - Algebra and Trigonometry

Question

Solve \(y=2 \sin ^{-1}(x-5)\) for \(x\) in terms of \(y\)

Step-by-Step Solution

Answer

The value of \(x\) in terms of \(y\) is \(x = \sin\left(\frac{y}{2}\right)+5\).

Step 1: Rewrite the equation

First, isolate the inverse sine function on one side of the equation, leaving \(y/2\) on the other side. It will look like this: \(\sin^{-1}(x-5) = \frac{y}{2}\).

Step 2: Apply the Sine Function

To remove the inverse sine function, apply the sine function on both sides of the equation. This will give \(\sin(\sin^{-1}(x-5)) = \sin\left(\frac{y}{2}\right)\). The sine and its inverse cancel out, simplifying the equation to \(x-5 = \sin\left(\frac{y}{2}\right)\).

Step 3: Solve for x

Lastly, solve for x by adding 5 on both sides of the equation. This gives \(x = \sin\left(\frac{y}{2}\right)+5\).