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Chapter 4: Problem 74

Write an algebraic expression that is equivalent to the given expression. $$\cos \left(\arcsin \frac{x-h}{r}\right)$$

Short Answer

Expert verified
The equivalent algebraic form of the expression \(\cos \left(\arcsin \left(\frac{x-h}{r}\right)\right)\) is \(1 - \frac{(x-h)^2}{r^2}\).

Step by step solution

01

Rewrite the expression

We start with the identity \(\cos^2\theta + \sin^2\theta = 1\) and isolate \(\cos\theta\) by subtracting \(\sin^2\theta\) from both sides. We get \(\cos\theta = \sqrt{1 - \sin^2\theta}\). This allows us to write \(\cos(\arcsin(x))\) as \(\sqrt{1 - x^2}\). Thus, the initial expression is rewritten as: \(\sqrt{1 - \left(\frac{x-h}{r}\right)^2}\).
02

Simplify the expression

Now we need to simplify our new expression. We square the fraction inside the square root to get \(\sqrt{1 - \frac{(x-h)^2}{r^2}}\). Now we need to get rid of the square root.
03

Remove the square root

As we are looking for an equivalent algebraic form, we can remove the square root if we square the whole expression. To do this we square both sides, which yields: \(1 - \frac{(x-h)^2}{r^2}\).

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