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Chapter 5: Problem 131

Write the trigonometric expression as an algebraic expression. $$\sin (2 \arcsin x)$$

Short Answer

Expert verified
The trigonometric expression \(\sin (2 \arcsin x)\) can be rewritten as an algebraic expression: \(2x \sqrt{1 - x^2}\).

Step by step solution

01

Recall the double-angle sine formula

The formula for the sine of a double angle is given by \(\sin(2a) = 2\sin(a)\cos(a)\). This formula will be required to simplify the given expression.
02

Apply the double-angle sine formula

Apply this double-angle sine identity to the expression \(\sin (2 \arcsin x)\). It becomes \(2 \sin (\arcsin x) \cos (\arcsin x)\).
03

Simplify the expression

We know that \(\sin(\arcsin x) = x\), therefore, the expression simplifies to \(2x \cos (\arcsin x)\).
04

Replace the cosine expression

Recall that for any angle \(a\), we have \(\cos^2(a) = 1 - \sin^2(a)\). Hence, we replace \(\cos(\arcsin x)\) with \(\sqrt{1 - x^2}\), given that \(x\) is in the domain of the arcsine function, which is \([-1, 1]\). This then further simplifies the trigonometric expression to \(2x \sqrt{1 - x^2}\).

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