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Chapter 4: Problem 69

Write an algebraic expression that is equivalent to the given expression. $$\sin (\arccos x)$$

Short Answer

Expert verified
\(\sin (\arccos x) = \sqrt{1 - x^2}\)

Step by step solution

01

Understand the properties of inverse trigonometric functions

One of the properties of inverse trigonometric functions is that \( \cos (\arccos x) = x \) for \( -1 \leq x \leq 1 \). The same property for sine can be expressed as \( \sin (\arcsin y) = y \), where \( -1 \leq y \leq 1 \). In the given expression, \( \arccos x \) is the angle \( y \) that satisfies these properties.
02

Apply the Pythagorean identity

By applying the Pythagorean identity of trigonometric functions, \( \sin^2 y + \cos^2 y = 1 \), to the function given, knowing that \( \cos y = x \), we can isolate the sine term: \( \sin^2 y = 1 - \cos^2 y \). Taking the square root gives \( \sin y = \sqrt{1 - \cos^2 y} \).
03

Substitute and simplify

Use the result from Step 2 to substitute back into the given expression. We know that \( \cos y \) is equivalent to \( x \), so the new expression becomes \( \sin y = \sqrt{1 - x^2} \). But we find that \( \sin (\arccos x) = \sqrt{1 - x^2} \).

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