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Chapter 5: Problem 59

Evaluate expression. $$\cos \left(\sin ^{-1} x\right)$$

Short Answer

Expert verified
\(\cos(\sin^{-1}x) = \sqrt{1-x^2}\), for -1 鈮� x 鈮� 1.

Step by step solution

01

Recognize the Trigonometric Identity

From the Pythagorean trigonometric identity, we know that \(\sin^2(胃) + \cos^2(胃) = 1\). We can rearrange the identity to solve for \(\cos(胃)\), giving \(\cos(胃) = \sqrt{1-\sin^2(胃)}\). We'll use this rearranged identity to make the substitution in the next step.
02

Substitute the Given Into the Identity

Substitute \(\sin^{-1}x\) for \(胃\) into the rearranged identity. This gives a new expression \(\sqrt{1-\sin^2(\sin^{-1}x)}\). However, \(\sin(\sin^{-1}x)\) is simple x, so the expression becomes \(\sqrt{1-x^2}\).
03

Finalize the Expression

Simplifying the expression, we get that \(\cos(\sin^{-1}x)\) equals to \(\sqrt{1-x^2}\). This is only true when -1 鈮� x 鈮� 1 as the arcsin function is defined in this range since sine of any angle is between -1 and 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Trigonometric identities are fundamental relationships between the trigonometric functions, like sine, cosine, and tangent. They are used to simplify complex expressions and can help you solve trigonometric equations. The most important identities are the Pythagorean identities, angle sum and difference identities, and double angle formulas.
  • These identities are core tools for transforming one trigonometric expression into another.
  • They allow for the conversion and simplification of trigonometric forms, which is crucial in solving equations.
Understanding and using these identities lets you link different trigonometric expressions together. This can make solving many problems much easier and quicker.
Pythagorean Identity
The Pythagorean identity is one of the most vital trigonometric relationships. It states that \( \sin^2(\theta) + \cos^2(\theta) = 1 \) for any angle \( \theta \). This identity arises from the Pythagorean theorem applied to the unit circle, where the radius is always 1.
  • Using this identity, you can express \( \cos(\theta) \) as \( \sqrt{1-\sin^2(\theta)} \).
  • It's a crucial tool in simplifying trigonometric expressions, allowing substitution for missing trigonometric functions.
For example, in the problem of finding \( \cos(\sin^{-1}(x)) \), knowing \( \sin^{-1}(x) \) is a specific angle \( \theta \), allows you to find the cosine using the adjusted identity: \( \cos(\theta) = \sqrt{1 - \sin^2(\theta)} \). This gives you \( \cos(\sin^{-1}(x)) = \sqrt{1 - x^2} \) when you replace \( \sin(\theta) \) with \( x \).
Arcsine Function
The arcsine function, denoted as \( \sin^{-1}(x) \), is the inverse of the sine function. It yields an angle whose sine is \( x \). Its range is limited to ensure it is a true function, typically from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
  • The arcsine function is essential when you want to retrieve the angle from its sine value.
  • The result of arcsine is always an angle in the principal range, making calculations predictable.
For instance, in evaluating \( \cos(\sin^{-1}(x)) \), you first determine what angle corresponds to \( \sin^{-1}(x) \), and then find the cosine of that angle. This equates to finding \( \sqrt{1-x^2} \) because it aligns with the Pythagorean identity and the properties of arcsine.

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