Problem 33 - Precalculus Mathematics for Calculus

Question

Rewrite the expression as an algebraic expression in \(x\). $$\cos \left(\sin ^{-1} x\right)$$

Step-by-Step Solution

Answer

\( \sqrt{1 - x^2} \)

Step 1: Identify the Function and Inverse Function Relationship

Given the expression \( \cos(\sin^{-1} x) \), recognize that \( \sin^{-1} x \) is the inverse sine function, which means if \( y = \sin^{-1} x \), then \( \sin(y) = x \). Our goal is to find \( \cos(y) \).

Step 2: Use Pythagorean Identity

From the relationship \( \sin(y) = x \), apply the Pythagorean identity: \( \sin^2(y) + \cos^2(y) = 1 \). Substitute \( \sin(y) = x \) into the identity to obtain \( x^2 + \cos^2(y) = 1 \), leading to \( \cos^2(y) = 1 - x^2 \).

Step 3: Solve for \( \cos(y) \)

Calculate \( \cos(y) \) as \( \cos(y) = \sqrt{1 - x^2} \). Since \( \sin^{-1} x \) outputs angles in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), where cosine is non-negative, we take the positive root.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are special functions that reverse the effect of the usual trigonometric functions. They often appear in problems where you start with the ratio of two sides of a right triangle and need to find the angle. In our original problem, we begin with \( \sin^{-1} x \), which is the inverse sine function. This function tells us what angle \( y \) such that \( \sin(y) = x \).

• Inverse Sine Function \( \sin^{-1} x \): Outputs the angle whose sine is \( x \).
• Has a range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\), meaning it only gives angles in the first and fourth quadrants where sine is either positive or negative, respectively.

This understanding is crucial because when we explore \( \cos(\sin^{-1} x) \), it implies finding the cosine of the angle that has a sine of \( x \). Realizing the role and behavior of inverse trigonometric functions helps us smoothly transform trigonometric expressions into algebraic ones.

Pythagorean Identity

The Pythagorean identity is an indispensable tool in trigonometry. It states that for any angle \( y \), the relationship \( \sin^2(y) + \cos^2(y) = 1 \) holds true. It is derived from the Pythagorean theorem and applies to any angle in terms of sine and cosine.This identity helps us find unknown sides or angles in triangle-related problems. In the original exercise, knowing \( \sin(y) = x \), we use the identity to relate sine and cosine:

• Substitute \( \sin(y) = x \) into the identity: \( \sin^2(y) + \cos^2(y) = 1 \) becomes \( x^2 + \cos^2(y) = 1 \).
• Solve for \( \cos^2(y) \), yielding \( \cos^2(y) = 1 - x^2 \).

This step is vital as it allows us to express the cosine in terms of \( x \) alone, which is our goal when converting the original trigonometric expression to an algebraic one.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and arithmetic operations. The task in our exercise is to convert the trigonometric expression \( \cos(\sin^{-1} x) \) into a purely algebraic form. Understanding how to manipulate these expressions is key to solving complex mathematical problems.After applying the Pythagorean identity, we find that \( \cos(y) \) can be expressed as \( \sqrt{1 - x^2} \). Since the inverse sine function restricts the angle range to where cosine is positive, we take the positive square root:

• The final algebraic expression is \( \cos(y) = \sqrt{1 - x^2} \).

This transformation from a trigonometric form to an algebraic expression enables easier calculations and better understanding of the relationships between different mathematical components. It's a fundamental skill useful in various applied mathematics fields and real-world problem-solving.