Problem 46 - Precalculus Mathematics for Calculus

Question

Write the given expression as an algebraic expression in \(x\). $$\cos \left(2 \sin ^{-1} x\right)$$

Step-by-Step Solution

Answer

The expression is \(1 - 2x^2\).

Step 1: Understanding the Expression

The given expression is \( \cos(2 \sin^{-1} x) \). We aim to express this entirely in terms of \(x\). Here, \( \sin^{-1} x \) is the inverse sine function, meaning that \( y = \sin^{-1} x \) such that \( x = \sin y \) and \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \).

Step 2: Use Double Angle Formula for Cosine

Apply the double angle formula for cosine: \( \cos 2\theta = 1 - 2\sin^2 \theta \). If we take \( \theta = \sin^{-1} x \), then \( \cos(2 \sin^{-1} x) = 1 - 2\sin^2(\sin^{-1} x) \).

Step 3: Simplify \( \sin(\sin^{-1} x) \)

Since \( y = \sin^{-1}x \), it follows that \( \sin y = x \). Therefore, \( \sin(\sin^{-1} x) = x \). Thus, \( \sin^2(\sin^{-1} x) = x^2 \).

Step 4: Write the Final Expression

Substitute \( \sin^2(\sin^{-1} x) = x^2 \) into the expression from Step 2: \( \cos(2 \sin^{-1} x) = 1 - 2x^2 \). This is the algebraic expression in terms of \(x\).

Key Concepts

Double Angle Formula

The double angle formula is a useful identity in trigonometry that helps simplify expressions involving angles that are double a certain angle. Specifically, for cosine, the double angle formula is given by:

• \( \cos 2\theta = 1 - 2\sin^2 \theta \)
• Alternatively, it can also be represented as \( \cos 2\theta = 2\cos^2 \theta - 1 \)
• And, \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)

Each version of the formula is handy depending on what information you have or need.For the exercise provided: If \( \theta = \sin^{-1} x \), the double angle formula for cosine converts the expression \( \cos(2 \sin^{-1} x) \) into a more manageable form using known values of \( \sin \theta \), which equals \( x \). This simplification is crucial for transforming trigonometric expressions into algebraic ones.

Inverse Sine Function

The inverse sine function, noted as \( \sin^{-1} x \) or arcsine, helps us find the angle whose sine is \( x \).

• It is defined such that \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \) and \( y = \sin^{-1} x \) when \( x = \sin y \).
• This means that if \( \sin y = x \), then \( y \), which is the angle, can be written as \( \sin^{-1} x \).

Understanding the properties of inverse sine is essential for manipulating expressions where trigonometric functions need to be expressed algebraically. In our context, knowing that \( \sin(\sin^{-1} x) = x \) helps plug \( x \) back into expressions, facilitating transformations and simplifications into algebraic form.

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. Transforming trigonometric expressions into algebraic expressions is a common exercise in pre-calculus to prepare for calculus applications.

• The task is to express the function \( \cos(2 \sin^{-1} x) \) entirely in terms of the variable \( x \).
• Using the trigonometric identity \( \cos 2\theta = 1 - 2\sin^2 \theta \), the expression becomes \( 1 - 2x^2 \), since \( \sin(\sin^{-1} x) = x \).

This transformation involves understanding and applying known identities and simplifications to eliminate trigonometric terms, resulting in a pure algebraic expression like the delivered answer \( 1 - 2x^2 \). This form is often more useful for further mathematical manipulation, outlining the bridge between trigonometry and algebra.