Problem 134 - Precalculus with Limits

Question

In Exercises 131 - 134, write the trigonometric expression as an algebraic expression. \( \sin\left(2 \arccos x\right) \)

Step-by-Step Solution

Answer

The trigonometric expression \( \sin\left(2 \arccos x\right) \) can be written as an algebraic expression \( = 2x\sqrt{1-x^2} \).

Step 1: Start With the Given Expression

The given expression is \( \sin\left(2 \arccos x\right) \). The goal is to write this as an algebraic expression.

Step 2: Use the Trigonometric Identity

We can use the double-angle formula which states that \( \sin(2θ) = 2\sin(θ)\cos(θ)\). Substituting the given \(θ = \arccos x\) into the formula, the expression becomes \( 2\sin(\arccos x)\cos(\arccos x)\).

Step 3: Transform to Algebraic Form

For a triangle with an angle \( θ = \arccos x \), the values of sine and cosine can be expressed in terms of \( x \). The cosine is by definition \( x \), and the sine is \( \sqrt{1-x^2} \) since for any angle \( θ \), \( \sin^2(θ) + \cos^2(θ) = 1 \). So, we can rewrite the expression as \( = 2x\sqrt{1-x^2} \).

Key Concepts

Double-angle Formula

The double-angle formula is a useful trigonometric identity that allows you to express trigonometric functions of double angles in terms of single angles. The formula for sine in this context is given by:

• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)

This formula is particularly handy when dealing with expressions like \( \sin(2 \text{arccos}(x)) \) where you want to convert a trigonometric expression involving an inverse function into a more algebraic form. By substituting \( \theta = \text{arccos}(x) \) into the double-angle formula, we simplify the problem into finding the sine and cosine of \( \text{arccos}(x) \).
The formula essentially helps break down complex angle calculations into simpler, more manageable parts, through the clever application of sine and cosine identities. This is a foundational tool in trigonometry, enabling transformations and simplifications needed across various mathematical and real-world applications.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to work backward from the value of a trigonometric function to determine the angle that corresponds to that value. Specifically, when you see an expression like \( \text{arccos}(x) \), it means you are looking for an angle \( \theta \) such that \( \cos(\theta) = x \).
This means, substituting this into the double-angle formula, \( \theta = \text{arccos}(x) \) gives \( \cos(\text{arccos}(x)) = x \), which is intuitive since arccos is the function that "undoes" cosine.
To find the sine of \( \text{arccos}(x) \), we use the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\). Knowing \( \cos(\theta) = x \), we solve for \( \sin(\theta) \) by:

• \( \sin(\theta) = \sqrt{1-x^2} \)

Hence, inverse trigonometric functions are pivotal for converting between trigonometric identities and their respective angles, aiding in complex problem solving.

Algebraic Expressions

Writing trigonometric expressions as algebraic expressions can simplify the computation and manipulation of these functions. In our solution, converting \( \sin(2 \text{arccos}(x)) \) to an algebraic expression involves utilizing identities to transform the trigonometric terms into functions of \( x \).
By substituting \( \sin(\text{arccos}(x)) = \sqrt{1-x^2} \) and \( \cos(\text{arccos}(x)) = x \) into the double-angle formula, we achieve:

• \( 2x\sqrt{1-x^2} \)

This results in an expression that is free from angle terms, purely in algebraic form, making it easier to analyze or compute for specific values of \( x \).
Algebraic expressions provide clarity and are fundamental in calculus and other higher mathematics, where simplifications like this enable deeper insights and straightforward computations.