Problem 116 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the trigonometric expression as an algebraic expression. $$\cos (2 \arccos x)$$

Step-by-Step Solution

Answer

The trigonometric expression \( \cos (2 \arccos x) \) can be written as an algebraic expression as \( 2x^2 - 1 \).

Step 1: Identify the double-angle identity

The relevant identity is \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). Here, \( \theta \) is \( \arccos x \).

Step 2: Replace sin in the formula

Since \( \arccos x\) is the angle in our formula, we need to express \( \sin(\arccos x) \) using an algebraic expression. We know that \( \sin^2(\theta) + \cos^2(\theta) = 1 \), so \( \sin(\arccos x) \) can be expressed as \( \sqrt{1 - x^2} \). Therefore, \( \sin^2(\arccos x) = 1 - x^2 \).

Step 3: Substitute into the original formula

Substitute the value of \( \sin^2(\arccos x) \) into the formula, we will get \( \cos (2 \arccos x) = 1 - 2(1 - x^2) \). Simplifying this gives: \( 2x^2 - 1 \).

Key Concepts

Double-Angle Identities

Double-angle identities are crucial in transforming and simplifying trigonometric expressions. They are relationships that involve trigonometric functions of double angles, such as \( \cos(2\theta) \) or \( \sin(2\theta) \).

In this exercise, we use the identity \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). This helps express \( \cos(2\theta) \) in terms of another trigonometric function, \( \sin \). These identities are especially useful in solving trigonometric equations and converting them to algebraic forms.

• Identities reduce complexity by simplifying expressions.
• They allow replacement of higher powers and angles with simpler terms.

Learning these identities improves problem-solving skills in mathematics.

Algebraic Expressions

Algebraic expressions involve numbers and variables combined using operations like addition, subtraction, and multiplication. They are often used to represent regular arithmetic operations algebraically.

In this solution, we transform the trigonometric function \( \cos(2 \arccos x) \) into an algebraic expression. This involves substituting \( \sin^2(\arccos x) \) with a square root expression derived from the Pythagorean identity.

• The expression \( \sqrt{1 - x^2} \) comes from \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Replacing trigonometric components helps simplify the final expression to \( 2x^2 - 1 \).

Through these conversions, complex trigonometric equations become more manageable.

Inverse Trigonometric Functions

Inverse trigonometric functions allow determination of angles given numerical values of trigonometric functions. For example, \( \arccos x \) is the angle whose cosine is \( x \).

In our problem, the process starts with expressing \( \arccos x \) and converting it through inverse functions, which connect angles to their trigonometric values. Understanding \( \arccos x \) helps transform \( \cos(2 \arccos x) \) by using the identity formula.

• These functions allow backtracking from a ratio to an angle.
• This conversion plays a critical role in achieving the final algebraic form.

Knowing how to manipulate inverse trigonometric functions widens the scope to solve more complex trigonometric problems and improves mathematical fluency.