Express each function as a trigonometric function of x. See Example 5.

cos 4x

검증된 단계별 안내

Recognize that the problem asks to express \( \cos 4x \) as a trigonometric function of \( x \), which typically involves using multiple-angle formulas or power-reduction formulas.

Recall the double-angle formula for cosine: \( \cos 2\theta = 2\cos^2 \theta - 1 \). This formula can be applied repeatedly to express \( \cos 4x \) in terms of \( \cos x \).

First, express \( \cos 4x \) as \( \cos(2 \cdot 2x) \) and apply the double-angle formula: \( \cos 4x = 2\cos^2 2x - 1 \).

Next, express \( \cos 2x \) in terms of \( \cos x \) using the double-angle formula again: \( \cos 2x = 2\cos^2 x - 1 \). Substitute this into the previous expression.

Combine the expressions to write \( \cos 4x \) fully in terms of \( \cos x \), resulting in a polynomial expression involving powers of \( \cos x \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Multiple-Angle Trigonometric Functions

Multiple-angle functions involve trigonometric expressions where the angle is multiplied by an integer, such as cos(4x). Understanding how to express these in terms of single angles or simpler functions is essential for simplification and solving equations.

Double-Angle and Power-Reduction Formulas

Double-angle formulas, like cos(2x) = 2cos²x - 1, help break down functions of multiple angles into expressions involving single angles. Power-reduction formulas further simplify powers of sine and cosine, aiding in expressing cos(4x) in terms of cos x.

Trigonometric Identities and Algebraic Manipulation

Using fundamental identities and algebraic techniques allows rewriting complex trigonometric functions into simpler forms. Mastery of identities like cos(A+B) and cos(2x) is crucial for expressing cos(4x) as a function of x.

Express each function as a trigonometric function of x. See Example 5.cos 4x