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


cos 3x

Verified step by step guidance

1. Recognize that the problem asks to express \( \cos 3x \) as a trigonometric function of \( x \). This typically involves using trigonometric identities to rewrite the expression in terms of \( \cos x \) and/or \( \sin x \).
2. Recall the triple-angle identity for cosine: \( \cos 3x = 4\cos^{3}x - 3\cos x \). This identity allows you to express \( \cos 3x \) purely in terms of \( \cos x \).
3. Alternatively, you can use the angle addition formula repeatedly: \( \cos(3x) = \cos(2x + x) = \cos 2x \cos x - \sin 2x \sin x \).
4. Use the double-angle formulas to express \( \cos 2x \) and \( \sin 2x \) in terms of \( \cos x \) and \( \sin x \): \( \cos 2x = 2\cos^{2}x - 1 \) and \( \sin 2x = 2\sin x \cos x \).
5. Substitute these back into the expression and simplify to get \( \cos 3x \) entirely in terms of \( \cos x \) and \( \sin x \), or use the triple-angle identity directly for a more straightforward expression.

Key Concepts

Multiple-Angle Formulas

Multiple-angle formulas express trigonometric functions of multiples of an angle, such as cos(3x), in terms of functions of the single angle x. For example, cos(3x) can be expanded using the formula cos(3x) = 4cos^3(x) - 3cos(x), which helps rewrite the function in simpler terms.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They allow the transformation and simplification of expressions, such as converting cos(3x) into a polynomial in cos(x), facilitating easier computation or integration.

Function Composition and Angle Multiplication

Understanding how trigonometric functions behave under angle multiplication is essential. This involves recognizing that functions like cos(3x) represent the cosine of an angle three times larger than x, and using identities to relate this to cos(x) helps in expressing and simplifying such functions.