Let f(x) = sin x, g(x) = cos x, and h(x) = 2x. Find the exact value of each expression. Do not use a calculator. (h o g) (17𝜋/3)

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Understand that the notation \( (h \circ g)(x) \) means the composition of functions \( h \) and \( g \), which is \( h(g(x)) \). So, you first apply \( g \) to \( x \), then apply \( h \) to the result.

Identify the given functions: \( g(x) = \cos x \) and \( h(x) = 2x \). Therefore, \( (h \circ g)(x) = h(g(x)) = 2 \cdot g(x) = 2 \cos x \).

Substitute \( x = \frac{17\pi}{3} \) into the expression: \( (h \circ g)\left( \frac{17\pi}{3} \right) = 2 \cos \left( \frac{17\pi}{3} \right) \).

Simplify the angle \( \frac{17\pi}{3} \) by reducing it within the standard interval \( [0, 2\pi) \) using the periodicity of cosine, which has period \( 2\pi \). Calculate \( \frac{17\pi}{3} - 2\pi \times n \) for an integer \( n \) to find an equivalent angle between 0 and \( 2\pi \).

Evaluate \( \cos \) of the simplified angle using known exact values of cosine for standard angles, then multiply the result by 2 to find \( (h \circ g)\left( \frac{17\pi}{3} \right) \).

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주요 개념

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

Function Composition

Function composition involves applying one function to the result of another, denoted as (f o g)(x) = f(g(x)). In this problem, (h o g)(x) means you first evaluate g(x), then use that output as the input for h. Understanding this process is essential to correctly evaluate the expression.

Trigonometric Values of Cosine

The function g(x) = cos x requires knowledge of cosine values at specific angles. Since the input is 17π/3, recognizing how to simplify angles using periodicity (cosine has period 2π) helps find an exact value without a calculator.

Periodicity and Angle Reduction

Trigonometric functions repeat their values in regular intervals called periods. For cosine, the period is 2π, so angles can be reduced by subtracting multiples of 2π to find equivalent angles within one cycle. This simplification is key to evaluating trigonometric expressions exactly.