Find the exact value of each expression. See Example 1.
sin 5π/9 cos π/18 - cos 5π/9 sin π/18 .

검증된 단계별 안내

Recognize that the expression \( \sin \frac{5\pi}{9} \cos \frac{\pi}{18} - \cos \frac{5\pi}{9} \sin \frac{\pi}{18} \) matches the sine difference identity: \( \sin A \cos B - \cos A \sin B = \sin (A - B) \).

Identify \( A = \frac{5\pi}{9} \) and \( B = \frac{\pi}{18} \) from the given expression.

Apply the sine difference formula to rewrite the expression as \( \sin \left( \frac{5\pi}{9} - \frac{\pi}{18} \right) \).

Find a common denominator to subtract the angles inside the sine function: \( \frac{5\pi}{9} = \frac{10\pi}{18} \), so \( \frac{10\pi}{18} - \frac{\pi}{18} = \frac{9\pi}{18} \).

Simplify the angle inside the sine function to \( \frac{9\pi}{18} = \frac{\pi}{2} \), so the expression becomes \( \sin \frac{\pi}{2} \).

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.

Trigonometric Angle Sum and Difference Identities

These identities express the sine or cosine of a sum or difference of angles in terms of sines and cosines of the individual angles. For example, sin(A - B) = sin A cos B - cos A sin B, which is directly applicable to the given expression.

Exact Values of Special Angles

Certain angles, often multiples of π/6, π/4, and π/3, have known exact sine and cosine values. Recognizing or converting angles to these special angles helps in finding precise trigonometric values without a calculator.

Simplification of Trigonometric Expressions

This involves manipulating and reducing trigonometric expressions using identities and algebraic techniques to find exact values. It includes combining like terms, applying identities, and recognizing patterns to simplify complex expressions.

Find the exact value of each expression. See Example 1.sin 5π/9 cos π/18 - cos 5π/9 sin π/18 .