In Exercises 87–92, find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. sin 𝜋/3 cos 𝜋 - cos 𝜋/3 sin 3𝜋/2

검증된 단계별 안내

Recognize that the expression is of the form \(\sin A \cos B - \cos A \sin B\), which matches the sine difference identity: \(\sin(A - B) = \sin A \cos B - \cos A \sin B\).

Identify the angles in the expression: \(A = \frac{\pi}{3}\) and \(B = \frac{2\pi}{3}\), so the expression simplifies to \(\sin\left(\frac{\pi}{3} - \frac{2\pi}{3}\right)\).

Calculate the difference inside the sine function: \(\frac{\pi}{3} - \frac{2\pi}{3} = -\frac{\pi}{3}\).

Use the odd property of sine: \(\sin(-\theta) = -\sin \theta\), so \(\sin\left(-\frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)\).

Recall the exact value of \(\sin\left(\frac{\pi}{3}\right)\), which is \(\frac{\sqrt{3}}{2}\), and write the final expression as a single fraction using this value.

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.

Unit Circle and Special Angles

The unit circle helps determine exact values of sine and cosine for special angles like π/3 and 2π/3. These angles correspond to well-known coordinates on the circle, allowing evaluation without a calculator.

Trigonometric Identities and Angle Multiplication

Understanding how to manipulate trigonometric expressions involving multiples of π is essential. Recognizing that sine and cosine values repeat or change sign at specific intervals aids in simplifying expressions.

Algebraic Simplification of Trigonometric Expressions

After substituting exact trigonometric values, combining terms into a single fraction requires careful algebraic manipulation. This includes factoring, common denominators, and simplifying radicals to express the answer neatly.