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

Verified step by step guidance

1. Recognize that the expression is of the form \(\sin A \cos B - \cos A \sin B\), which matches the sine difference identity: \(\sin(A - B) = \sin A \cos B - \cos A \sin B\).
2. Identify the angles in the expression: \(A = \frac{\pi}{3}\) and \(B = \frac{2\pi}{3}\), so the expression simplifies to \(\sin\left(\frac{\pi}{3} - \frac{2\pi}{3}\right)\).
3. Calculate the difference inside the sine function: \(\frac{\pi}{3} - \frac{2\pi}{3} = -\frac{\pi}{3}\).
4. Use the odd property of sine: \(\sin(-\theta) = -\sin \theta\), so \(\sin\left(-\frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)\).
5. 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.

Key Concepts

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.