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

Verified step by step guidance

1. 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: \( \sin A \cos B - \cos A \sin B = \sin (A - B) \).
2. Identify \( A = \frac{5\pi}{9} \) and \( B = \frac{\pi}{18} \) from the given expression.
3. Apply the sine difference formula to rewrite the expression as \( \sin \left( \frac{5\pi}{9} - \frac{\pi}{18} \right) \).
4. Find a common denominator to subtract the angles inside the sine function: \( \frac{5\pi}{9} = \frac{10\pi}{18} \), so \( \frac{10\pi}{18} - \frac{\pi}{18} = \frac{9\pi}{18} \).
5. Simplify the angle inside the sine function to \( \frac{9\pi}{18} = \frac{\pi}{2} \), so the expression becomes \( \sin \frac{\pi}{2} \).

Key Concepts

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.