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

Guida verificata passo dopo passo

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 \( \sin \frac{\pi}{2} \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.

Video consigliato:

Percorso guidato

2:25

Verifying Identities with Sum and Difference Formulas

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.

Video consigliato:

Percorso guidato

04:39

45-45-90 Triangles

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.

Video consigliato:

Percorso guidato