Problem 36 - Precalculus Essentials

Question

Use an identity to find the value of each expression. Do not use a calculator. $$\sin ^{2} \frac{\pi}{3}+\cos ^{2} \frac{\pi}{3}$$

Step-by-Step Solution

Answer

The value of \( \sin^{2} \frac{\pi}{3} + \cos^{2} \frac{\pi}{3} \) is 1.

Step 1: Identify the Pythagorean Trigonometric Identity

Recognize that the given expression fits the form of the Pythagorean Trigonometric Identity, which is \( \sin^{2} \theta + \cos^{2} \theta = 1 \) for any angle \( \theta \).

Step 2: Substitute the Angle into the Identity

Substitute \( \frac{\pi}{3} \) for \( \theta \) in the identity, getting \( \sin^{2} \frac{\pi}{3} + \cos^{2} \frac{\pi}{3} \).

Step 3: Solve the Substituted Identity

Evaluate the expression \( \sin^{2} \frac{\pi}{3} + \cos^{2} \frac{\pi}{3} \) using the Identity rule, which equals to 1.