Problem 35 - Precalculus Essentials

Question

Use an identity to find the value of each expression. Do not use a calculator. $$\sin ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{6}$$

Step-by-Step Solution

Answer

The value of the expression \(\sin ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{6}\) is 1.

Step 1: Identify the Identity

Notice that the expression is in the form of a Pythagorean identity, which states that for any angle \(a\), \(\sin^2(a) + \cos^2(a) = 1\). This identity is used to find the value of the given expression \(\sin^2(\frac{\pi}{6}) + \cos^2(\frac{\pi}{6})\).

Step 2: Applying the Identity

The Pythagorean identity equates the sum of the squares of sin and cos of any angle to 1. Here the angle is \(\frac{\pi}{6}\), so when substituting this into the identity of \(\sin^2(a) + \cos^2(a) = 1\), you have \(\sin^2(\frac{\pi}{6}) + \cos^2(\frac{\pi}{6}) = 1\).

Step 3: Evaluate the Expression

Applying the Pythagorean identity proves that \(\sin^2(\frac{\pi}{6}) + \cos^2(\frac{\pi}{6})\) is equal to 1. So, \(1\) is the value of the given expression.