Question

your turn!
choose the option that best answers the question.
find the exact value of $\sin\frac{\pi}{6}$ using the fact that $\frac{\pi}{6}=\frac{\pi}{2}-\frac{\pi}{3}$.
a. $\frac{1}{2}$
b. $\frac{\sqrt{3}}{2}$
c. $\frac{1+\sqrt{3}}{2}$

Explanation:

Step1: Use the sine subtraction formula

The formula for \(\sin(A - B)=\sin A\cos B-\cos A\sin B\). Here \(A = \frac{\pi}{2}\) and \(B=\frac{\pi}{3}\).
So \(\sin(\frac{\pi}{2}-\frac{\pi}{3})=\sin\frac{\pi}{2}\cos\frac{\pi}{3}-\cos\frac{\pi}{2}\sin\frac{\pi}{3}\).

Step2: Substitute the values of trigonometric functions

We know that \(\sin\frac{\pi}{2} = 1\), \(\cos\frac{\pi}{2}=0\), \(\cos\frac{\pi}{3}=\frac{1}{2}\), and \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\).
Substituting these values: \(\sin\frac{\pi}{6}=(1)\times(\frac{1}{2})-(0)\times(\frac{\sqrt{3}}{2})\).

Step3: Simplify the expression

\(\sin\frac{\pi}{6}=\frac{1}{2}-0=\frac{1}{2}\).

Answer:

A. \(\frac{1}{2}\)