Chapter 3.5, Problem 3QR

To determine

To find:The exactvalue of $\sin \frac{\pi }{3}$ without a calculator.

Answer to Problem 3QR

The exact value of $\sin \frac{\pi }{3}$ is $\frac{\sqrt{3}}{2}$ .

Explanation of Solution

Given information: Given problem is about the sine function..

Calculation:

  $\begin{array}{l}\pi \mathrm{radians}=180\mathrm{degree}\\ 1\mathrm{radians}=\frac{180}{\pi }\mathrm{degree}\\ \frac{\pi }{3}\mathrm{radians}=\frac{\pi }{3}(\frac{180}{\pi })=60\mathrm{degree}\end{array}$

To prove this considers an equilateraltriangle $\mathrm{ABC}$ of side $2$ units as diagram below.

  

The bisector of angle $\mathrm{A}$ meets side $\mathrm{BC}$ at $\mathrm{D}$ which is the midpoint of $\mathrm{BC}$ .

By Pythagoras:

  $\begin{array}{l}{P}^2+B^2=H^2\\ P^2+1^2=2^2\\ P^2+1=4\\ P^2=4-1\\ P^2=3\\ P=\sqrt{3}\end{array}$

  $\mathrm{Angle}B=60\mathrm{Degree}$$\sin {60}^{\circ }=\frac{P}{H}=\frac{\sqrt{3}}{2}$

Therefore the value of $\sin {60}^{\circ }=\frac{\sqrt{3}}{2}$ .