Q. 60 - Precalculus Enhanced with Graphing Utilities

Question

Use the even–odd properties to find the exact value of each expression. Do not use a calculator.

$\cos (-{30}^{\mathrm{o}})$

Step-by-Step Solution

Answer

The exact value of the function is $\frac{\sqrt{3}}{2}.$

Step 1. Given Information

Given that the trigonometric function is $\cos (-{30}^{\mathrm{o}}).$ To find the exact value of trigonometric function using even odd property.

Step 2. Even odd property

Cosine function is an even function

$$\begin{gathered} \cos (-\mathrm{\theta })=\mathrm{cos\theta } \\ \cos (-{30}^{\mathrm{o}})=\cos \left({30}^{\mathrm{o}}\right) \\ =\frac{\sqrt{3}}{2} \end{gathered}$$

Step 3. Proof for equilateral triangle

On taking equilateral triangle,

Step 4. Proof using equilateral triangle

On taking the equilateral triangle as ABC and draw a line bisects of A. On taking triangle as $\mathrm{ADC},\angle \mathrm{CAD}={30}^{\mathrm{o}}$. On using Pythagorean triangle,

$$\begin{gathered} {\mathrm{AC}}^2={\mathrm{AD}}^2+{\mathrm{DC}}^2 \\ {\mathrm{a}}^2=\frac{{\mathrm{a}}^2}{4}+{\mathrm{AD}}^2 \\ \mathrm{AD}=\frac{\sqrt{3}}{2}\mathrm{a} \\ \cos \left({30}^{\mathrm{o}}\right)=\frac{\mathrm{AD}}{\mathrm{AC}} \\ \cos \left({30}^{\mathrm{o}}\right)=\frac{{\displaystyle \frac{\sqrt{3}}{2}}\mathrm{a}}{\mathrm{a}} \\ \cos \left({30}^{\mathrm{o}}\right)=\frac{\sqrt{3}}{2} \end{gathered}$$