Q. 67 - Precalculus Enhanced with Graphing Utilities

Question

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

$\tan (-\frac{\mathrm{\pi }}{4}).$

Step-by-Step Solution

Answer

The exact value of given expression is $-1.$

Step 1. Given Information

Given that the trigonometric function is $\tan (-\frac{\mathrm{\pi }}{4}).$To find the exact value of trigonometric function using even odd property.

Step 2. Even odd property substitution

Tangent is the odd function and it can be denoted using even odd function as,

$$\begin{gathered} \tan (-\mathrm{\theta })=-\mathrm{tan\theta } \\ =\frac{-\mathrm{sin\theta }}{\mathrm{cos\theta }} \\ \tan (-\frac{\mathrm{\pi }}{4})=-\tan \frac{\mathrm{\pi }}{4} \\ =\frac{-\sin {\displaystyle \frac{\mathrm{\pi }}{4}}}{\cos {\displaystyle \frac{\mathrm{\pi }}{4}}} \\ =\frac{{\displaystyle \frac{-\sqrt{2}}{2}}}{{\displaystyle \frac{\sqrt{2}}{2}}} \\ \tan (-\frac{\mathrm{\pi }}{4})=-1 \end{gathered}$$