Question

In Problems 19–28, use the Half-angle Formulas to find the exact value of each expression.

$\tan \frac{7\mathrm{\pi }}{8}$

Short answer

The exact value of the expression $\tan \frac{7\mathrm{\pi }}{8}$by using Half angle formula is $1-\sqrt{2}$

Step-by-step solution

Step 1. Given

An expression$\tan \frac{7\mathrm{\pi }}{8}$

To find the exact value of the expression

Step 2. Half angle formula

The half angle formula is,

$\tan \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}$

Step 3. Consider an angle

Consider,

$$\begin{gathered} \tan \alpha =\tan \frac{7\mathrm{\pi }}{8} \\ \alpha =\frac{7\mathrm{\pi }}{8} \\ \frac{\alpha }{2}=\frac{{\displaystyle \frac{7\mathrm{\pi }}{4}}}{2} \end{gathered}$$

Let $\alpha =\frac{7\mathrm{\pi }}{4}$

We know that, $\tan \frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }$

Step 4. Use the half-angle formula

By Half-angle formula,

$\tan \frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }$

Compute for $\cos \alpha ,\sin \alpha$

$$\begin{gathered} \cos \frac{7\mathrm{\pi }}{4}=\cos (2\mathrm{\pi }-\frac{\mathrm{\pi }}{4}) \\ =\cos (-\frac{\mathrm{\pi }}{4}) \\ =\cos \frac{\mathrm{\pi }}{4} \\ =\frac{\sqrt{2}}{2} \end{gathered}$$

$$\begin{gathered} \sin \frac{7\mathrm{\pi }}{4}=\sin (2\mathrm{\pi }-\frac{\mathrm{\pi }}{4}) \\ =\sin (-\frac{\mathrm{\pi }}{4}) \\ =-\sin \frac{\mathrm{\pi }}{4} \\ =-\frac{\sqrt{2}}{2} \end{gathered}$$

Step 5. Find the exact value

Since $\tan \frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }$

$$\begin{gathered} \tan \frac{{\displaystyle \frac{7\mathrm{\pi }}{4}}}{2}=\frac{1-{\displaystyle \frac{\sqrt{2}}{2}}}{-{\displaystyle \frac{\sqrt{2}}{2}}} \\ =\frac{{\displaystyle \frac{2-\sqrt{2}}{2}}}{-{\displaystyle \frac{\sqrt{2}}{2}}} \\ =-\left(\frac{2-\sqrt{2}}{\sqrt{2}}\right) \\ =1-\sqrt{2} \end{gathered}$$