91Ó°ÊÓ

To find the angle θ, we will need to set \(\frac{\theta}{2} = 3 \pi / 8\). We will multiply both sides by 2 to get our angle. $$ \theta = 2 \left(\frac{3 \pi}{8}\right) = \frac{6 \pi}{8} = \frac{3 \pi}{4} $$ So, \(\theta = \frac{3 \pi}{4}\).

Next, we will need to find the sine and cosine values of \(\theta = \frac{3 \pi}{4}\). The angle is in the second quadrant, and we can use the values of a 45-degree triangle to find the values of \(\sin \theta\) and \(\cos \theta\). $$ \sin \frac{3 \pi}{4} = \sin \left(\frac{\pi}{4} + \frac{\pi}{2}\right) = \sin \frac{\pi}{4} \cos \frac{\pi}{2} + \cos \frac{\pi}{4} \sin \frac{\pi}{2} = 0 + \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} $$ $$ \cos \frac{3 \pi}{4} = \cos \left(\frac{\pi}{4} + \frac{\pi}{2}\right) = \cos \frac{\pi}{4} \cos \frac{\pi}{2} - \sin \frac{\pi}{4} \sin \frac{\pi}{2} = 0 - \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2} $$ So, \(\sin \frac{3 \pi}{4} = \frac{\sqrt{2}}{2}\), and \(\cos \frac{3 \pi}{4} = -\frac{\sqrt{2}}{2}\).

Now that we have values for \(\sin \theta\) and \(\cos \theta\), we can apply the half-angle formula for tangent: $$ \tan \frac{\theta}{2} = \frac{1-\cos \theta}{\sin \theta} = \frac{1 - \left(-\frac{\sqrt{2}}{2}\right)}{\frac{\sqrt{2}}{2}} \\ $$ Simplify the expression: $$ =\frac{1 + \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = \frac{1 + \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} \cdot \frac{2}{\sqrt{2}} = \frac{\sqrt{2}+2}{\sqrt{2}} $$ Thus, the exact value of \(\tan \left(\frac{3 \pi}{8}\right) = \frac{\sqrt{2}+2}{\sqrt{2}}\).