Use the half-angle formulas to determine the exact value of the given trigonometric expression. $\tan \frac { 3 \pi } { 8 }$
A) $\quad \tan \frac { 3 \pi } { 8 } = \sqrt { 2 } - 1$
B)$\tan \frac { 3 \pi } { 8 } = \frac { \sqrt { 2 + \sqrt { 2 } } } { 4 }$
C) $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } + 1$
D)$\tan \frac { 3 \pi } { 8 } = - \frac { \sqrt { 2 - \sqrt { 2 } } } { 4 }$
E) $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 - \sqrt { 2 } }$

Question

Use the half-angle formulas to determine the exact value of the given trigonometric expression. $\tan \frac { 3 \pi } { 8 }$ 
A) $\quad \tan \frac { 3 \pi } { 8 } = \sqrt { 2 } - 1$
B)$\tan \frac { 3 \pi } { 8 } = \frac { \sqrt { 2 + \sqrt { 2 } } } { 4 }$
C) $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } + 1$
D)$\tan \frac { 3 \pi } { 8 } = - \frac { \sqrt { 2 - \sqrt { 2 } } } { 4 }$
E) $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 - \sqrt { 2 } }$

Quizplus · Accepted Answer

The half-angle formula for tangent is $\tan \left( \frac{\theta}{2} \right) = \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$. For $\tan \left( \frac{3\pi}{8} \right)$, we can consider it as $\tan \left( \frac{\frac{3\pi}{4}}{2} \right)$. Since $\cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2}$, substituting into the formula gives $\tan \left( \frac{3\pi}{8} \right) = \sqrt{\frac{1 - (-\frac{\sqrt{2}}{2})}{1 + (-\frac{\sqrt{2}}{2})}} = \sqrt{2} + 1$.

Use the Half-Angle Formulas to Determine the Exact Value of the Given