Question

Find the exact value by using a half-angle identity.

Answer 1

Answer:

The exact form of $\tan(\frac{7\pi}{8})$ is $-\sqrt{2}+1$

Step-by-step explanation:

We need to find the exact value of $\tan(\frac{7\pi}{8})$ using half angle identity.

Since, $\frac{7\pi}{8}$  is not an angle where the values of the six trigonometric functions are known, try using half-angle identities.

$\frac{7\pi}{8}$  is not an exact angle.

First, rewrite the angle as the product of $\frac{1}{2}$ and an angle where the values of the six trigonometric functions are known. In this case,

$\frac{7\pi}{8}$ can be written as ;

$(\frac{1}{2})\frac{7\pi}{4}$

Use the half-angle identity for tangent to simplify the expression. The formula states that $\tan \frac{\theta}{2}=\frac{\sin \theta}{1+ \cos \theta}$

$=\frac{\sin(\frac{7\pi}{4})}{1+ \cos (\frac{7\pi}{4})}$

Simplify the numerator.

$=\frac{\frac{-\sqrt{2}}{2}}{1+ \cos (\frac{7\pi}{4})}$

Simplify the denominator.

$=\frac{\frac{-\sqrt{2}}{2}}{\frac{2+\sqrt{2}}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{-\sqrt{2}}{2}\times \frac{2}{2+\sqrt{2}}$

cancel the common factor of 2.

$\frac{-\sqrt{2}}{1}\times \frac{1}{2+\sqrt{2}}$

Simplify,

$\frac{-\sqrt{2}(2-\sqrt{2})}{2}$

$\frac{-(2\sqrt{2}-\sqrt{2}\sqrt{2})}{2}$

$\frac{-(2\sqrt{2}-2)}{2}$

simplify terms,

$-\sqrt{2}+1$

Therefore, the exact form of $\tan(\frac{7\pi}{8})$ is $-\sqrt{2}+1$

Answer 2

Answer:

-sqrt((1-(sqrt(2)/2)/1+(sqrt(2)/2))

Step-by-step explanation:

tan (7pi/8) = tan (7pi/4/2)

Which equals...

-sqrt((1-cos((7pi/4))/1+cos((7pi/4))

And just simplify into...

-sqrt((1-(sqrt(2)/2)/1+(sqrt(2)/2))