Question

What is the exact value of tan ( 5 pi/8 )

Answer 1

Answer: Choice D) $\tan\left(\frac{5\pi}{8}\right) = - \sqrt{\frac{2+\sqrt{2}}{1-\sqrt{2}}}$

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Explanation:

Make sure your calculator is in radian mode. Use your calculator to find that tan(5pi/8) = -2.41421 which is approximate.

Since the value is negative, this means the answer is between choices C and D. You can use your calculator to compute those expressions given and you should find it matches with choice D.

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Further explanation:

We can apply the half angle identity for tangent like so

$\tan\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\\\\\\\tan\left(\frac{5\pi/4}{2}\right) = -\sqrt{\frac{1-\cos(5\pi/4)}{1+\cos(5\pi/4)}}\\\\\\\tan\left(\frac{5\pi}{8}\right) = -\sqrt{\frac{1-\left(-\frac{\sqrt{2}}{2}\right)}{1+\left(-\frac{\sqrt{2}}{2}\right)}}$

Simplifying further, we get

$\tan\left(\frac{5\pi}{8}\right) = -\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{1-\frac{\sqrt{2}}{2}}}\\\\\\\tan\left(\frac{5\pi}{8}\right) = -\sqrt{\frac{2+\sqrt{2}}{1-\sqrt{2}}}$

In the last step, I multiplied top and bottom of the outer fraction by 2 to clear out the denominators of the inner fraction.