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# How do you find Tan 22.5 using the half angle formula?

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#### Explanation

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#### Explanation:

I want someone to double check my answer

69
Apr 1, 2017

Find tan (22.5)

Answer: $- 1 + \sqrt{2}$

#### Explanation:

Call tan (22.5) = tan t --> tan 2t = tan 45 = 1

Use trig identity: $\tan 2 t = \frac{2 \tan t}{1 - {\tan}^{2} t}$ (1)

$\tan 2 t = 1 = \frac{2 \tan t}{1 - {\tan}^{2} t}$ -->
--> ${\tan}^{2} t + 2 \left(\tan t\right) - 1 = 0$
Solve this quadratic equation for tan t.

$D = {d}^{2} = {b}^{2} - 4 a c = 4 + 4 = 8$ --> $d = \pm 2 \sqrt{2}$
There are 2 real roots:
tan t = -b/2a +- d/2a = -2/1 + 2sqrt2/2 = - 1 +- sqrt2
$\tan t = \tan \left(22.5\right) = - 1 \pm \sqrt{2}$
Since tan 22.5 is positive, then take the positive answer:
tan (22.5) = - 1 + sqrt2

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