Given that tan ⁡ u = 0.4 &amp; π &lt; u &lt; 3 π 2 .Finding the value of tan ⁡ u 2

Question

Given that   tan  ⁡  u  =  0.4 &amp;amp;   π  &amp;lt;  u  &amp;lt;            3      π        2  .Finding the value of   tan  ⁡      u    2

Harper Heath · Accepted Answer

Use the formula for the tangent of a sum to obtain

\tan (2 x) = \frac{\tan (x) + \tan (x)}{1 - \tan (x) \tan (x)} = \frac{2 \tan (x)}{1 - \tan^{2} (x)} .

Sustitute u=2x and get

\frac{2}{5} = 0.4 = \tan (u) = \tan (2 x) = \frac{2 \tan (x)}{1 - \tan^{2} (x)} = \frac{2 \tan \frac{u}{2}}{1 - \tan^{2} \frac{u}{2}} .

Rename

w = \tan \frac{u}{2}

, to get the equation

\frac{2}{5} = \frac{2 w}{1 - w^{2}} .

This becomes

1 - w^{2} = 5 w

, which leads

w^{2} + 5 w - 1 = 0

with roots

w_{1} = \frac{- 5 + \sqrt{29}}{2}, w_{2} = \frac{- 5 - \sqrt{29}}{2} .

Up to this point, the two posibilities are correct. However,

w_{1}

is positive, while

w_{2}

is negative. Since

π < u < \frac{3 π}{2}

, we have

\frac{π}{2} < \frac{u}{2} < \frac{3 π}{4}

. On that interval, the tangent function is negative. Thus,

w_{1}

cannot be a solution, so it is

w_{2}

.

Given that tan ⁡ u = 0.4 & π < u