Question

what is tan142 1/2?

(mixed fraction)

Answer 1

$\tan 142.5=\tan \left(180-37.5\right)=-\tan 37.5$
Let 37.5 be x. Therefore,75 = 2x
$\tan 75=\tan 2x=\frac{2\tan x}{1-{\tan }^{2}x}$
$\left(1-{\tan }^{2}x\right)\tan 75=2\tan x\Rightarrow \tan 75{\tan }^{2}x+2\tan x-\tan 75=0$
$\tan x=\frac{-2\pm \sqrt{4+4{\tan }^{2}75}}{2\tan 75}=\frac{-1\pm {\displaystyle \frac{1}{\cos 75}}}{{\displaystyle \frac{\sin 75}{\cos 75}}}=\frac{-\cos 75\pm 1}{\sin 75}$
cos 75 = cos (45 + 30) = $\frac{\sqrt{3}-1}{2\sqrt{2}}$and sin 75 = sin (45 + 30) = $\frac{\sqrt{3}+1}{2\sqrt{2}}$
Therefore, tan 37.5 = $\frac{1-\sqrt{3}+2\sqrt{2}}{\sqrt{3}+1}$as tan 37.5 > 0
Therefore, tan 142.5 = - tan 37.5 = $\frac{\sqrt{3}-1-2\sqrt{2}}{\sqrt{3}+1}$