# How do you find the exact value of tan(cos^-1 (1/2))?

Feb 17, 2017

tan (pi/3) = sqrt3
tan (-pi/3) = - sqrt3

#### Explanation:

${\cos}^{-} 1 \left(\frac{1}{2}\right)$ --> $\arccos \left(\frac{1}{2}\right)$
Trig table and unit circle -->
$\cos x = \frac{1}{2}$ --> arc $x = \pm \frac{\pi}{3}$
There are 2 answers for $\left(0 , 2 \pi\right)$
$\tan \left(\frac{\pi}{3}\right) = \sqrt{3}$
$\tan \left(- \frac{\pi}{3}\right) = - \sqrt{3}$

Mar 20, 2017

see below

#### Explanation:

From the color(red)(cos^-1(1/2)  we have the side color(red)(adjacent to color(red)(color(red)(angle theta=1 and the side color(red)(hypote n use = 2 . So this is a color(red)(30^@-60^@-90^@ triangle and color(red)(theta = 60^@therefore the color(red)(opposite to color(red)(angle theta = sqrt3.

Note that from the restrictions for the range of inverse circular functions color(red)(cos^-1 x is restricted to quadrants 1 and 2 and since the argument is positive $\textcolor{red}{\frac{1}{2}}$ our answer will be in quadrant 1 only.

Hence,
color(blue)(tan(cos^-1(1/2))=tan theta=(opposite)/(adjacent)=sqrt3/1=sqrt3