# How do you evaluate tan(cos^-1((2sqrt5)/5)) without a calculator?

Evaluate the triangle represented by the ${\cos}^{-} 1$ function, solve for the opposite, then find tan to be
$\tan = \text{opp"/"adj} = \frac{\sqrt{5}}{2 \sqrt{5}} = \frac{1}{2}$

#### Explanation:

Starting with the original:

$\tan \left({\cos}^{-} 1 \left(\frac{2 \sqrt{5}}{5}\right)\right)$

The first thing to do is evaluate ${\cos}^{-} 1 \left(\frac{2 \sqrt{5}}{5}\right)$

So we are dealing with a triangle with an angle that has adjacent side $2 \sqrt{5}$ and hypotenuse 5. We're going to need to find the opposite side to satisfy the second part of this - the tan function.

We can find the opposite by using the pythagorean theorem:

${a}^{2} + {b}^{2} = {c}^{2}$

We know a and c:

${\left(2 \sqrt{5}\right)}^{2} + {b}^{2} = {5}^{2}$

Solving for b:

$20 + {b}^{2} = 25$

${b}^{2} = 5$

$b = \sqrt{5}$

We can now find the tan function:

$\tan = \text{opp"/"adj} = \frac{\sqrt{5}}{2 \sqrt{5}} = \frac{1}{2}$