Question

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

Answer 1

Evaluate the triangle represented by the `cos^-1` function, solve for the opposite, then find tan to be
`tan="opp"/"adj"=sqrt5/(2sqrt5)=1/2`

Explanation:

Starting with the original:

`tan(cos^-1((2sqrt5)/5))`

The first thing to do is evaluate `cos^-1((2sqrt5)/5)`

So we are dealing with a triangle with an angle that has adjacent side `2sqrt5` 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:

`(2sqrt5)^2+b^2=5^2`

Solving for b:

`20+b^2=25`

`b^2=5`

`b=sqrt5`

We can now find the tan function:

`tan="opp"/"adj"=sqrt5/(2sqrt5)=1/2`