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

1 Answer

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#