Question

How do you find the exact value of `tan(arccos(3/5))`?

Answer 1

`4/3`

Explanation:

Geometric method
Draw a right triangle with an angle `theta`. The adjacent side has length `3`, and hypotenuse has length `5`. Notice that `theta=arccos(3/5)`.

By Pythagoras' Theorem, the opposite side has length `sqrt(5^2-3^2)=4`. Therefore, `tan(arccos(3/5))=tan(theta)=4/3`.

Algebraic method
Remember that `tan^2(theta)+1=sec^2(theta)`, or `tan(theta)=sqrt(sec^2(theta)-1)=sqrt(1/cos^2(theta)-1)`.

Therefore, `tan(arccos(3/5))=sqrt(1/cos^2(arccos(3/5))-1)=sqrt(1/(3/5)^2-1)=4/3`.