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

1 Answer
Apr 17, 2017

#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#.