Calculus inverse trig question

Question

Find arctan( tan 4π/7)

 

how would you do this?

 

thankyou

Answer 1

(4/7)pi is about 102.85 degrees. Its tangent is -4.385.

The angle that has that value as tangent is -77.15 degrees. You may convert back to radian measure if you prefer.

 

I did this decimally in degrees in order to show that the (4/7)pi is in second quadrant.  But the arctan and tan do not just cancel one another because of the restriction that arctan yields output in the interval (-pi/2,pi/2).

Answer 2

The derivative of arctan(x) is 1/(1 + x²).  However, we have a composition of functions, so we need to use the chain rule.

 

Take the derivative of the inner function times the derivative of the outer function.

 

 

sec²(4π/7) / [1 + tan²(4π/7)]