Let's write atan for tan^-1.Originally Posted byjp25111

Then

sin(atan(p/q)) = p/sqrt(p^2+q^2)

(draw a right triangle with legs p and q and hypotenuse sqrt(p^2+q^2), the question is what is the sine of the angle whose tangent is p/q )

and

cos(atan(p/q)) = q/sqrt(p^2+q^2)

(same argument).

Also, by the double angle formula for the sine,

sin(2theta) = 2sin(theta)cos(theta).

Thus sin(2atan(p/q)) = 2pq/(p^2+q^2).

You can generalize this to cos(2atan(p/q)) or sin(3atan(p/q)) etc. and anticipate what other questions your teacher might come up with