Given that tan x = p/q, find sin x. Solve by using right-angled triangles.

We know that the tangent, in a right angle triangle, is the ratio between the opposite cathetus and the joined cathetus.

In this case, is given the tan x=p/q, so we'll conclude that one cathetus is p and the other one is q.

Also, in a right angle triangle,

sin x = opposite cathetus/hypotenuse=p/hypotenuse

But in a right angle triangle, by applying Pythagorean theorem:

(hypotenuse)^2=(cathetus)^2 + (cathetus)^2

(hypotenuse)^2 = p^2 + q^2

We'll substitute the formula above

sin x =p/hypotenuse = p/(p^2 + q^2)^1/2