Trigonometrical Proof Question

Question

You have a triangle RST. Prove SinR^2 + CosR^2 = 1 given ST is perpendicular to RT.

Answer

Firstly,

sin R = opposite/hypotenuse
and
cos R = adjacent/hypotenuse

Square and add, to obtain,

sin^2 R + cos^2 R = (ST)^2/(SR)^2 + (RT)^2/(SR)^2

sin^2 R + cos^2 R = [(ST)^2 + (RT)^2 ] / [(SR)^2] …(1)

Now, By Pythagoras’ theorem, we have, ( note that SR is the hypotenuse because right angle is at T ),

(SR)^2 = (ST)^2 + (RT)^2 …(2)

Substitute (2) into (1), to get,

sin^2 R + cos^2 R = (SR)^2 / (SR)^2

sin^2 R + cos^2 R = 1

QED