Please explain Sin(x+y) = SinxCosy+SinyCosx

Question

I'm trying to derive Sin x and would like to know this.

Answer 1

This is a well known formula from trigonometry.  One of the ways in which it is developed uses the distance formula in connection with the Unit Circle.  The derivation is generally found in Trigonometry textbooks.

When x is substituted for y, BTW, you get the convenient formula

sin 2x = 2 sin x cos x.

 

I think that a better way is to memorize it as Sin(x+y) = sin x  cos y + cos x sin y;  x & y are in alphabetical order in each of the product terms, and Sines are in the outer positions, Cosines in the inner positions.

 

This parallels the formula cos(x+y) = cos x cos y - sin x cos y; from this latter formula, by setting y = x, we obtain

cos 2x = cos²x - sin²x (and from the Pythagorean identity we can then derive two alternative forms for cos 2x).