# How do you derive the trigonometric sum and difference formulas for sin, cos, and tan? I.e: How do I derive something like sin(x+y)=sinxcosy+cosxsiny?

Oct 30, 2017

#### Explanation:

these are compound angle identities :

1. sin(A+B) $\equiv$ sinAcosB + cosAsinB

from that:

put B=A
therefore: sin (A+A) = sin (2A)

= sinAcosA+cosAsinA

= 2 sinAcosA(thats the double angle formula) <<<<<

it goes on for all the other compound angles..

:try them and let me know if you had any difficulty, i'll be glad to help :)

Oct 30, 2017

Use a diagram and some reasoning...

#### Explanation:

...The best math teacher I ever had taught me: memorize as little as possible in mathematics.

Words to live by, as I can never be 100% sure I remember these trig identities correctly. But refer to the diagram:

Angle AOE is the sum of angles x and y.
Furthermore, segment OA has length 1.
Therefore,
$\frac{A D}{O A} = A D = \sin y$

Note now that triangles AFD and OFB are similar.

Line Segments $\frac{A C}{A D} = \cos x$

...but we previously deduced that $A D = \sin y$. Therefore,

$A C = \sin y \cdot \cos x$

Now, note that segment $\frac{O D}{O A} = O D = \cos y$

And you can see from the diagram that $\frac{D E}{O D} = \sin x$.

Therefore segment $D E = \sin x \cos y$

Segments DE and CB are equal.

Therefore, Segments $C B + A C = \sin \left(x + y\right) = \cos x \sin y + \sin x \cos y$

GOOD LUCK