Can sin (A+B) = sin A+ sin B

Question

Is it ever possible that sin sin⁡ sin⁡ ? If so, explain how. If not, explain why not.

Answer 1

Alyssa,

Using the addition formula sin(a+b)=cos(b)sin(a)+cos(a)sin(b) you can see the only time this is equal to sin(a)+sin(b) is when cos(b)=cos(a)=1 The cos is equal to one when the angle is 2πn where n=0,1,2,.... So as long as a or b is = 2πn where n=0,1,2,.... the sin(a+b)= sin(a)+sin(b) which is 0=0

Jim

Answer 2

For certain angles, yes.

Here is an example:

Let A=90^o and B=-90^o

sin(90-90) = sin(0) = 0

and

sin(90) + sin(-90) = 1 - 1 = 0

Note that -90 is a CW rotation, but even if we use CCW rotation (i.e. a positive angle of 270^o) we get:

sin(90+270) = sin(360) = 0

and

sin(90) + sin(270) = 1 - 1 = 0

I am sure there are other solutions (graphing on an interactive graphing site will help you find others).

Also note that B=0 (or A=0) is a trivial solution.

Probably the most important thing to remember is that these are just special cases, the equation does not hold in general.