Question

Is it ever possible that sin (A+B)= sin⁡ A+sin⁡ B? If so, explain how. If not, explain why not.

Answer 1

Answer:  yes, when A & B are on the quadrantals (0°, 90°, 180°, 270°)          which results in a sum of 0

Step-by-step explanation:

sin (A + B) = cos A · sin B  +  cos B · sin A

If cos A = 1 and cos B = 1, then you will end up with sin A + sin B.

The only angle where cos is equal to 1 is at 0°. Notice that sin is equal to 0.

It follows that when cos is equal to -1, sin is also 0. This occurs at 180°

The other option is when cos is equal to 0. This occurs at 90° and 270°.

All of these options result in sin A + sin B = 0

So, the only possibility when sin (A + B) = sin A + sin B is when it equals zero.

Answer 2

The formula for the sine sum is

$\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$

So, equation

$\sin(a)\cos(b)+\cos(a)\sin(b) = \sin(a)+\sin(b)$

Cannot be solved with standard ways. However, we can build special cases to make it work.

The most trivial example would be A=B=0. In this case, we have

$\sin(0+0)=0 \sin(0) = 0,\quad\sin(0)+\sin(0)=0+0=0$

If we work on this idea a little bit, we can produce examples like

$A=2k_1\pi,\quad B=2k_2\pi$

And again we have

$\sin(A+B)=\sin(2\pi(k_1+k_2))=0$

$\sin(A)+\sin(B)=\sin(2k_1\pi)+\sin(2k_2\pi)=0+0=0$

So, yes, it is possible that $\sin(A+B)=\sin(A)+\sin(B)$, although solving the equation directly would be quite difficult.