Question

Is it right to say that $\sin (A+B)=\sin A+\sin B$. Justify your answer?

Answer 1

Given statement is,

$\sin (A+B)=\sin A+\sin B$

Let $A={30}^{\circ },B={60}^{\circ }$

Now, $LHS=\sin (A+B)$

$=\sin ({30}^{\circ }+{60}^{\circ })$

$=\sin {90}^{\circ }$

$=1$---(1)

$RHS=\sin A+\sin A$

$=\sin A+\sin B$

$=\sin {30}^{\circ }+\sin {60}^{\circ }$

$=\frac{1}{2}+\frac{\sqrt{3}}{2}$

$=\frac{1+\sqrt{3}}{2}$ ---(2)

From (1) and (2),

Hence, $LHS\ne RHS.$

Hence, The given statement is false.

Since, $\sin (A+B)=\sin A\cos B+\cos A\sin B$