Question

State whether the following are true or false. Justify your answer.
$\sin \left( A+B \right)=\sin A+\sin B$.

Answer 1

Hint:To state the above statement as true or false. Take any combination of angle of A and B and substitute in the given equation $\sin \left( A+B \right)=\sin A+\sin B$ and then see whether the combination of angles satisfies this equation or not.

Complete step-by-step answer:
The equation given in the question is:
$\sin \left( A+B \right)=\sin A+\sin B$ ……… Eq. (1)
We have to state whether this equation is true or false.
Let us take a combination of angle A and angle B.
$\angle A={{30}^{\circ }}\And \angle B={{60}^{\circ }}$
Substitute these values of angles in the eq. (1).
$\sin \left( {{30}^{\circ }}+{{60}^{\circ }} \right)=\sin {{30}^{\circ }}+\sin {{60}^{\circ }}$
L.H.S of the above equation yields:
$\sin {{90}^{\circ }}=1$
R.H.S of the above equation yields:
$\begin{align}
  & \sin {{30}^{\circ }}+\sin {{60}^{\circ }} \\
 & =\dfrac{1}{2}+\dfrac{\sqrt{3}}{2} \\
 & =\dfrac{\sqrt{3}+1}{2} \\
\end{align}$
From the above calculation we have found that,
L.H.S = 1
$\text{R}\text{.H}\text{.S}=\dfrac{\sqrt{3}+1}{2}$
From the above solution we can see that L.H.S ≠ R.H.S. Hence, this equation $\sin \left( A+B \right)=\sin A+\sin B$ is false.

Note: You can check the truth value of this equation $\sin \left( A+B \right)=\sin A+\sin B$ by taking different angles also.
Like if we take $A={{45}^{0}}\text{ and B=4}{{\text{5}}^{0}}$ and then substituting these values of A and B in the given equation $\sin \left( A+B \right)=\sin A+\sin B$ we get,
$\sin \left( {{45}^{\circ }}+{{45}^{\circ }} \right)=\sin {{45}^{\circ }}+\sin {{45}^{\circ }}$
Solving L.H.S of the given equation:
$\begin{align}
  & \sin \left( {{45}^{\circ }}+{{45}^{\circ }} \right) \\
 & =\sin {{90}^{\circ }}=1 \\
\end{align}$
Solving R.H.S of the given equation:
$\begin{align}
  & \sin {{45}^{\circ }}+\sin {{45}^{\circ }} \\
 & =2\sin {{45}^{\circ }} \\
 & =2\left( \dfrac{1}{\sqrt{2}} \right)=\sqrt{2} \\
\end{align}$
From the above calculations we have found that:
L.H.S = 1
$R.H.S=\sqrt{2}$
From the above solution we can see that L.H.S ≠ R.H.S. Hence, this equation $\sin \left( A+B \right)=\sin A+\sin B$ is false.
Hence, this combination of angles of A and B also states that the given equation is false.We can also check by using the formula of $\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$.Comparing the formula with given equation we can say that the statement is false.