Question

The value of $\sin \left(50°\right)-\sin \left(70°\right)+\sin \left(10°\right)$ is

• A. $0$
• B. $1$
• C. $\frac{1}{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

The correct option is A

$0$

The explanation for the correct option

The given trigonometric expression: $\sin \left(50°\right)-\sin \left(70°\right)+\sin \left(10°\right)$.

It is known that, $\mathbf{sin}\left(A\right)\mathbf{-}\mathbf{sin}\left(B\right)\mathbf{=}\mathbf{2}\mathbf{cos}\left(\frac{A+B}{2}\right)\mathbf{sin}\left(\frac{A-B}{2}\right)$.

Thus, $\sin \left(50°\right)-\sin \left(70°\right)+\sin \left(10°\right)=2\cos \left(\frac{50°+70°}{2}\right)\sin \left(\frac{50°-70°}{2}\right)+\sin \left(10°\right)$

$$\begin{gathered} \Rightarrow \sin \left(50°\right)-\sin \left(70°\right)+\sin \left(10°\right)=2\cos \left(\frac{120°}{2}\right)\sin \left(-\frac{20°}{2}\right)+\sin \left(10°\right) \\ \Rightarrow \sin \left(50°\right)-\sin \left(70°\right)+\sin \left(10°\right)=2\cos \left(60°\right)\sin \left(-10°\right)+\sin \left(10°\right) \\ \Rightarrow \sin \left(50°\right)-\sin \left(70°\right)+\sin \left(10°\right)=-2\left(\frac{1}{2}\right)\sin \left(10°\right)+\sin \left(10°\right) \\ \Rightarrow \sin \left(50°\right)-\sin \left(70°\right)+\sin \left(10°\right)=-\sin \left(10°\right)+\sin \left(10°\right) \\ \Rightarrow \sin \left(50°\right)-\sin \left(70°\right)+\sin \left(10°\right)=0 \end{gathered}$$

Therefore, the value of $\sin \left(50°\right)-\sin \left(70°\right)+\sin \left(10°\right)$ is $0$.

Hence, the correct option is (A).