Question

If $\alpha ,\beta$ belongs to $(0,\pi /2),\sin \alpha =4/5and\cos (\alpha +\beta )=-(12/13)$, then $\sin \beta$is equal to

• A. $\frac{63}{65}$
• B. $\frac{61}{65}$
• C. $\frac{3}{5}$
• D. $\frac{5}{13}$
• E. $\frac{8}{65}$

Answer 1

The correct option is A

$\frac{63}{65}$

Step 1. Finding the value of $\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\beta }\mathbf{:}$:

Given that $\alpha ,\beta$ belongs to $(0,\pi /2)$

and $\sin \alpha =4/5$

$\Rightarrow$ $\cos \alpha =3/5$

And

$\alpha ,\beta \in (0,\pi /2)$

$\Rightarrow$$0<\alpha +\beta <\pi$

$\cos (\alpha +\beta )=-(12/13)$

$\Rightarrow$$\sin (\alpha +\beta )=5/13$

Step 2.The required value of $\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\beta }\mathbf{}\mathbf{=}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{\left[}\mathbf{\right(}\mathit{\alpha }\mathbf{}\mathbf{+}\mathbf{}\mathit{\beta }\mathbf{)}\mathbf{}\mathbf{–}\mathbf{}\mathit{\alpha }\mathbf{]}$

$$\begin{gathered} =\sin (\alpha +\beta )\cos \alpha –\cos (\alpha +\beta )\sin \alpha \\ =(5/13)(3/5)–(-12/13)(4/5) \\ =(15/65)+(48/65) \\ =(15+48)/65 \\ =\mathbf{63}\mathbf{/}\mathbf{65} \end{gathered}$$

Hence, Option ‘A’ is Correct.