Question

The value of $\sin {{60}^{\circ }}\cos {{30}^{\circ }}+\cos {{60}^{\circ }}\sin {{30}^{\circ }}$ is equal to:
(A) $\sin {{90}^{\circ }}$
(B) $\sin {{45}^{\circ }}$
(C) $\sin {{180}^{\circ }}$
(D) $\sin {{30}^{\circ }}$

Answer 1

Hint: For answering this question we will observe the given expression $\sin {{60}^{\circ }}\cos {{30}^{\circ }}+\cos {{60}^{\circ }}\sin {{30}^{\circ }}$ and apply the suitable formulae $\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$ . Then we will simplify it and obtain the required answer. Then we will mark the correct answer among the given options.

Complete step by step answer:
Now considering from the question we have the expression $\sin {{60}^{\circ }}\cos {{30}^{\circ }}+\cos {{60}^{\circ }}\sin {{30}^{\circ }}$ .
If we observe this expression it is in the form of the formulae $\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$ .
By applying this formula we will have the simplified expression $\sin {{60}^{\circ }}\cos {{30}^{\circ }}+\cos {{60}^{\circ }}\sin {{30}^{\circ }}=\sin \left( {{60}^{\circ }}+{{30}^{\circ }} \right)$ .
After performing the calculations we will have $\sin \left( {{60}^{\circ }}+{{30}^{\circ }} \right)=\sin {{90}^{\circ }}$ .
Hence we have a conclusion that the value of $\sin {{60}^{\circ }}\cos {{30}^{\circ }}+\cos {{60}^{\circ }}\sin {{30}^{\circ }}$ is equal to $\sin {{90}^{\circ }}$ .

So, the correct answer is “Option A”.

Note: While answering questions of this type we should be sure with the calculations and formulae. If we perform a mistake then we will end up having a different answer. We can answer this question in another way that is by substituting the values. Now we will substitute the respective values: $\sin {{60}^{\circ }}=\dfrac{\sqrt{3}}{2}$ , $\sin {{30}^{\circ }}=\dfrac{1}{2}$ , $\cos {{60}^{\circ }}=\dfrac{1}{2}$ and $\cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2}$ . After substituting the respective values in respective places we will have $\sin {{60}^{\circ }}\cos {{30}^{\circ }}+\cos {{60}^{\circ }}\sin {{30}^{\circ }}=\dfrac{\sqrt{3}}{2}\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}\dfrac{1}{2}$ . By simplifying the expression we will have $\dfrac{3}{4}+\dfrac{1}{4}=1$ . We know that the value of this expression is equal to the value of $\sin {{90}^{\circ }}$ . Hence we can conclude that $\sin {{60}^{\circ }}\cos {{30}^{\circ }}+\cos {{60}^{\circ }}\sin {{30}^{\circ }}=\sin {{90}^{\circ }}$. So option A is the correct answer. We are having the same answer in both the procedures. Hence any one of them can be used for answering this question.