Question

Find the value of $\sin {{330}^{\circ }}\cos {{120}^{\circ }}+\cos {{210}^{\circ }}\sin {{300}^{\circ }}$.

Answer 1

Let us assume the value of $\sin {{330}^{\circ }}\cos {{120}^{\circ }}+\cos {{210}^{\circ }}\sin {{300}^{\circ }}$ is equal to I. Now we should first express 330 in terms of $2\pi -\theta$. We know that $\sin \left( n\pi -\theta \right)=-\sin \theta$ if n is even. Now by using this concept, we will find the value of $\sin {{330}^{\circ }}$. Now we should express 120 in terms of $\pi -\theta$. We know that $\sin \left( n\pi -\theta \right)=-\sin \theta$ if n is even. Now by using this concept, we will find the value of $\cos {{120}^{\circ }}$. Now we should express 210 in terms of $\pi +\theta$. We know that $\cos \left( n\pi +\theta \right)=-\cos \theta$ if n is odd. Now by using this concept, we will find the value of $\cos {{210}^{\circ }}$. Now we should first express 300 in terms of $2\pi -\theta$. We know that $\sin \left( n\pi -\theta \right)=-\sin \theta$ if n is even. Now by using this concept, we will find the value of $\sin {{300}^{\circ }}$. In this way, we can find the value of $\sin {{330}^{\circ }}\cos {{120}^{\circ }}+\cos {{210}^{\circ }}\sin {{300}^{\circ }}$.

Complete step-by-step answer:
From the question, it is clear that we should find the value of $\sin {{330}^{\circ }}\cos {{120}^{\circ }}+\cos {{210}^{\circ }}\sin {{300}^{\circ }}$.
Let us assume the value of $\sin {{330}^{\circ }}\cos {{120}^{\circ }}+\cos {{210}^{\circ }}\sin {{300}^{\circ }}$ is equal to I.
$I=\sin {{330}^{\circ }}\cos {{120}^{\circ }}+\cos {{210}^{\circ }}\sin {{300}^{\circ }}.....(1)$
Now we have to find the value of $\sin {{330}^{\circ }}$.
We know that $\sin \left( n\pi -\theta \right)=-\sin \theta$ if n is even.

& \Rightarrow sin{{330}^{\circ }}=\sin \left( 2\pi -{{30}^{\circ }} \right) \\\ & \Rightarrow \sin {{330}^{\circ }}=-\sin {{30}^{\circ }} \\\ \end{aligned}$$ We know that $$\sin {{30}^{\circ }}=\dfrac{1}{2}$$. $$\Rightarrow \sin {{330}^{\circ }}=-\dfrac{1}{2}......(2)$$ Now we should find the value of $$\cos {{120}^{\circ }}$$. We know that $$\cos \left( n\pi -\theta \right)=-\cos \theta $$ if n is odd. $$\begin{aligned} & \Rightarrow \cos {{120}^{\circ }}=\cos \left( \pi -{{60}^{\circ }} \right) \\\ & \Rightarrow \cos {{120}^{\circ }}=-\cos \left( {{60}^{\circ }} \right) \\\ \end{aligned}$$ We know that $$\cos {{60}^{\circ }}=\dfrac{1}{2}$$. $$\Rightarrow \cos {{120}^{\circ }}=-\dfrac{1}{2}.......(3)$$ Now we should find the value of $$\cos {{210}^{\circ }}$$. We know that $$\cos \left( n\pi +\theta \right)=-\cos \theta $$ if n is odd. $$\begin{aligned} & \Rightarrow \cos {{210}^{\circ }}=\cos \left( \pi +{{30}^{\circ }} \right) \\\ & \Rightarrow \cos {{210}^{\circ }}=-\cos \left( {{30}^{\circ }} \right) \\\ \end{aligned}$$ We know that $$\cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2}$$. $$\Rightarrow \cos {{210}^{\circ }}=-\dfrac{\sqrt{3}}{2}.......(4)$$ Now we have to find the value of $$\sin {{300}^{\circ }}$$. We know that $$\sin \left( n\pi -\theta \right)=-\sin \theta $$ if n is even. $$\begin{aligned} & \Rightarrow sin{{300}^{\circ }}=\sin \left( 2\pi -{{60}^{\circ }} \right) \\\ & \Rightarrow \sin {{300}^{\circ }}=-\sin {{60}^{\circ }} \\\ \end{aligned}$$ We know that $$\sin {{60}^{\circ }}=\dfrac{\sqrt{3}}{2}$$. $$\Rightarrow \sin {{300}^{\circ }}=-\dfrac{\sqrt{3}}{2}......(5)$$ Now let us substitute equation (2), equation (3), equation (4) and equation (5) in equation (1), then we get $$\begin{aligned} & I=\sin {{330}^{\circ }}\cos {{120}^{\circ }}+\cos {{210}^{\circ }}\sin {{300}^{\circ }} \\\ & \Rightarrow I=\left( \dfrac{-1}{2} \right)\left( \dfrac{-1}{2} \right)+\left( \dfrac{-\sqrt{3}}{2} \right)\left( \dfrac{-\sqrt{3}}{2} \right) \\\ & \Rightarrow I=\dfrac{1}{4}+\dfrac{3}{4} \\\ & \Rightarrow I=1.....(6) \\\ \end{aligned}$$ From equation (6), it is clear that the value of $$\sin {{330}^{\circ }}\cos {{120}^{\circ }}+\cos {{210}^{\circ }}\sin {{300}^{\circ }}$$ is equal to 1. **Note:** Students may have a misconception that $$\sin \left( n\pi -\theta \right)=\sin \theta $$ if n is even. If this misconception is followed, then the final answer may get interrupted. In the same way, students may have a misconception that $$\cos \left( n\pi +\theta \right)=\cos \theta $$ if n is odd. If even this misconception is followed, then also the final answer will get interrupted. So, these misconceptions should get avoided.