Question

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

Hint: 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 $\sin {{330}^{\circ }}\cos {{120}^{\circ }}+\cos {{210}^{\circ }}\sin {{300}^{\circ }}$.

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.
\begin{align} & \Rightarrow sin{{330}^{\circ }}=\sin \left( 2\pi -{{30}^{\circ }} \right) \\ & \Rightarrow \sin {{330}^{\circ }}=-\sin {{30}^{\circ }} \\ \end{align}
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{align} & \Rightarrow \cos {{120}^{\circ }}=\cos \left( \pi -{{60}^{\circ }} \right) \\ & \Rightarrow \cos {{120}^{\circ }}=-\cos \left( {{60}^{\circ }} \right) \\ \end{align}
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{align} & \Rightarrow \cos {{210}^{\circ }}=\cos \left( \pi +{{30}^{\circ }} \right) \\ & \Rightarrow \cos {{210}^{\circ }}=-\cos \left( {{30}^{\circ }} \right) \\ \end{align}
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{align} & \Rightarrow sin{{300}^{\circ }}=\sin \left( 2\pi -{{60}^{\circ }} \right) \\ & \Rightarrow \sin {{300}^{\circ }}=-\sin {{60}^{\circ }} \\ \end{align}
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{align} & 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{align}
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.