Question

Find the value of $\sin 330{}^\circ $

Answer 1

Hint: Rewrite the given expression as $\sin \left( 330{}^\circ \right)=\sin \left( 360{}^\circ -30{}^\circ \right)$. Then use the property $\sin \left( 360{}^\circ -x \right)=-\sin x$ to get $\sin \left( 330{}^\circ \right)=-\sin \left( 30{}^\circ \right)$ which gives us $\sin \left( 330{}^\circ \right)=-\dfrac{1}{2}$. This is the final answer.

Complete step by step answer:
In this question, we need to find the value of $\sin 330{}^\circ $.
We notice in this question that $330{}^\circ $ is a very big angle. It is typically a non-conventional angle and students may not directly know the sine of this angle. To counter this problem, we will simplify the given expression to a form which will be easy to solve and students will be aware of.
We can write $\sin 330{}^\circ $ as $\sin \left( 360{}^\circ -30{}^\circ \right)$
i.e. $\sin \left( 330{}^\circ \right)=\sin \left( 360{}^\circ -30{}^\circ \right)$
Now, as we can see from the diagram below that sine is positive in the first quadrant and that it is negative in the fourth quadrant.

Owing to this nature, we have the following property:
For any angle x, the sine of the angle (360 – x) will be equal to the negative of the sine of angle x.
i.e. $\sin \left( 360{}^\circ -x \right)=-\sin x$
Using this property, we can rewrite the given expression as:
$\sin \left( 330{}^\circ \right)=\sin \left( 360{}^\circ -30{}^\circ \right)$
$\sin \left( 330{}^\circ \right)=-\sin \left( 30{}^\circ \right)$
We, also know that the value of $\sin \left( 30{}^\circ \right)=\dfrac{1}{2}$
Substituting this in the above equation, we will get the following:
$\sin \left( 330{}^\circ \right)=-\sin \left( 30{}^\circ \right)$
$\sin \left( 330{}^\circ \right)=-\dfrac{1}{2}$
This is our final answer.

Note: In this question, it is very important to notice that $330{}^\circ $ is a very big angle. It is typically a non-conventional angle and students may not directly know the sine of this angle. To counter this problem, we have to simplify the given expression to a form which will be easy to solve and students will be aware of.