Question

Find the value of\[\sin {{120}^{\circ }}\].
A) \[\dfrac{\sqrt{3}}{2}\]
B) \[-\dfrac{\sqrt{3}}{2}\]
C) \[\dfrac{1}{2}\]
D) \[-\dfrac{1}{2}\]

Answer 1

Hint: Draw a unit circle and chart out the trigonometric values on each quadrant of the circle. Find the value of \[\sin {{120}^{\circ }}\] or the value of \[\sin {{120}^{\circ }}\]can be taken from the trigonometric table. Find either sine function or cosine function.

Complete step-by-step answer:
By using a unit circle we can find the value of \[\sin {{120}^{\circ }}\]. Now let us draw a cartesian plane with \[x=\cos \theta \] and \[y=\sin \theta \].
 Let us draw the trigonometric table as well:

	sin	cos	tan	cot	sec	Cosec
0	0	1	0	N.A	1	N.A
30	\[\dfrac{1}{2}\]	\[\dfrac{\sqrt{3}}{2}\]	\[\dfrac{1}{\sqrt{3}}\]	\[\sqrt{3}\]	\[\dfrac{2\sqrt{3}}{3}\]	2
45	\[\dfrac{1}{\sqrt{2}}\]	\[\dfrac{1}{\sqrt{2}}\]	1	1	\[\sqrt{2}\]	\[\sqrt{2}\]
60	\[\dfrac{\sqrt{3}}{2}\]	\[\dfrac{1}{2}\]	\[\sqrt{3}\]	\[\dfrac{\sqrt{3}}{3}\]	2	\[\dfrac{2\sqrt{3}}{3}\]
90	1	0	N.A	0	N.A	1

Now let us mark their values in the unit circle.
 

Here, \[x=\cos \theta ,y=\sin \theta \].
Eg: -\[\left( \cos {{30}^{\circ }},\sin {{30}^{\circ }} \right)=\left( x,y \right)\]
\[\left( x,y \right)=\left( \dfrac{\sqrt{3}}{2},\dfrac{1}{2} \right)\].
In the first quadrant, the values of \[\cos \theta \] and \[\sin \theta \] are positive.
In the second quadrant, the value of \[\cos \theta \] is negative and \[\sin \theta \] is positive.
In the third quadrant, both are negative.
In the fourth quadrant, \[\cos \theta \] is positive and \[\sin \theta \] is negative.
By looking into the figure, you can find that \[\sin 60=\sin 120\].
i.e. \[\sin 60=\sin 120=\dfrac{\sqrt{3}}{2}\]
Or if we are directly taking value from the trigonometric table, we need to find the value of \[\sin {{120}^{\circ }}\] by using other angles of sin functions such as \[{{60}^{\circ }}\] and \[\sin {{180}^{\circ }}\].
We know that \[{{180}^{\circ }}-{{60}^{\circ }}={{120}^{\circ }}\].
We also know that the trigonometric identity:
\[\sin \left( 180-\theta \right)=\sin \theta \].
Put, \[\theta ={{120}^{\circ }}\].
\[\begin{align}
  & \Rightarrow \sin \left( 180-120 \right)=\sin {{120}^{\circ }} \\
 & \Rightarrow \sin {{60}^{\circ }}=\sin {{120}^{\circ }}=\dfrac{\sqrt{3}}{2} \\
\end{align}\]
From the trigonometric table, find the value of \[\sin {{60}^{\circ }}=\dfrac{\sqrt{3}}{2}\].
\[\therefore \]Value of \[\sin {{120}^{\circ }}=\dfrac{\sqrt{3}}{2}\]
\[\therefore \]Option (a) is the correct answer.
Note:
We can also find the value of \[\sin {{120}^{\circ }}\] by using cosine function.
Using the trigonometry formula,
\[\sin \left( 90+\theta \right)=\cos \theta \]
Thus to find the values of \[\sin {{120}^{\circ }}\], put \[\theta ={{30}^{\circ }}\]
as, \[{{90}^{\circ }}+{{30}^{\circ }}={{120}^{\circ }}\]
\[\begin{align}
  & \Rightarrow \sin \left( 90+30 \right)=\cos 30 \\
 & \sin {{120}^{\circ }}=\cos {{30}^{\circ }} \\
\end{align}\]
From trigonometric table, value of \[\cos 30=\dfrac{\sqrt{3}}{2}\]
\[\therefore \sin {{120}^{\circ }}=\dfrac{\sqrt{3}}{2}\].