# Find the value of\[\sin {{120}^{\circ }}\].

A) \[\dfrac{\sqrt{3}}{2}\]

B) \[-\dfrac{\sqrt{3}}{2}\]

C) \[\dfrac{1}{2}\]

D) \[-\dfrac{1}{2}\]

Last updated date: 17th Mar 2023

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Answer

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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:

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 \[\cos 30=\dfrac{\sqrt{3}}{2}\]

\[\therefore \sin {{120}^{\circ }}=\dfrac{\sqrt{3}}{2}\].

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 \[\cos 30=\dfrac{\sqrt{3}}{2}\]

\[\therefore \sin {{120}^{\circ }}=\dfrac{\sqrt{3}}{2}\].

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