Question

What is the value of $\sin \left( {{{1920}^ \circ }} \right)$?
A.$\dfrac{1}{2}$
B.$\dfrac{1}{{\sqrt 2 }}$
C.$\dfrac{{\sqrt 3 }}{2}$
D.$\dfrac{1}{3}$

Answer 1

Here in this question, we have to find the exact value of a given trigonometric function. For this, first we need to write the given angle in terms of the sum of difference of standard angle, then apply a periodic function and ASTC rule of trigonometry and further simplify by using a table of standard angle of trigonometry we get the required solution.

Complete answer:
Consider the given function
$\Rightarrow \,\,\sin \left( {{{1920}^ \circ }} \right)$-------(1)
The angle ${1920^ \circ }$ can be written as ${1800^ \circ } + {120^ \circ }$, then
Equation (1) becomes
$\Rightarrow \,\sin \left( {{{1800}^ \circ } + {{120}^ \circ }} \right)$ ------(2)
The angle ${1800^ \circ }$ can be written as $5 \times {360^ \circ }$, then
Equation (2) becomes
$\Rightarrow \,\sin \left( {5{{\left( {360} \right)}^ \circ } + {{120}^ \circ }} \right)$
In radians angle ${360^ \circ }$ can be written as $2{\pi ^c}$, then we have
$\Rightarrow \,\sin \left( {5\left( {2\pi } \right) + {{120}^ \circ }} \right)$ ------(3)
By the Periodic function of trigonometry, since $\sin \left( {2n\pi + \theta } \right) = \sin \theta$, for all values of $\theta$ and $n \in N$. Then equation (3) becomes
$\Rightarrow \,\sin \left( {{{120}^ \circ }} \right)$
The angle ${120^ \circ }$ can be written as ${90^ \circ } + {30^ \circ }$, then
$\Rightarrow \,\sin \left( {{{90}^ \circ } + {{30}^ \circ }} \right)$ ---(4)
Let us by the complementary angles and ASTC rule of trigonometric ratios:
The angle can be written as
$\sin \left( {90 + \theta } \right) = \cos \theta$
Then equation (4) becomes
$\Rightarrow \,\cos \left( {{{30}^ \circ }} \right)$
By using specified cosine and sine angle i.e., $cos\,\,\dfrac{\pi }{6} = \cos {30^0} = \dfrac{{\sqrt 3 }}{2}$.
$\therefore \,\,\sin \left( {{{1920}^ \circ }} \right) = \dfrac{{\sqrt 3 }}{2}$
Hence, the value of $\,\sin \left( {{{1920}^ \circ }} \right) = \dfrac{{\sqrt 3 }}{2}$.
Therefore, option (C) is the correct answer.

Note:
$2n\pi$ is the total angle of a circle. If we have taken a triangle with an angle $\theta$ and added $2n\pi$ to that angle, the triangle would rotate ${360^ \circ }$ and would be at the same coordinates. As the triangle is at the same coordinates, the length of the sides remains the same and so will be the ratios. So, $\sin \left( {2n\pi + \theta } \right) = \sin \theta$.
Remember, when the sum of two angles is ${90^ \circ }$, then the angles are known as complementary angles at that time the ratios will change like $\sin \leftrightarrow \cos$, $\sec \leftrightarrow cosec$ and $\tan \leftrightarrow \cot$ then should know the value of standard angles and basic three trigonometric identities.