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

Hint: 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.