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How do you find the value of \[\csc \left( {\dfrac{{5\pi }}{3}} \right)\] ?

Last updated date: 14th Jun 2024
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Hint: In this question, we need to find the value of \[\csc \left( {\dfrac{{5\pi }}{3}} \right)\]. The cosecant angle is the inverse of the reciprocal of the secant angle, where the sine angle is the ratio of perpendicular to the hypotenuse in a triangle. Since the cosecant is the reciprocal of sine angle then, it will be the ratio of the hypotenuse to perpendicular in a triangle.

Complete step by step solution:
In this question, we have given a trigonometric ratio that is \[\csc \left( {\dfrac{{5\pi }}{3}} \right)\] and we need to calculate the value of the trigonometric ratio at $\dfrac{{5\pi }}{3}$.
The first step is to write the angle as the sum or difference of the angle so that the sine and the cosine angle can easily be identified from it. This is given as,
\[ \Rightarrow \dfrac{{5\pi }}{3} = \dfrac{{6\pi - \pi }}{3}\]
After splitting we will get,
\[ \Rightarrow \dfrac{{5\pi }}{3} = 2\pi - \dfrac{\pi }{3}\]
Here, \[2\pi \] and \[\dfrac{\pi }{3}\] are the angles in which the sine and the cosine angle are easily identified while solving.
Then, we will write the given term as,
\[ \Rightarrow \csc \left( {\dfrac{{5\pi }}{3}} \right) = \dfrac{1}{{\sin \left( {\dfrac{{5\pi }}{3}} \right)}}\]
Now, we will substitute \[2\pi - \dfrac{\pi }{3}\] for \[\dfrac{{5\pi }}{3}\]in the above equation as,
\[ \Rightarrow \csc \left( {\dfrac{{5\pi }}{3}} \right) = \dfrac{1}{{\sin \left( {2\pi - \dfrac{\pi }{3}} \right)}}\]
Now we will consider the trigonometric identity \[\sin \left( {a - b} \right) = \sin a\cos b - \sin b\cos a\]. Apply this identity to above equation as,
\[ \Rightarrow \csc \left( {\dfrac{{5\pi }}{3}} \right) = \dfrac{1}{{\sin 2\pi \cos \dfrac{\pi }{3} - \cos 2\pi \sin \dfrac{\pi }{3}}}\]
As we know that the value of \[\sin 2\pi = 0\], the value of \[\sin \dfrac{\pi }{3} = \dfrac{{\sqrt 3 }}{2}\], and the value of \[\cos 2\pi = 1\]. Now we will substitute the values in the above equation as,
\[ \Rightarrow \csc \left( {\dfrac{{5\pi }}{3}} \right) = \dfrac{1}{{\left( 0 \right)\cos \dfrac{\pi }{3} - \left( 1 \right)\left( {\dfrac{{\sqrt 3 }}{2}} \right)}}\]
After calculation we will get,
\[ \Rightarrow \csc \left( {\dfrac{{5\pi }}{3}} \right) = \dfrac{1}{{ - \left( {\dfrac{{\sqrt 3 }}{2}} \right)}}\]
\[\therefore \csc \left( {\dfrac{{5\pi }}{3}} \right) = - \dfrac{2}{{\sqrt 3 }}\]

From above, the value of \[\csc \left( {\dfrac{{5\pi }}{3}} \right) = - \dfrac{2}{{\sqrt 3 }}\]

As we know that the trigonometry is a part of calculus and the basic ratio of trigonometric are sine and cosine which have their application in sound and light wave theories. The trigonometric have vast applications in naval engineering such as determining the height of the wave and the tide in the ocean.