Question

How do you evaluate $\cos ec\left( { - \dfrac{{4\pi }}{3}} \right)$ ?

Answer 1

Hint : In order to determine the exact value of any trigonometric function of an angle, first convert the angle of the trigonometric function to an angle whose trigonometric ratio is known to us or is comparatively easier to find. We can change the angle by using the periodicity of the trigonometric functions or using some trigonometric formulae or identities. Then, we simplify the value of the expression to get the simplified value of the desired trigonometric function.

Complete step by step solution:
So, we are required to evaluate the value of $\cos ec\left( { - \dfrac{{4\pi }}{3}} \right)$ in the question given to us.
Firstly, we know that the cosecant function is periodic in nature and $2\pi$ is the fundamental period of the function.
So, $\cos ec\left( { - \dfrac{{4\pi }}{3}} \right) = \cos ec\left( { - \dfrac{{4\pi }}{3} + 2\pi } \right)$
$\Rightarrow \cos ec\left( {\dfrac{{ - 4\pi }}{3}} \right) = \cos ec\left( {\dfrac{{ - 4\pi + 6\pi }}{3}} \right)$
$\Rightarrow \cos ec\left( {\dfrac{{ - 4\pi }}{3}} \right) = \cos ec\left( {\dfrac{{2\pi }}{3}} \right)$
So, we get the value of $\cos ec\left( {\dfrac{{ - 4\pi }}{3}} \right)$ is same as $\cos ec\left( {\dfrac{{2\pi }}{3}} \right)$ .
Now, we know that the cosecant trigonometric function is the reciprocal of the sine function. So, we have, $\cos ec\left( x \right) = \dfrac{1}{{\sin \left( x \right)}}$ .
So, we get, $\cos ec\left( {\dfrac{{2\pi }}{3}} \right) = \dfrac{1}{{\sin \left( {\dfrac{{2\pi }}{3}} \right)}}$
Now, we know that the values of sine and cosecant trigonometric function are positive in the second quadrant.
Also, the value of $\sin \left( {\dfrac{{2\pi }}{3}} \right) = \left( {\dfrac{{\sqrt 3 }}{2}} \right)$
So, we get the value of $\cos ec\left( {\dfrac{{2\pi }}{3}} \right)$ as $\left( {\dfrac{2}{{\sqrt 3 }}} \right)$.
Hence, the value of $\cos ec\left( { - \dfrac{{4\pi }}{3}} \right)$ is $\left( {\dfrac{2}{{\sqrt 3 }}} \right)$.
For rationalising the denominator, we have to multiply the numerator and denominator by $\sqrt 3$ . So, we get the value of $\cos ec\left( { - \dfrac{{4\pi }}{3}} \right)$ as $\left( {\dfrac{{2\sqrt 3 }}{3}} \right)$.
So, the correct answer is “$\left( {\dfrac{{2\sqrt 3 }}{3}} \right)$.”.

Note : One must be careful while taking values from the trigonometric table and cross-check at least once to avoid any error in the answer. Trigonometric ratios are the ratios of the sides of a triangle and thus the trigonometric ratios can be found by expressing the ratios in the terms of the sides of a triangle.