Question

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

Answer 1

Hint: Cosecant is the reciprocal of sin that is ${\text{cosec x}} = \dfrac{1}{{\sin x}}$. Now, note that when the numerator is perpendicular and denominator is hypotenuse is known as sin x that means ${\text{sin x}} = \dfrac{{Perpendicular}}{{Hypotenuse}}$ . So as Cosecant is reciprocal of sin x that means when the numerator is hypotenuse and numerator is perpendicular is known as csc x then${\text{csc x}} = \dfrac{{Hypotenuse}}{{Perpendicular}}$ . As we know that the value of sin x ranges between -1 < x < 1. So, the range of csc x lies between 1 < x < -1.

Complete step by step solution:
We have to find the value of csc ( \[\dfrac{\pi }{3}\] ). As we know that ${\text{csc x = }}\dfrac{1}{{\sin x}}$. Then we just need to recall that the value of sin( $\dfrac{\pi }{3}$ ) = $\dfrac{{\sqrt 3 }}{2}$[where $\dfrac{\pi }{3}$in degrees is ${60^ \circ }$]
So, the csc( $\dfrac{\pi }{3}$ ) which is reciprocal of sin( $\dfrac{\pi }{3}$ ) , we get,
$ \Rightarrow {\text{csc x}} = \dfrac{1}{{{\text{sin x}}}}$
Replace the x with $\dfrac{\pi }{3}$ , we get,
$ \Rightarrow {\text{csc }}\dfrac{\pi }{3} = \dfrac{1}{{{\text{sin}}\dfrac{\pi }{3}}}$
Now just place the value of sin( $\dfrac{\pi }{3}$ ) = $\dfrac{{\sqrt 3 }}{2}$, we get,
$ \Rightarrow {\text{csc }}\dfrac{\pi }{3} = \dfrac{1}{{\dfrac{{\sqrt 3 }}{2}}}$
So, when the right-hand side get reciprocal, we get
$ \Rightarrow {\text{csc }}\dfrac{\pi }{3} = \dfrac{2}{{\sqrt 3 }}$

So, the value of csc ( \[\dfrac{\pi }{3}\] ) will be $\dfrac{2}{{\sqrt 3 }}$.

Additional Information:
We can get the value of sin x by dividing the perpendicular with hypotenuse. And csc x is the reciprocal of sin x. So, we can find the value of csc x by dividing the hypotenuse with perpendicular in a right-angle triangle.

Note:
We can find csc x when the length of the hypotenuse is divided with the perpendicular that is the length of the opposite side of the hypotenuse, it gives the csc x angle in a right triangle. And we also know that sin x is the reciprocal of csc x.