Question

How do you evaluate $\csc \left( {\dfrac{{7\pi }}{6}} \right)$ ?

Answer 1

Hint : Here the question is related to the trigonometry, we use the trigonometry ratios and we are going to solve this question. By using the trigonometry properties we are going to solve this problem. To find the value we need the table of trigonometry ratios for standard angles.

Complete step-by-step answer :
The question is related to trigonometry and it includes the trigonometry ratios. The trigonometry ratios are sine, cosine, tan, cosec, sec, and cot.
Now consider the given question
$\csc \left( {\dfrac{{7\pi }}{6}} \right)$
The cosecant is the reciprocal of sine.
So let we find the value of $\sin \left( {\dfrac{{7\pi }}{6}} \right)$
This term can be written as
$\Rightarrow \sin \left( {\pi + \dfrac{\pi }{6}} \right)$
This will lie in the third quadrant. In trigonometry we have ASTC rules for the trigonometry ratios. The above inequality lies in the third quadrant and the sine trigonometry ratio is negative in the third quadrant.
As we know that $\sin \left( {\pi + \theta } \right) = - \sin \theta$ , by using this condition the above equation is written as
$\Rightarrow - \sin \left( {\dfrac{\pi }{6}} \right)$
So we have a table for the trigonometry ratio sine for the standard angles.

Angle	0	30	45	60	90
sine	0	$\dfrac{1}{2}$	$\dfrac{1}{{\sqrt 2 }}$	$\dfrac{{\sqrt 3 }}{2}$	1

The above table is given for the standard degree. But in the above equation the value is in the radians. $\dfrac{\pi }{6}$ represents the angle value ${30^ \circ }$ . By considering the table we have
$\Rightarrow - \dfrac{1}{2}$
The cosine is the reciprocal of sine so we have
$\Rightarrow \csc \left( {\dfrac{{7\pi }}{6}} \right) = \dfrac{1}{{\sin \left( {\dfrac{{7\pi }}{6}} \right)}} = \dfrac{1}{{\sin \left( {\dfrac{\pi }{6}} \right)}}$
On substituting the value we have
$\Rightarrow \csc \left( {\dfrac{{7\pi }}{6}} \right) = \dfrac{1}{{\left( {\dfrac{{ - 1}}{2}} \right)}}$
On simplifying we have
$\Rightarrow \csc \left( {\dfrac{{7\pi }}{6}} \right) = - 2$
Hence we have solved the trigonometry ratio and found the value.
So, the correct answer is “ - 2”.

Note : In trigonometry to find the value of angles we have a table of trigonometry ratios for the standard angles. ASTC rule is applicable for the highest values. Whether the value of angle is in degree or radians the value for the standard angles will not change. Where ASTC rule is abbreviated as ALL SINE TANGENT COSINE.