Question

What is the exact value of \[\cot\ 210^{o}\] ?

Answer 1

Hint: In this question, we need to find the value of \[\cot\ 210^{o}\] . In this type of question we have to use the concepts of trigonometry. We know that the cotangent is the ratio of the cosine and sine functions so we can write this cotangent expression as the ratio of cosine and sine. Then we can rewrite the angle \[210^{o}\] in the form of \[(180^{o} + \theta)\] . Then by using trigonometric identities , we can simplify the expression. Using the value of \[\sin(30^{o})\] and \[\cos(30^{o})\] , we can find the value of cot \[210^{o}\] .
Identities used :
1. \[\sin(180^{o} + \theta)\ = - \sin\theta\]
2. \[\cos\left( 180^{o} + \theta \right) = - \cos\theta\]
Trigonometry table :

Angles	\[0^{o}\]	\[30^{o}\]	\[45^{o}\]	\[60^{o}\]	\[90^{o}\]
Sine	\[0\]	\[\dfrac{1}{2}\]	\[\dfrac{1}{\sqrt{2}}\]	\[\dfrac{\sqrt{3}}{2}\]	\[1\]
Cosine	\[1\]	\[\dfrac{\sqrt{3}}{2}\]	\[\dfrac{1}{\sqrt{2}}\]	\[\dfrac{1}{2}\]	\[0\]

Complete step-by-step answer:
Given, \[\cot\ 210^{o}\]
Here we need to find the exact value of \[\cot\ 210^{o}\] .
We know that the cotangent function is the ratio of the cosine and sine functions.
\[\cot\theta = \dfrac{\cos\theta}{\sin\theta}\]
Here angle \[\theta\] is \[210^{o}\] .
\[\cot\ 210^{o} = \dfrac{\cos\left( 210^{o} \right)}{\sin\left( 210^{o} \right)}\]
We can rewrite \[210^{o}\] as \[(180^{o}+ 30^{o})\] ,
We get,
\[\Rightarrow \ \cot\ 210^{o} = \dfrac{\cos\left( 180^{o} + 30^{o} \right)}{\sin\left( 180^{o} + 30^{o} \right)}\]
By using the identities , \[\sin(180^{o} + \theta)\ = - \sin\theta\] and \[\cos(180^{o} + \theta) = - \cos\theta\]
We can write,
\[\Rightarrow \ \cot\ 210^{o} = \dfrac{- \cos\left( 30^{o} \right)}{- \sin\left( 30^{o} \right)}\]
From the trigonometric table, the value of \[\sin(30^{o})\] is \[\dfrac{1}{2}\] and the value of \[\cos(30^{o})\] is \[\dfrac{\sqrt{3}}{2}\] .
\[\Rightarrow \ \cot\ 210^{o} = \dfrac{- \dfrac{\sqrt{3}}{2}}{- \dfrac{{1}}{2}}\]
On multiplying the numerator by the reciprocal of the denominator,
We get,
\[\Rightarrow \ \cot\ 210^{o} = \left( - \dfrac{\sqrt{3}}{2} \right) \times \left( - \dfrac{2}{{1}} \right)\]
On simplifying,
We get,
\[\cot\ 210^{o} = \dfrac{ - \sqrt{3}}{-1} \]
On further simplifying,
We get,
\[\cot\ 210^{o} = \sqrt{3}\]
Thus the value of \[\cot\ 210^{o}\] is equal to \[\sqrt{3}\] .
Final answer :
The exact value of \[\cot\ 210^{o}\] is equal to \[\sqrt{3}\] .

Note: The concept used in this problem is use of trigonometric identities and ratios. The given angle \[210^{o}\] will be in the third quadrant and in the third quadrant tangent and cotangent functions will be positive. Also we have to be careful while taking the value of \[\sin(30^{o})\] and \[\cos(30^{o})\] . If the value of \[\sin(30^{o})\] and \[\cos(30^{o})\] is not known then it is hard to solve this question. If it is not known then it is hard to solve this question.