Question

How do you evaluate $\cot 30^{\circ}$?

Answer 1

' cot ' is the word used in trigonometry for cotangent which is the reciprocal of a tangent. The cotangent of an angle is also defined as the cosine of the angle divided by the sin of the angle. Therefore to find the cotangent of an angle we can either use the value of the tangent of that particular angle or cosine divided sine for that particular angle. So cotangent formula requires the study of the tangent of angles as it is the reciprocal of a tangent. There are a number of trigonometric formulas that can be used here.

Complete step by step solution:
Method 1:
We can find the value of $cot\text{ }30$ directly by the definition which says it is the reciprocal of $\tan$ .
Assuming $30$ to be $30$ degrees
Here $\theta =30{}^\circ$
Thus it can be written in equation form as
$cot\text{ }30{}^\circ =\dfrac{1}{\tan 30{}^\circ }$
Now to find the value of $\tan 30{}^\circ$ , we need to have the value of $\sin 30{}^\circ$ and $\cos 30{}^\circ$ as
$tan\text{ }\theta =\text{ }\dfrac{sin\text{ }\theta }{cos\theta }\text{ }$
Here the value of $\theta$ is $30{}^\circ$ .
We know that the value of $\sin 30{}^\circ$ is $1/2$ and that the value of $\cos 30{}^\circ$ is $\dfrac{\sqrt{3}}{2}$ .
So we can use this to first find the value of tan 30°

tan30{}^\circ =sin30{}^\circ /cos30{}^\circ \\\ =\left( \surd 1/2 \right)/\left( \surd 3/2 \right) \\\ =1/\surd 3 \\\ \end{array}$$ $$\begin{array}{*{35}{l}} cot\text{ }30{}^\circ =1/tan30{}^\circ \\\ =1/\left( 1/\surd 3 \right) \\\ =\surd 3 \\\ \end{array}$$ Method 2: We can use the formula to find the value of $\cot 30{}^\circ $ The formula says $$cot\text{ }\theta =cos\text{ }\theta /sin\text{ }\theta $$ Here $$\theta =30{}^\circ $$ And we know that the value of $$\sin 30{}^\circ $$ is $1/2$ and that the value of $$\cos 30{}^\circ $$ is $\dfrac{\sqrt{3}}{2}$ . So , by the formula $$\begin{array}{*{35}{l}} cot\text{ }30{}^\circ =\text{ }cos30{}^\circ /sin30{}^\circ \\\ =\left( \surd 3/2 \right)/\left( {\scriptscriptstyle 1\\!/\\!{ }_2} \right) \\\ =\surd 3 \\\ \end{array}$$ Hence by both the methods , we get the same answer. **Therefore, $$cot30{}^\circ =\surd 3$$ is the right answer.** **Note:** The value of sin is natural and changes with change in angles. It is the most common trigonometric ratio that can be used to find the value of all other trigonometric ratios. None of sine, cosine, tangent, etc. has any unit since these are all the ratios of the same quantity which is usually longer.