Question

How do I find the value of $\cot {330^ \circ }?$

Answer 1

Here we will use a method to find the exact value of $\cot {330^ \circ }$. Also, putting the value for the term and after some simplification we get the required answer.

Formula used: We will use the following formulas:
$\tan ( - \theta ) = - \tan \theta$ and $\cot ( - \theta ) = - \cot \theta$
And, $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
Also, $\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}$
And, $\cot ({360^ \circ } - \theta ) = - \cot \theta$
Also we will use the following chart of $All - \sin - \tan - \cos$ :

So, it is clear that in the first quadrant all are positive.
But in the second quadrant $\sin$ and $\cos ec$ are positive but all others are negative by sign.
In the third quadrant $\tan$ and $\cot$ are positive but all others are negative by sign.
In the fourth quadrant $\cos$ and $\sec$ are positive but all others are negative by sign.

Complete step-by-step solution:
Now we can write $\cot {330^ \circ }$ as $\cot ({360^ \circ } - {30^ \circ })$.
So, if we try to simulate these values in the quadrants, then we can say that it will come under the ${4^{th}}$ quadrant.
But in the ${4^{th}}$ quadrant, only $\cos$ and $\sec$ are positive and all other parameters are negative.
So, the value $\cot \theta$ will be negative in ${4^{th}}$ quadrant.
So, we can say that $\cot ({360^ \circ } - {30^ \circ }) = - \cot {30^ \circ }$.
But the value of $\cot {30^ \circ }$ is $\sqrt 3$.
So, the value of $\cot {30^ \circ } = - \sqrt 3$.
So, we can say that $\cot {330^ \circ } = \cot ({360^ \circ } - {30^ \circ }) = - \cot {30^ \circ } = - \sqrt 3$.

$\therefore$ The value $\cot {330^ \circ }$ is $- \sqrt 3$.

Note: To find the value of $\cot \theta$, we can use the following formula also:
$\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}$.
So, we can write $\cot {30^ \circ }$ as $\dfrac{{\cos {{30}^ \circ }}}{{\sin {{30}^ \circ }}}$.
But we know the value of $\cos {30^ \circ }$is $\dfrac{{\sqrt 3 }}{2}$ and the value of $\sin {30^ \circ }$ is $\dfrac{1}{2}$.
So, the value of $\cot {30^ \circ }$ can be written as:
$\cot {30^ \circ } = \dfrac{{\cos {{30}^ \circ }}}{{\sin {{30}^ \circ }}} = \dfrac{{\dfrac{{\sqrt 3 }}{2}}}{{\dfrac{1}{2}}}$.
By solving it, we can state that:
$\cot {30^ \circ } = \dfrac{{\sqrt 3 }}{2} \times \dfrac{2}{1} = \sqrt 3$.
So, we can write it again as:
$\cot {330^ \circ } = \cot ({360^ \circ } - {30^ \circ }) = - \cot {30^ \circ } = - \sqrt 3$.