Question

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

Answer 1

Hint: 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:
\[\cot {330^ \circ } = \cot ({360^ \circ } - {30^ \circ }) = - \cot {30^ \circ } = - \sqrt 3 \].