Question

How do you find the exact value of \[\cot {{510}^{\circ }}\]?

Answer 1

Hint: In This problem, we have to find the exact value of \[\cot {{510}^{\circ }}\]. We can first remove the full rotation of \[{{360}^{\circ }}\] until the angle is between \[{{0}^{\circ }}\]and \[{{360}^{\circ }}\]. We can then apply the reference angle by finding the angle with equivalent trigonometric values in the first quadrant. We can write it as negative, as cotangent is negative in the second quadrant. We can then find the value of the given cotangent.

Complete step by step solution:
We know that the given cotangent is,
\[\cot {{510}^{\circ }}\]
We can now first remove the full rotation of \[{{360}^{\circ }}\] until the angle is between \[{{0}^{\circ }}\]and \[{{360}^{\circ }}\].
We can now subtract the given degree from \[{{360}^{\circ }}\], we get
\[\Rightarrow \cot \left( {{510}^{\circ }}-{{360}^{\circ }} \right)=\cot {{150}^{\circ }}\]
We can then apply the reference angle by finding the angle with equivalent trigonometric values in the first quadrant.
Here the reference angle for \[{{150}^{\circ }}\] will be \[{{30}^{\circ }}\].
We can write it as,
\[\Rightarrow \cot {{30}^{\circ }}\]
Which will be equal to,
 \[\Rightarrow \cot {{30}^{\circ }}=\sqrt{3}\]
 We can write it as negative, as cotangent is negative in the second quadrant.
\[\Rightarrow -\sqrt{3}\]
Therefore, the exact value of \[\cot {{510}^{\circ }}\] is \[-\sqrt{3}\].

Note: We should always remember that we should write the result in negative as the cotangent is negative in the second quadrant. We should know some trigonometric degree values to solve these types of problems. We can also write the exact value for the result in the decimal form, if needed
Therefore, the exact value of \[\cot {{510}^{\circ }}\]is -1.73205080.