Question

How do you find the value of \[\cot 0\]?

Answer 1

Hint: Real functions which relate any angle of a right angled triangle to the ratio of any two of its sides are Trigonometric functions. We can also use geometric definitions to evaluate trigonometric values. Here, it’s important that we know the sine of theta is the ratio of the opposite side to the hypotenuse and the cosine of theta is the ratio of the adjacent side (base) to the hypotenuse. Also, it is known that the ratio of cosine of theta to the sine of theta is cot of theta.

Complete step-by-step answer:
According to the given data, we need to evaluate \[\cot 0\]
If in a right angled triangle \[\theta \] represents one of its acute angle then by definition we can write
\[\sin \theta = \dfrac{{Opposite}}{{Hypotenuse}}\]
Also,\[\cos \theta = \dfrac{{Base}}{{Hypotenuse}}\]
Thereafter we know that,
\[\cot \theta = \dfrac{1}{{\tan \theta }} = \dfrac{{\cos \theta }}{{\sin \theta }} = \dfrac{{Base}}{{Opposite}}\]
Therefore,
\[\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}\]
\[ \Rightarrow \cot \theta = \dfrac{{\cos 0}}{{\sin 0}}\].
Hence, it is known to us that \[\cos 0 = 1\] and \[\sin 0 = 0\].
When we substitute the values in the expression we get,
\[ \Rightarrow \cot \theta = \dfrac{1}{0} = \infty \]
This gives rise to the fact that \[\cot x\] doesn't exist for \[x = n\pi \].
Hence, the value of \[\cot 0\] is not defined (Infinity).

Note: One should be careful while evaluating trigonometric values and rearranging the terms to convert from one function to the other. Trigonometric functions are real functions which relate any angle of a right angled triangle to the ratio of any two of its sides. The widely used ones are sin, cos and tan. While the rest can be referred to as the reciprocal of the others, i.e., cosec, sec and cot respectively. If in a right angled triangle θ represents one of its acute angles then, \[\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}\].