What is the correct definition for [tex]\cot \theta[/tex]?

A. [tex]\cot \theta = \tan^{-1} \theta[/tex]

B. [tex]\cot \theta = \frac{\sin \theta}{\cos \theta}[/tex]

C. [tex]\cot \theta = \frac{\cos \theta}{\sin \theta}[/tex]

D. [tex]\cot \theta = \frac{1}{\cos \theta}[/tex]

E. [tex]\cot \theta = \frac{1}{\sin \theta}[/tex]



Answer :

To determine the correct definition for [tex]\(\cot \theta\)[/tex], we start by recalling the basic definitions and properties of trigonometric functions in terms of sine ([tex]\(\sin\)[/tex]) and cosine ([tex]\(\cos\)[/tex]).

[tex]\(\tan \theta\)[/tex] is defined as:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]

Given this, the cotangent function, denoted [tex]\(\cot \theta\)[/tex], is known as the reciprocal of the tangent function. Therefore:
[tex]\[ \cot \theta = \frac{1}{\tan \theta} \][/tex]

Using the definition of [tex]\(\tan \theta\)[/tex]:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]

So,
[tex]\[ \cot \theta = \frac{1}{\frac{\sin \theta}{\cos \theta}} \][/tex]

By taking the reciprocal of the fraction, we get:
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \][/tex]

Now, we compare this result with the given options:

A. [tex]\(\cot \theta = \tan^{-1} \theta\)[/tex]
B. [tex]\(\cot \theta = \frac{\sin \theta}{\cos \theta}\)[/tex]
C. [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex]
D. [tex]\(\cot \theta = \frac{1 \theta}{\cos \theta}\)[/tex]
E. [tex]\(\cot \theta = \frac{1}{\sin \theta}\)[/tex]

From our derivation, the correct definition that matches our result is:

C. [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex]

So, the correct choice is option C.