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.
