91Ó°ÊÓ

The cotangent function, often abbreviated as \(\text{cot}\), is an important trigonometric function. It is defined as the reciprocal of the tangent function. This means that for any angle \(\theta\), \(\text{cot}(\theta) = \frac{1}{\text{tan}(\theta)}\).

For example, if you know the value of \(\text{tan}(\theta)\), finding \(\text{cot}(\theta)\) is straightforward. Simply take the reciprocal. However, you need to be careful when \(\text{tan}(\theta)\) is zero. Division by zero is undefined, meaning \(\text{cot}(\theta)\) will also be undefined in such cases.

In the given exercise, we are to find \(\text{cot}(\theta)\) when \(\theta = \pi\). We use the relationship between sine, cosine, and tangent from the unit circle to solve it.

Understanding the unit circle is fundamental in trigonometry. The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is used to define sine, cosine, and tangent for all angles.

Each angle \(\theta\) on the unit circle corresponds to a point (x, y) where \(\text{x}\) is \(\text{cos}(\theta)\) and \(\text{y}\) is \(\text{sin}(\theta)\).

For the angle \(\theta = \pi\):

• \(\text{sin}(\theta) = \text{sin}(\pi) = 0\)
• \(\text{cos}(\theta) = \text{cos}(\pi) = -1\)

With this information, you can find tangent and subsequently cotangent using their definitions.

The reciprocal of tangent is key to defining cotangent. If \(\text{tan}(\theta)\) equals the ratio of sine to cosine for a given angle \(\theta\), the cotangent \(\text{cot}(\theta)\) is the ratio of cosine to sine.

Consider the relationship: \(\text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}\). Therefore, \(\text{cot}(\theta) = \frac{\text{cos}(\theta)}{\text{sin}(\theta)}\).

In the case of angle \(\theta = \pi \):

• \(\text{sin}(\pi) = 0\)
• \(\text{cos}(\pi) = -1\)

Using these values, \(\text{tan}(\pi) = \frac{0}{-1} = 0\), which leads to \(\text{cot}(\pi) = \frac{1}{0}\). Since division by zero is undefined, \(\text{cot}(\pi)\) is also undefined.