Question

For what numbers \(\theta\) is \(f(\theta)=\cot \theta\) not defined?

Short answer

\(\theta eq k\text{\textpi}\) where k is an integer.

Step-by-step solution

Understanding the cotangent function

The cotangent function is defined as \(\f(\theta)=\frac{\text{cos}\theta}{\text{sin}\theta}\). This means it is only defined when \(\text{sin}\theta eq 0\).

Identify when \(\text{sin}\theta\) is zero

The sine function equals zero at integer multiples of \(\theta=\frac{k\text{\textpi}}{a}\), represented as \(\text{\texttheta} = k \text{\textpi}\), where k is an integer (\(k \in \mathbb{Z}\)).

State the values of \(\theta\)

Therefore, \(\theta\) cannot be any multiple of \(\text{\textpi}\): \(\theta eq k\text{\textpi}\), where k is any integer.

Key concepts

trigonometric functions

Understanding trigonometric functions is crucial in solving problems involving angles and their relationships. The main trigonometric functions are sine (\text{sin}), cosine (\text{cos}), tangent (\text{tan}), cotangent (\text{cot}), secant (\text{sec}), and cosecant (\text{csc}). These functions help describe the properties of triangles and relate angles to side lengths.

In the context of the cotangent function, defined as \( \text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta} \), it is essential to recognize its relationship with sine and cosine. Cotangent can be thought of as the reciprocal of the tangent function (\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}). By understanding these fundamental relationships, you can easily navigate through problems involving these functions.

sine function

The sine function (\text{sin} \theta) is one of the most well-known trigonometric functions. It represents the ratio of the length of the opposite side of an angle to the hypotenuse in a right-angled triangle.

The sine function has distinct properties:

• Its values range from -1 to 1.
• It has a periodic behavior, repeating every 2\text{\textpi}.
• Key points on its graph include \( \text{sin} 0 = 0 \), \( \text{sin} \frac{\text{\textpi}}{2} = 1 \), and \( \text{sin} \text{\textpi} = 0 \).

Understanding when the sine function equals zero is crucial in problems involving undefined values. In this case, sine equals zero at integer multiples of \text{\textpi}, i.e., \( \theta = k \text{\textpi} \), where k is an integer.

undefined values

Spotting undefined values in trigonometric functions can save you from common pitfalls. For the cotangent function, which is defined as \( \text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta} \), it becomes undefined whenever \text{sin} \theta equals zero.

Mathematically, \text{sin} \theta is zero at \( \theta = k \text{\textpi} \), where k is any integer (\text{k} ∈ \text{\textbb{Z}}). This means that the cotangent function is not defined at these points. For example, \( \text{sin} (0) = 0 \) and \( \text{sin} (\text{\textpi}) = 0 \), making the cotangent function undefined at \( \theta = 0 \) and \( \theta = \text{\textpi} \), respectively.

Identifying these undefined values helps in solving problems accurately and avoids potential errors when dealing with trigonometric functions.