Question

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

Short answer

\theta\text{ is an integer multiple of } \pi \.

Step-by-step solution

Understand the definition of \(\text{csc} \theta \)

The cosecant function, \( \text{csc} \theta \), is defined as \( \text{csc} \theta = \frac{1}{\text{sin} \theta} \). It is the reciprocal of the sine function.

Determine when \(\text{csc} \theta \) is not defined

Since \( \text{csc} \theta \) is the reciprocal of the sine function, it will be undefined when \( \text{sin} \theta = 0 \). This is because division by zero is undefined.

Find the angles where \(\text{sin} \theta = 0 \)

The sine function equals zero at integer multiples of \( \pi \). Therefore, \( \text{sin} \theta = 0 \) when \( \theta = n\pi \), where \( n \) is any integer.

Key concepts

reciprocal trigonometric functions

In trigonometry, reciprocal trigonometric functions play a key role by providing alternate ways to express relationships between angles and sides in right triangles. For the cosecant function, denoted as \text{csc} \theta\, it is defined as the reciprocal of the sine function. This means: \[ \text{csc} \theta = \frac{1}{\text{sin} \theta} \]
This relationship shows that for any given angle \(\theta\), the cosecant value is calculated by taking 1 and dividing it by the sine of \(\theta\). There are other reciprocal trigonometric functions like secant (\(\text{sec} \theta\)) and cotangent (\(\text{cot} \theta\)) that have similar reciprocal relationships with cosine and tangent respectively.
Understanding the reciprocal nature of these functions helps in simplifying complex trigonometric problems by transforming them into more familiar terms.

undefined trigonometric values

Trigonometric functions can sometimes result in undefined values. This typically happens when the function involves division by zero, which is mathematically undefined. For the cosecant function, \(\text{csc} \theta\), it becomes undefined when the sine of \(\theta\) equals zero: \[ \text{csc} \theta = \frac{1}{\text{sin} \theta} \]
Specifically, when \(\text{sin} \theta = 0\), the calculation of cosecant involves dividing by zero, which is not possible. Therefore, \(\text{csc} \theta\) does not exist when \(\text{sin} \theta = 0\).
Always be cautious about these specific values to avoid erroneous computations in trigonometric problems.

sine function zeros

The sine function, \(\text{sin} \theta\), equals zero at specific angles. Recognizing these angles is crucial for understanding when cosecant becomes undefined. For sine, the function zeros occur at integer multiples of \(\pi\): \[ \text{sin} \theta = 0 \ \text{where}\ \theta = n\pi,\ n \in\mathbb{Z}\ \] This means:

• \(\sin 0 = 0\)
• \(\sin \pi = 0\)
• \(\sin 2\pi = 0\)
• and so on...

Whenever \(\theta\) is any multiple of \(\pi\), the sine function reaches zero. Consequently, the cosecant function, being the reciprocal of sine, is undefined at these values.
Understanding here simplifies the identification of angles where trigonometric functions are not defined.