Question

Show that the period of \(f(\theta)=\csc \theta\) is \(2 \pi\)

Short answer

The period of \(\csc \theta\) is \(2 \pi\) because \(\csc(\theta + 2 \pi) = \csc \theta\).

Step-by-step solution

- Understand the function

The function given is \(f(\theta) = \csc \theta\). Recall that the cosecant function is defined as the reciprocal of the sine function: \(\csc \theta = \frac{1}{\sin \theta}\).

- Identify the periodicity of sine

The sine function, \(\sin \theta\), is periodic with a period of \(2 \pi\). This means that \(\sin(\theta + 2 \pi) = \sin \theta\) for any angle \(\theta\).

- Use the periodicity of sine to find the period of cosecant

Since \(\csc \theta\) is defined in terms of \(\sin \theta\), the periodicity of \(\sin \theta\) determines the periodicity of \(\csc \theta\). If \(\sin(\theta + 2 \pi) = \sin \theta\), then it follows that \(\csc(\theta + 2 \pi) = \frac{1}{\sin(\theta + 2 \pi)} = \frac{1}{\sin \theta} = \csc \theta\).

- Conclude the period

Since \(\csc(\theta + 2 \pi) = \csc \theta\) for any angle \(\theta\), this shows that the period of the cosecant function \(\csc \theta\) is \(2 \pi\).

Key concepts

cosecant function

The cosecant function, denoted as \(\text{csc} \theta\), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function. This means that \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\). Therefore, wherever the sine function is zero, the cosecant function is undefined since division by zero is not possible.

Key points to understand:

• The graph of cosecant mirrors the sine graph but with vertical asymptotes where sine equals zero.
• It has the same periodicity as the sine function.

These characteristics make it easier to understand the behavior and transformations of the cosecant function.

reciprocal functions

Reciprocal functions are those that are defined as the inverse of another function. In trigonometry, reciprocal functions are common and most trigonometric functions have corresponding reciprocals. For example, the reciprocal of the sine function is the cosecant function.

Other commonly used pairs include:

• Cosine and secant: \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\)
• Tangent and cotangent: \(\text{cot} \theta = \frac{1}{\text{tan} \theta}\)

Understanding reciprocal functions helps in analyzing various trigonometric identities and properties, as well as in comprehending how different trigonometric graphs behave.

trigonometric periodicity

Periodicity in trigonometry refers to the repeating nature of trigonometric functions after a certain interval. For the cosecant function, its periodic nature is inherited from the sine function since \(\text{csc} \theta\) depends on \(\text{sin} \theta\). The period of a function is the smallest positive value for which the function repeats its values.

Important aspects to remember include:

• The sine and cosine functions have periods of \(\text{2} \pi\).
• Tangent and cotangent functions have periods of \(\text{𝜋}\).

Since \(\text{sin} (\theta + 2 \pi) = \text{sin} \theta\), it implies for \(\text{csc} \theta\) that \(\text{csc} (\theta + 2 \pi) = \text{csc} \theta\), proving that the period of \(\text{csc} \theta\) is also \(\text{2} \pi\).

sine function

The sine function, denoted as \(\text{sin} \theta\), is perhaps the most well-known trigonometric function. It maps an angle \(\theta\) to its y-coordinate on the unit circle. Understanding the sine function is crucial for comprehending other trigonometric functions since many of them are defined through sine.

Some crucial points:

• It oscillates between \(-1\) and \(+1\).
• Sine function is periodic with a period of \(\text{2} \pi\).
• Sine of an angle \(0, \text{𝜋}\) and \(\text{2} \pi\) is zero, causing undefined values for \(\text{csc} \theta\) at these points.

The sine function's periodicity helps in determining the characteristics of the cosecant function, as seen in this exercise.