Problem 47 - Precalculus

Question

Use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$\csc \theta \text { is undefined, } 0 \leq \theta \leq 2 \pi$$

Step-by-Step Solution

Answer

\(\csc \theta\) is undefined at \(\theta = 0, \pi, \text{ and } 2\pi\).

Step 1: Understanding the Function

The cosecant function, \( ext{csc } heta\), is the reciprocal of the sine function: \( ext{csc } heta = \frac{1}{\sin \theta}\). This means \( ext{csc } \theta\) is undefined wherever \( ext{sin } \theta = 0\), since division by zero is undefined.

Step 2: Identifying Zero Points of Sine

The sine function, \( ext{sin } \theta\), is zero at the points \(\theta = 0, \pi, ext{ and } 2\pi\) within the interval \(0 \leq \theta \leq 2\pi\). This is due to the sine of 0, \(\pi\) and \(2\pi\) all being 0.

Step 3: Finding Undefined Points for Cosecant

Therefore, \(\text{csc } \theta\) is undefined at the points where \(\sin \theta = 0\), which are \(\theta = 0, \pi, ext{ and } 2\pi\).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is a trigonometric function that is the reciprocal of the sine function. This means that \( \csc \theta = \frac{1}{\sin \theta} \). Understanding this relationship is crucial for solving problems involving the cosecant function. If the sine of an angle is known, the cosecant can be easily found by taking the reciprocal of this sine value.

• If \( \sin \theta = 1 \), then \( \csc \theta = 1 \).
• If \( \sin \theta = -1 \), then \( \csc \theta = -1 \).
• If \( \sin \theta = 0.5 \), \( \csc \theta = \frac{1}{0.5} = 2 \).

These examples demonstrate how the cosecant function changes with different sine values. Understanding this reciprocal relationship makes it more intuitive to determine the cosecant of various angles.

Sine Function

The sine function, \( \sin \theta \), is fundamental in trigonometry. It is one of the primary functions used to understand relationships in circles and triangles. The sine of an angle in a unit circle represents the y-coordinate of the point on the circle that corresponds to that angle.

• On the unit circle, the radius is \(1\).
• \( \sin \theta \) refers to the vertical distance from the x-axis.

The sine function has specific properties important for calculations:

• It reaches its maximum value of \(1\) when \( \theta = \frac{\pi}{2} \).
• The minimum value is \(-1\) when \( \theta = \frac{3\pi}{2} \).
• The sine of \(0\), \(\pi\), and \(2\pi\) is \(0\).

Knowing these key values helps when solving problems related to sin \( \theta \), as it allows for quick calculations without needing a calculator for angle values.

Undefined Values

Certain trigonometric functions, like the cosecant, can become undefined. The cosecant function \( \csc \theta \) is undefined where \( \sin \theta = 0 \). This is because division by zero is not possible. The sine function is exactly \(0\) at specific angles:

• \( \theta = 0\)
• \( \theta = \pi\)
• \( \theta = 2\pi\)

At these points, since \( \sin \theta = 0 \), \( \csc \theta = \frac{1}{0} \), which makes it undefined.
When dealing with trigonometric equations or functions, recognizing where these undefined values occur is essential. It helps prevent errors in calculations and ensures a better understanding of the function behaviors within the given intervals.