Problem 10 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$\csc \theta \sin \theta=1$$

Step-by-Step Solution

Answer

All angles except \( \theta = n\pi \), where \( n \) is an integer.

Step 1: Recall the Definitions

To solve the equation \( \csc \theta \sin \theta = 1 \), remember that the cosecant function is defined as the reciprocal of the sine function: \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Simplify the Equation

Substitute \( \csc \theta \) with \( \frac{1}{\sin \theta} \) into the equation, giving us \( \frac{1}{\sin \theta} \cdot \sin \theta = 1 \).

Step 3: Solve the Simplified Equation

The left side simplifies because \( \frac{1}{\sin \theta} \cdot \sin \theta = 1 \) simplifies to \( 1 = 1 \), which is always true. This means \( \theta \) can be any angle where \( \sin \theta \) is defined and not equal to zero.

Step 4: Identify Undefined Points

The sine function is undefined or zero at the points where it equals zero: \( \theta = n\pi \), where \( n \) is an integer. However, due to the definition of the functions, \( \csc \theta \) is undefined when \( \sin \theta = 0 \).

Step 5: State the Range of Valid Solutions

The solutions to the equation are all real numbers except those where \( \sin \theta = 0 \). Therefore, the solution set excludes \( \theta = n\pi \), where \( n \) is an integer, specifically the points where csc(θ) becomes undefined.

Key Concepts

The Cosecant Function Unpacked

The cosecant function, denoted as \( \csc \theta \), is a fundamental trigonometric function derived from the sine function. It is defined as the reciprocal of the sine function. Therefore, \( \csc \theta = \frac{1}{\sin \theta} \). This definition means that any property of the sine function, when inverted, affects the cosecant function accordingly.

There are specific properties of the cosecant function that arise due to this reciprocal relationship:

• \( \csc \theta \) is undefined whenever \( \sin \theta = 0 \), because division by zero is undefined in mathematics.
• Since the sine function oscillates between -1 and 1, its reciprocal, the cosecant function, will have a range of \((-\infty, -1] \cup [1, \infty)\).

Remembering these properties is crucial when solving equations involving \( \csc \theta \), as they will help us understand where the function is undefined and what values \( \theta \) cannot take.

Exploring the Sine Function

The sine function, \( \sin \theta \), is one of the primary trigonometric functions. It describes the y-coordinate of a point on the unit circle, corresponding to a given angle \( \theta \).

Here are a few key characteristics of the sine function that are important to consider:

• It oscillates between -1 and 1, inclusive. This implies that all values of the sine function are within this range.
• \( \sin \theta \) is zero at integral multiples of \( \pi \), meaning it intersects the x-axis at these points: \( \theta = n\pi \), where \( n \) is an integer.
• It has a period of \( 2\pi \), meaning the function repeats its values every \( 2\pi \) radians.

Understanding these aspects helps when dealing with trigonometric equations, since certain solutions might be straightforward to identify simply by recognizing where \( \sin \theta \) equals specific values, especially zero.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles \( \theta \) that satisfy a given equation involving trigonometric functions. They often require a deep understanding of the properties and behaviors of these functions.

In this exercise, we dealt with the equation \( \csc \theta \sin \theta = 1 \). By recalling the definition of the cosecant function, substituting \( \csc \theta \) with \( \frac{1}{\sin \theta} \), we simplified the equation to \( \frac{1}{\sin \theta} \cdot \sin \theta = 1 \). This simplifies further because the terms cancel out, leading to the tautology \( 1 = 1 \).

However, the fun part is identifying where these cancellations are invalid or undefined:

• The sine function is not defined at \( \theta = n\pi \), meaning it's zero and doesn’t affect the reciprocal (cosecant) to validly cancel out with \( \sin \theta \).
• Consequently, the solution set excludes these undefined points: \( \theta eq n\pi \).

Practicing solving these equations can enormously improve one's ability to look out for these solution nuances and better understand not only how to find solutions but also why certain angles must be excluded.