Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ (\cos \theta-1)(\sin \theta+1)=0 $$

Step-by-Step Solution

Answer

The solutions are \(\theta = 2k\pi\) and \(\theta = \frac{3\pi}{2} + 2k\pi\), where \(k\) is an integer.

Step 1: Understand the Zero Product Property

We have the equation \((\cos \theta - 1)(\sin \theta + 1) = 0\). According to the zero product property, if the product of two terms is zero, then at least one of the terms must be zero. So, either \(\cos \theta - 1 = 0\) or \(\sin \theta + 1 = 0\) must be true.

Step 2: Solve \(\cos \theta - 1 = 0\)

Start by solving the first equation: \(\cos \theta - 1 = 0\).This implies:\[ \cos \theta = 1 \]The cosine of \(\theta\) equals 1 at angles \(\theta = 2k\pi\), where \(k\) is an integer.

Step 3: Solve \(\sin \theta + 1 = 0\)

Now solve the second equation: \(\sin \theta + 1 = 0\).This implies:\[ \sin \theta = -1 \]The sine of \(\theta\) equals -1 at angles \(\theta = \frac{3\pi}{2} + 2k\pi\), where \(k\) is an integer.

Step 4: Combine the solutions

The solution to the original equation includes both sets of solutions from Step 2 and Step 3. Therefore, the complete solution set is:\[ \theta = 2k\pi \quad\text{or}\quad \theta = \frac{3\pi}{2} + 2k\pi, \quad k \in \mathbb{Z} \]

Key Concepts

Zero Product Property

The zero product property is a fundamental concept in mathematics, particularly useful when solving equations that involve a product of terms. It states that if the product of two or more factors equals zero, then at least one of the factors must be zero.

In the context of the given exercise, the equation is \( (\cos \theta - 1)(\sin \theta + 1) = 0 \). To solve this equation, we apply the zero product property by setting each factor to zero independently:

• \( \cos \theta - 1 = 0 \)
• \( \sin \theta + 1 = 0 \)

This approach simplifies a potentially complex problem into smaller, more manageable pieces. Essentially, we're breaking down the equation to find the individual conditions that satisfy the original statement. Each condition leads to its own set of solutions.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is a periodic function that represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It has important properties that we utilize when solving trigonometric equations.

For the solution \( \cos \theta - 1 = 0 \), we're interested in the angles where the cosine of \( \theta \) equals 1. This happens at specific points on the unit circle.

• The angle \( \theta = 0 \) is one such solution
• Because cosine is a periodic function with a period of \( 2\pi \), this repeats for every integer \( k \), yielding \( \theta = 2k\pi \)

Thus, the set of all angles where \( \cos \theta = 1 \) includes all multiples of \( 2\pi \), capturing the cycle of the cosine function.

Sine Function

The sine function, represented as \( \sin \theta \), describes the ratio of the opposite side to the hypotenuse in a right triangle. Like cosine, sine is also periodic and critical in solving equations involving angles.

In the context of the equation \( \sin \theta + 1 = 0 \), we seek angles where the sine of \( \theta \) is -1.

• The angle \( \theta = \frac{3\pi}{2} \) is our starting point, where sine equals -1
• Due to the function's periodicity with a cycle of \( 2\pi \), this result generalizes to \( \theta = \frac{3\pi}{2} + 2k\pi \) for any integer \( k \)

The behavior of the sine function provides a repeating pattern of solutions across the angular spectrum, showing how negative values fit within its wave form.

Solution Sets

Combining the individual solutions derived from both the cosine and sine equations forms the complete solution set for the problem. We use all the angles where conditions from the zero product property hold true.

The solution set encompasses:

• Angles from the cosine equation: \( \theta = 2k\pi \), where \( k \) is any integer
• Angles from the sine equation: \( \theta = \frac{3\pi}{2} + 2k\pi \), where \( k \) is any integer

Together, these identify every angle \( \theta \) meeting the original equation's criteria.
The symmetry and repetition in trigonometric functions make it possible to express an infinite number of solutions concisely. By using parameterization with integer values, we effectively describe all possible scenarios where the equation holds true.