Problem 23 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The solutions are given by \(\theta = 2k\pi\) and \(\theta = \frac{3\pi}{2} + 2m\pi\), where \(k, m\) are integers.

Step 1: Understand the Zero Product Property

The equation \((\cos \theta - 1)(\sin \theta + 1) = 0\) suggests the use of the zero product property. According to this property, if \(a \cdot b = 0\), then either \(a = 0\) or \(b = 0\). This means that either \(\cos \theta - 1 = 0\) or \(\sin \theta + 1 = 0\).

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

Set \(\cos \theta - 1 = 0\) to find \(\theta\). Solving gives: \[\cos \theta = 1\] The values of \(\theta\) that satisfy \(\cos \theta = 1\) are \(\theta = 2k\pi\) where \(k\) is any integer.

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

Set \(\sin \theta + 1 = 0\) to find \(\theta\). Solving gives: \[\sin \theta = -1\] The values of \(\theta\) that satisfy \(\sin \theta = -1\) are \(\theta = \frac{3\pi}{2} + 2m\pi\), where \(m\) is any integer.

Step 4: Combine the Solutions

Combining the solutions from Steps 2 and 3 gives us two sets of solutions: \(\theta = 2k\pi\) and \(\theta = \frac{3\pi}{2} + 2m\pi\), where \(k\) and \(m\) are integers.

Key Concepts

Zero Product Property

When working with equations like \((\cos \theta - 1)(\sin \theta + 1) = 0\), the zero product property is essential. This property states that if the product of two factors is zero,one or both of the factors must themselves be zero. This helps us break down complex equations into simpler ones.

• If we have two expressions, \(a\) and \(b\), then \(a \cdot b = 0\) implies that \(a = 0\) or \(b = 0\).
• In this exercise, this means either \(\cos \theta - 1 = 0\) or \(\sin \theta + 1 = 0\).

The zero product property turns the original equation into two separate equations that are easier to solve.This concept is often used to simplify solving quadratic equations and in our trigonometric setup, highlights potential angles that satisfy our given condition.

Cosine Function

The cosine function, denoted as \(\cos \theta\), is one of the primary trigonometric functions.It describes the x-coordinate of a point on the unit circle at an angle \(\theta\).This concept is pivotal in understanding trigonometric equations because it provides information about angle positions and measurements.
Some crucial points about the cosine function include:

• The function is periodic with a period of \(2\pi\).
• Values range from -1 to 1 inclusively.
• \(\cos \theta = 1\) corresponds to angles where \(\theta = 2k\pi\), with \(k\) as any integer representing the complete cycles around the circle.

When the cosine function equals 1, it indicates a specific position directly on the positive x-axis of the unit circle.For trigonometric equations,identifying such special positions simplifies finding solutions.

Sine Function

The sine function, symbolized as \(\sin \theta\),is another key trigonometric function that signifies the y-coordinate of a point on the unit circle at angle \(\theta\).Understanding the sine function is vital for solving trigonometric equations because it helps detect particular angles' corresponding vertical positions.
Here are some facts about the sine function:

• It also has a periodic cycle of \(2\pi\).
• The sine function's values are bounded between -1 and 1, inclusively.
• \(\sin \theta = -1\) describes angles like \(\theta = \frac{3\pi}{2} + 2m\pi\), where \(m\) represents any integer, indicating patterns of vertical positions below the center of the circle.

When \(\sin \theta = -1\),we are directly beneath the origin on the unit circle.These observations assist in determining concrete solutions within the equation and simplify understanding vertical angular occurrences.

Solution Set of Trigonometric Equations

Combining solutions derived from different parts of an equation forms the complete solution set of trigonometric equations.This involves identifying all possible angles that satisfy both sections of an equation,understanding that they can form sequences or patterns across the unit circle.
Here’s how this works effectively:

• Solve each segment separately using trigonometric identities or properties, e.g., zero product property.
• Identify periodic solutions,on account of functions like sine and cosine being periodic.
• For the given problem,the complete solution set combines angles from each part, \(\theta = 2k\pi\) and \(\theta = \frac{3\pi}{2} + 2m\pi\), where \(k\) and \(m\) are integers.

Bringing together these solutions constructs the set of all possible angles \(\theta\) that satisfy the overall equation. This methodology is crucial for comprehensively resolving trigonometric equations and understanding their broad implications across angles.