Problem 46 - Trigonometry

Question

\(3 \sin \theta+2 \sin \theta \cos \theta=0\)

Step-by-Step Solution

Answer

The solutions are \( \theta = n\pi \), where \( n \) is an integer.

Step 1: Factor the Equation

Start by factoring the common factor \( \sin \theta \) from both terms in the equation.\[ 3 \sin \theta + 2 \sin \theta \cos \theta = \sin \theta (3 + 2 \cos \theta) \]

Step 2: Set Each Factor to Zero

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. So we have two cases to solve:1. \( \sin \theta = 0 \)2. \( 3 + 2 \cos \theta = 0 \)

Step 3: Solve for \( \theta \) in First Case

Solve \( \sin \theta = 0 \).The sine function is zero at \( \theta = n\pi \) where \( n \) is any integer. Hence, one solution set is \( \theta = n\pi \).

Step 4: Solve for \( \theta \) in Second Case

Solve \( 3 + 2 \cos \theta = 0 \) for \( \theta \).Rearrange the equation:\[ 2 \cos \theta = -3 \]\[ \cos \theta = -\frac{3}{2} \]Since the range of the cosine function is \([-1, 1]\), \( \cos \theta = -\frac{3}{2} \) has no solution in the real numbers.

Key Concepts

Factorization

Factorization is a common method used in solving equations, particularly when dealing with polynomials or combinations of trigonometric terms. It involves breaking down an expression into simpler "factors" that, when multiplied together, yield the original equation. In the case of the trigonometric equation provided, the expression \(3 \sin \theta + 2 \sin \theta \cos \theta\) can be factored by identifying a common factor shared by both terms: \(\sin \theta\). This can be rewritten as:

• \( \sin \theta (3 + 2 \cos \theta) \)

This step is crucial as it simplifies the equation into a form where you can apply other mathematical properties, such as the zero-product property, to solve for the unknowns.

Zero-Product Property

The zero-product property is a fundamental concept in algebra used to solve equations that have been factored. It states that if the product of two terms equals zero, at least one of the terms must be zero. For the equation \( \sin \theta (3 + 2 \cos \theta) = 0 \), this means:

• \( \sin \theta = 0 \)
• or \( 3 + 2 \cos \theta = 0 \)

This property allows us to break apart the problem into simpler ones by setting each factor to zero separately. Thus, we can explore solutions for each condition independently, which often makes finding solutions more straightforward.

Sine Function

The sine function, denoted as \( \sin \theta \), is a fundamental trigonometric function that describes the ratio of the length of the opposite side to the hypotenuse in a right triangle. On the unit circle, \( \sin \theta \) represents the y-coordinate of a point where the terminal side of an angle \( \theta \) intersects the circle.
This function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) units. Since \( \sin \theta = 0 \) happens when \( \theta = n\pi \), where \(n\) is an integer, these are considered solutions to the equation stemming from the zero-product property. These specific solutions occur at multiples of \(\pi\).

Cosine Function

The cosine function, denoted as \( \cos \theta \), is another key trigonometric function. It describes the ratio of the adjacent side to the hypotenuse in a right triangle. On the unit circle, \( \cos \theta \) corresponds to the x-coordinate of where the terminal side intersects the circle.
The cosine function has a range of

• \([-1, 1]\)

which means it doesn't produce values outside of this interval. This characteristic is crucial, particularly when solving equations involving cosine. In the problem, \(3 + 2 \cos \theta = 0\) leads to \(\cos \theta = -\frac{3}{2}\), which falls outside the valid range for the cosine function, thus showing no real solution exists for this particular case.