Problem 53 - Algebra and Trigonometry

Question

\(53-56\) a Solve the equation by first using a Sum-to-Product Formula. \(\sin \theta+\sin 3 \theta=0\)

Step-by-Step Solution

Answer

\(\theta = \frac{n\pi}{2}\) and \(\theta = \frac{\pi}{2} + m\pi\) where \(n, m \in \mathbb{Z}\).

Step 1: Understand Sum-to-Product Identities

Sum-to-Product identities are trigonometric identities that allow you to convert a sum of sines or cosines into a product involving sine and cosine. For sine, the identity is given by: \[ \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \] We will use this identity to rewrite the given equation.

Step 2: Apply Sum-to-Product Formula

Given the equation \( \sin \theta + \sin 3\theta = 0 \), let's identify \( A = \theta \) and \( B = 3\theta \). Applying the sum-to-product identity:\[ \sin \theta + \sin 3\theta = 2 \sin \left( \frac{\theta + 3\theta}{2} \right) \cos \left( \frac{\theta - 3\theta}{2} \right) \]This simplifies to:\[ 2 \sin (2\theta) \cos (-\theta) = 0 \]Since \( \cos(-\theta) = \cos(\theta) \), this becomes:\[ 2 \sin (2\theta) \cos (\theta) = 0 \]

Step 3: Solve the Product Equation

The equation \( 2 \sin (2\theta) \cos (\theta) = 0 \) implies that either \( \sin (2\theta) = 0 \) or \( \cos (\theta) = 0 \). We will solve both cases separately.

Step 4: Solve \( \sin (2\theta) = 0 \)

The equation \( \sin (2\theta) = 0 \) has solutions when:\[ 2\theta = n\pi \quad \text{where} \quad n \in \mathbb{Z} \]Solving for \( \theta \), we get:\[ \theta = \frac{n\pi}{2} \]

Step 5: Solve \( \cos (\theta) = 0 \)

The equation \( \cos (\theta) = 0 \) has solutions when:\[ \theta = \frac{\pi}{2} + m\pi \quad \text{where} \quad m \in \mathbb{Z} \]

Step 6: Combine Solutions

The general solutions to the equation \( \sin \theta + \sin 3\theta = 0 \) are:\( \theta = \frac{n\pi}{2} \) and \( \theta = \frac{\pi}{2} + m\pi \). Both scenarios provide different sets of solutions based on the integer parameters \( n \) and \( m \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are universally true for any angle involved. They're like mathematical shortcuts used to simplify and rearrange expressions in trigonometry. In the context of our problem, we focus on the Sum-to-Product identities which are extremely useful for converting the sum of sine functions into a product form. This transformation allows us to simplify complex trigonometric equations, making them more accessible for solving.

In particular, the Sum-to-Product identity for sine looks like this:

• \( \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \)

Applying these identities can turn an equation like \( \sin \theta + \sin 3\theta = 0 \) into a form that's easier to manage by reducing it to a product of sine and cosine. This process is often the first step in solving trigonometric equations because it sets the stage for finding the exact values of \( \theta \).

Solving Trigonometric Equations

To solve trigonometric equations means to find all possible values of the variable that satisfy the given trigonometric expression. Once the equation is simplified through the use of Sum-to-Product identities, like transforming \( \sin \theta + \sin 3\theta = 0 \) into \( 2 \sin (2\theta) \cos (\theta) = 0 \), we can solve the equation by setting each factor of the product to zero separately.

This is based on the principle that if a product equals zero, then at least one of the factors must be zero. Thus, we split the trigonometric product equation into two separate equations:

• \( \sin (2\theta) = 0 \)
• \( \cos (\theta) = 0 \)

Solving each equation gives a set of solutions. For instance:

• From \( \sin (2\theta) = 0 \), we obtain \( 2\theta = n\pi \) where \( n \in \mathbb{Z} \). Solving for \( \theta \) gives \( \theta = \frac{n\pi}{2} \).
• From \( \cos (\theta) = 0 \), we find \( \theta = \frac{\pi}{2} + m\pi \) where \( m \in \mathbb{Z} \).

These solutions are the angles for which the original trigonometric equation holds true.

Sum of Sines

The sum of sines is a concept that frequently emerges when dealing with trigonometric identities and equations. It involves adding two or more sine functions together, which can often make the problem more complex. However, using identities like the Sum-to-Product identity simplifies this process significantly.

When addressing the sum such as \( \sin \theta + \sin 3\theta \), rewriting it using the identity \( \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \) provides a clearer perspective. This makes it easier to identify the next steps in solving the equation or simplifying expressions.

This conversion is particularly powerful because:

• It reduces the complication associated with multiple sine functions.
• It transforms the expression into a more straightforward product form.
• This form is easier to equate to zero, so the solution to the equation can be found using simple trigonometric laws.

Thus, mastering the sum of sines and the related identities is crucial for efficiently tackling and solving trigonometric equations in a clear and concise manner.