Problem 56 - Algebra and Trigonometry

Question

\(53-56\) a Solve the equation by first using a Sum-to-Product Formula. \(\sin 5 \theta-\sin 3 \theta=\cos 4 \theta\)

Step-by-Step Solution

Answer

The solutions are \(\theta = \frac{\pi}{6} + 2n\pi\) and \(\theta = \frac{5\pi}{6} + 2n\pi\).

Step 1: Identify the Sum-to-Product Formula

The sum-to-product formulas help convert a sum or difference of sines or cosines into a product. Here, for the difference of sines, we use the formula: \(\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)\).

Step 2: Apply the Formula to the Left Side

Apply the formula \(\sin 5\theta - \sin 3\theta = 2 \cos\left(\frac{5\theta + 3\theta}{2}\right) \sin\left(\frac{5\theta - 3\theta}{2}\right)\). This simplifies to \(2 \cos 4\theta \sin \theta\).

Step 3: Substitute into the Original Equation

Substitute the sum-to-product expression back into the equation: \(2 \cos 4\theta \sin \theta = \cos 4\theta\).

Step 4: Simplify the Equation

Since \(2 \cos 4\theta \sin \theta = \cos 4\theta\), divide both sides by \(\cos 4\theta\), assuming \(\cos 4\theta eq 0\). This gives \(2 \sin \theta = 1\).

Step 5: Solve for \(\theta\)

Solve the equation \(2 \sin \theta = 1\) by dividing both sides by 2, resulting in \(\sin \theta = \frac{1}{2}\). The solutions are \(\theta = \frac{\pi}{6} + 2n\pi\) and \(\theta = \frac{5\pi}{6} + 2n\pi\) for integer values of \(n\).

Step 6: Verify with \(\cos 4\theta = 0\)

If \(\cos 4\theta = 0\), then \(4\theta = \frac{(2m+1)\pi}{2}\) where \(m\) is an integer. Solving for \(\theta\), \(\theta = \frac{(2m+1)\pi}{8}\). Verify these solutions do not satisfy the original equation unless \(\sin\left(\frac{(2m+1)\pi}{8}\right) = 0\).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate various trigonometric functions. They are useful in simplifying equations, solving trigonometric problems, and transforming expressions. These identities include relations for sines, cosines, tangents, and other trigonometric functions.
One important category of these identities is the sum-to-product formulas. These formulas allow us to convert a sum or difference of trigonometric functions into a product, making them easier to manipulate. For example, the difference of sines formula is used when you have expressions like \( \sin A - \sin B \) and can be converted using:
\[\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)\]
This transformation helps to simplify complex expressions and is particularly helpful when dealing with equation solving.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all the angles that satisfy the equation under consideration. These equations typically involve trigonometric functions like sine, cosine, or tangent.
To solve these equations effectively, you'll often need to manipulate them using various trigonometric identities or algebraic techniques, like dividing or isolating terms. Consider the equation \( 2 \sin \theta = 1 \). To isolate \( \sin \theta \), divide both sides by 2, yielding \( \sin \theta = \frac{1}{2} \).
This equation can be solved by determining the angles \( \theta \) within a certain range that have a sine value of \( \frac{1}{2} \). These values are well-known: \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{5\pi}{6} \) for the principal values, and repeating every cycle at periods of \( 2\pi \) with additional terms involving \( 2n\pi \) for integer \( n \).
It is important to take into account all possible solutions in the domain of interest.

Difference of Sines Formula

The difference of sines formula is a specific trigonometric identity that can streamline the process of working with sine differences. It is used to convert expressions like \( \sin A - \sin B \) into a more manageable product form.
This formula states:

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

This identity transforms a potentially cumbersome difference into a straightforward product. This new form is easier to handle and may often reveal simpler solutions or further manipulations.
Using this formula helps simplify both the process of solving trigonometric problems and the underlying calculations, providing meaningful insights into the behavior of trigonometric expressions. It is particularly useful in solving equations involving the sine function by allowing for easier simplification and resolution.