Questions: Express the given sum or difference as a product of sines and/or cosines. sin 5θ - sin θ

Solution Steps

To express the given difference as a product of sines and/or cosines, we can use the trigonometric identity for the difference of sines: \[ \sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \] Here, \( A = 5\theta \) and \( B = \theta \).

1. Identify \( A \) and \( B \) in the given expression.
2. Apply the trigonometric identity for the difference of sines.
3. Simplify the resulting expression.

We start with the expression: \[ \sin(5\theta) - \sin(\theta) \]

Using the identity for the difference of sines: \[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \] we set \( A = 5\theta \) and \( B = \theta \). Thus, we have: \[ \sin(5\theta) - \sin(\theta) = 2 \cos\left(\frac{5\theta + \theta}{2}\right) \sin\left(\frac{5\theta - \theta}{2}\right) \]

Calculating the averages: \[ \frac{5\theta + \theta}{2} = \frac{6\theta}{2} = 3\theta \] \[ \frac{5\theta - \theta}{2} = \frac{4\theta}{2} = 2\theta \] Substituting these back into the identity gives: \[ \sin(5\theta) - \sin(\theta) = 2 \cos(3\theta) \sin(2\theta) \]

Final Answer