Problem 14 - Precalculus: Functions and Graphs

Question

Express as a product. $$\cos \theta-\cos 5 \theta$$

Step-by-Step Solution

Answer

The expression as a product is \( 2 \sin(3\theta) \sin(2\theta) \).

Step 1: Use the Cosine Difference Formula

The difference of cosines formula states that \( \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \). In this problem, \( A = \theta \) and \( B = 5\theta \).

Step 2: Substitute into the Formula

Substitute \( A = \theta \) and \( B = 5\theta \) into the formula: \[\cos \theta - \cos 5\theta = -2 \sin\left(\frac{\theta + 5\theta}{2}\right) \sin\left(\frac{\theta - 5\theta}{2}\right) \].

Step 3: Simplify the Expression

Calculate inside the sine functions: \( \frac{\theta + 5\theta}{2} = \frac{6\theta}{2} = 3\theta \), and \( \frac{\theta - 5\theta}{2} = \frac{-4\theta}{2} = -2\theta \). Thus, the expression becomes \[-2 \sin(3\theta) \sin(-2\theta)\].

Step 4: Use the Identity for Negative Angles in Sine

Using the identity \( \sin(-x) = -\sin(x) \), rewrite \( \sin(-2\theta) \) as \(-\sin(2\theta)\). The expression becomes \[-2 \sin(3\theta)(-\sin(2\theta))\].

Step 5: Simplify the Final Expression

Multiply out to obtain \[2 \sin(3\theta) \sin(2\theta)\]. This is the final expression as a product.

Key Concepts

Cosine Difference Formula

The cosine difference formula is a very useful trigonometric identity. It is applied when you need to find the difference between the cosines of two angles. This formula is:

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

It essentially transforms the subtraction of two cosine terms into a product of sines.

This conversion often simplifies trigonometric expressions and allows for easier manipulation in solving problems. In the earlier exercise, \( A = \theta \) and \( B = 5\theta \) were used, successfully converting \( \cos \theta - \cos 5\theta \) into a more manageable form using this formula.

Sine Function

The sine function is one of the primary trigonometric functions and is integral to understanding circular and oscillating phenomena.

• It is represented as \( \sin(x) \), where \( x \) can be any angle.
• Sine functions describe relationships in right triangles, namely the ratio of the length of the opposite side to the hypotenuse.

The sine function is periodic with a period of \( 2\pi \), meaning it repeats its values in regular intervals.

In this context, the sine function was used within the cosine difference formula to express cosine differences as a product of sine terms, allowing for the simplification and further manipulation of trigonometric expressions and equations.

Angle Subtraction Identity

The angle subtraction identity is crucial in simplifying and calculating trigonometric expressions. It helps to break down the difference between two angles into components.

• For cosine and sine, this identity allows us to express differences using formulas that can simplify complex expressions involving angles.
• Specifically, the identity helps us use quantities like \( \frac{A+B}{2} \) and \( \frac{A-B}{2} \), to evaluate the sine functions involved in the cosine difference formula.

In the problem from the exercise, the angle subtraction identities played a pivotal role in moving from \( \theta \) and \( 5\theta \) to simpler terms, allowing us to perform further substitutions that made the original expression more compact and manageable.

Negative Angle Identity

The negative angle identity for sine is a simple yet powerful tool in trigonometry. It tells us that

• \( \sin(-x) = -\sin(x) \).

This identity is significant because it reflects the odd nature of the sine function. Negative angles effectively "flip" the sine value over the origin.

In solving the problem, the negative angle identity was used to transform \( \sin(-2\theta) \) into \(-\sin(2\theta)\).

This conversion helped us simplify the expression to its final form by allowing multiplication of negative terms, ultimately leading to the neat transformation into \( 2 \sin(3\theta) \sin(2\theta) \). Using this identity reduces the complexity in calculating and understanding the behavior of trigonometric expressions.