Problem 62 - Algebra and Trigonometry

Question

\(61-66\) Write the sum as a product. $$ \sin x-\sin 4 x $$

Step-by-Step Solution

Answer

\( \sin x - \sin 4x = -2 \cos \left( \frac{5x}{2} \right) \sin \left( \frac{3x}{2} \right) \)

Step 1: Identify the Trigonometric Identity

To transform a difference of sines into a product, we need to use a trigonometric identity. The identity is: \[ \sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \]

Step 2: Assign Values for A and B

In the given expression \( \sin x - \sin 4x \), assign \( A = x \) and \( B = 4x \).

Step 3: Calculate \(A+B\) and \(A-B\)

Find \( A+B \) and \( A-B \):\[ A + B = x + 4x = 5x \]\[ A - B = x - 4x = -3x \]

Step 4: Plug Values into the Identity

Substitute \( A+B \) and \( A-B \) into the identity. This gives:\[ \sin x - \sin 4x = 2 \cos \left( \frac{5x}{2} \right) \sin \left( \frac{-3x}{2} \right) \]

Step 5: Use Sin(-θ) = -Sin(θ) Property

Since the sine function is an odd function, \( \sin(-\theta) = -\sin(\theta) \). Apply this to simplify:\[ \sin x - \sin 4x = -2 \cos \left( \frac{5x}{2} \right) \sin \left( \frac{3x}{2} \right) \]

Key Concepts

Difference of Sines

When dealing with the difference of sines, such as \(\sin x - \sin 4x\), a useful approach involves a trigonometric identity designed for this very scenario. The identity is

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

This formula helps convert the expression from a difference into a product of sine and cosine. Here's how it works:

• First, consider the terms \(A\) and \(B\) as the angles in the sine functions.
• In our example, \(A = x\) and \(B = 4x\).
• Next, calculate \(A + B\) and \(A - B\) to use in the identity.
• For \(\sin x - \sin 4x\), these would be \(5x\) and \(-3x\), respectively.

The identity aids in simplifying what might otherwise be a complex algebraic or trigonometric problem. It opens the door to further simplifications or analyses by converting it into a simpler product form.

Cosine Function

The cosine function is a fundamental part of many trigonometric identities and calculations. In our identity for the difference of sines, the cosine function plays a crucial role:

• The identity uses \( \cos \left( \frac{A+B}{2} \right) \).
• In the expression \(\sin x - \sin 4x\), this translates to \( \cos \left( \frac{5x}{2} \right) \).

Here's what makes cosine stand out:

• It is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• The cosine function represents the horizontal coordinate of a point on the unit circle.

This function is involved in measuring angles from the x-axis. In the context of our identity, it affects the amplitude of the resulting product term.

Sine Function

The sine function often appears in identities such as the difference of sines, being an essential part of these trigonometric transformations. It is important to understand a few key properties:

• The sine function is an odd function, meaning \( \sin(-\theta) = -\sin(\theta) \).
• It relates to the vertical component of a point on the unit circle.

In the identity used for \(\sin x - \sin 4x\), the final simplification step notably employs the property of odd functions:

• When we calculated \( \sin \left( \frac{-3x}{2} \right) \), the result is transformed to \(-\sin \left( \frac{3x}{2} \right)\).
• This transforms the expression into \(-2 \cos \left( \frac{5x}{2} \right) \sin \left( \frac{3x}{2} \right) \).

The sine function is integral not only in basic calculations but also in understanding deeper identities in trigonometry. Its behavior helps tame and transform complex expressions into easier forms.