Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \sin (-4 x) \cos 8 x $$

Step-by-Step Solution

Answer

\( \sin(-4x)\cos(8x) = \frac{1}{2}[\sin(4x) - \sin(12x)] \)

Step 1: Identify the Trigonometric Identity

We need to express the product \( \sin(-4x) \cos(8x) \) as a sum or difference. A useful identity for this situation is the product-to-sum identity for sine and cosine: \( \sin(A)\cos(B) = \frac{1}{2}[\sin(A + B) + \sin(A - B)] \).

Step 2: Substitute the Values into the Identity

Using the identity, we substitute \( A = -4x \) and \( B = 8x \) into \( \frac{1}{2}[\sin(A + B) + \sin(A - B)] \). This results in: \[\sin(-4x)\cos(8x) = \frac{1}{2}[\sin((-4x) + 8x) + \sin((-4x) - 8x)]\]

Step 3: Simplify the Sums Inside the Sine Functions

Calculate the expressions inside each of the sine functions: \((-4x) + 8x = 4x\) and \((-4x) - 8x = -12x\). Substitute these simplified expressions back into the equation: \[\frac{1}{2}[\sin(4x) + \sin(-12x)]\]

Step 4: Use the Sine Negative Angle Identity

Using the identity \( \sin(-A) = -\sin(A) \), simplify \( \sin(-12x) \) to \( -\sin(12x) \). Thus, the expression becomes: \[\frac{1}{2}[\sin(4x) - \sin(12x)]\]

Step 5: Write the Final Answer

The expression \( \sin(-4x)\cos(8x) \) rewritten as a sum or difference using trigonometric identities is: \( \frac{1}{2}[\sin(4x) - \sin(12x)] \).

Key Concepts

Sine and Cosine Functions

Sine and cosine functions are fundamental to trigonometry. They describe the relationships between the angles and sides of right triangles, but go far beyond that into describing waves and oscillatory motions.
The sine function, noted as \( \sin \theta \), gives the ratio of the length of the opposite side to the hypotenuse in a right triangle. The cosine function, \( \cos \theta \), provides the ratio of the adjacent side to the hypotenuse. These functions are periodical with a period of \( 2\pi \), meaning they repeat their values in regular intervals.
Understanding sine and cosine functions is crucial for solving trigonometric equations. Their graphs are useful for visual interpretation, showing the wave-like characteristics. This knowledge is foundational for employing various trigonometric identities.

Product-to-Sum Formulas

The product-to-sum formulas are trigonometric identities that convert the product of sine and cosine functions into a sum or difference of trigonometric functions. These identities make certain calculations easier, especially when dealing with integrals and simplifying expressions.

• The formula \( \sin(A)\cos(B) = \frac{1}{2}[\sin(A+B) + \sin(A-B)] \) is particularly handy when you have a product like \( \sin(-4x)\cos(8x) \).
• By expressing the product as a sum, we can simplify computation in trigonometric problem solving.

Understanding and applying this formula allows you to handle complex trigonometric functions effectively by breaking them down into simpler components. This is crucial in both theoretical and applied mathematics.

Negative Angle Identity

Trigonometric identities that handle negative angles are vital because they help simplify expressions involving negative arguments in trigonometric functions. Specifically, the sine of a negative angle is handled by the identity \( \sin(-A) = -\sin(A) \).
This identity states that the sine of an angle is an odd function, meaning that flipping the angle across the origin changes the sign of the output value.

• For example, in the expression \( \sin(-12x) \) which simplifies to \(-\sin(12x)\), this identity helps to convert the sine of a negative angle to a more manageable form.

This allows for consistent simplification and aids in finding solutions particularly when combined with product-to-sum formulas or other trigonometric identities.