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Chapter 19: Problem 15

Express \(\sin 4 x \cos 3 x\) as a sum or difference of sines and cosines.

Short Answer

Expert verified
\(\sin 4x \cos 3x = \frac{1}{2} (\sin 7x + \sin x)\).

Step by step solution

01

Identify the Product-to-Sum Formula

To express the product \(\sin A \cos B\) as a sum or difference, we use the product-to-sum identity: \(\sin A \cos B = \frac{1}{2} \left( \sin(A+B) + \sin(A-B) \right)\). In this exercise, \(A = 4x\) and \(B = 3x\).
02

Apply the Formula

Substitute \(A = 4x\) and \(B = 3x\) into the product-to-sum identity: \(\sin 4x \cos 3x = \frac{1}{2} \left( \sin((4x+3x)) + \sin((4x-3x)) \right)\).
03

Simplify the Expression

Combine the terms in the parentheses: \(\sin(4x+3x)\) becomes \(\sin(7x)\) and \(\sin(4x-3x)\) becomes \(\sin(x)\). So the expression is \(\sin 4x \cos 3x = \frac{1}{2} \left( \sin 7x + \sin x \right)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product-to-Sum Formulas
The product-to-sum formulas are a key tool in trigonometry that help convert products of sine and cosine functions into sums or differences. This makes complex trigonometric expressions easier to work with and solve. The formula we use here is:
  • \( \sin A \cos B = \frac{1}{2} \left( \sin(A+B) + \sin(A-B) \right) \)
This formula allows us to express the product of a sine and a cosine function as a simpler addition of sines, making it more straightforward for calculations.
The conversion into a sum is particularly useful in integration and solving equations, simplifying what might otherwise be a challenging problem.
Trigonometric Simplification
Trigonometric simplification involves transforming a trigonometric expression into a simpler form. This process often requires applying identities like the product-to-sum formulas. In our exercise, by using the product-to-sum identity, the original product \( \sin 4x \cos 3x \) is transformed into a sum: \[ \frac{1}{2} \left( \sin(7x) + \sin(x) \right) \]This simplification decreases the complexity of the original expression.
Making such transformations is crucial for solving trigonometric equations, as it helps in finding solutions more efficiently by using simpler terms often seen as standard forms.
Sine and Cosine Functions
The sine and cosine functions are fundamental in trigonometry, representing periodic oscillations. Writing trigonometric expressions in terms of sine and cosine can simplify problem-solving and understanding.
  • Sine Function: \( \sin(\theta) \) measures the vertical position of a point on the unit circle.
  • Cosine Function: \( \cos(\theta) \) measures the horizontal position on the unit circle.
When expressions like \( \sin 4x \cos 3x \) are rewritten using identities, it turns into a more interpretable format with just a combination of these basic sine functions. Understanding these relationships not only aids in academic problems but also in real-world applications involving waves and oscillations.

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