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Chapter 5: Problem 16

Write each expression as a product of sines and/or cosines. $$ \sin (10 x)+\sin (5 x) $$

Short Answer

Expert verified
The expression is rewritten as \( 2 \sin\left(\frac{15x}{2}\right) \cos\left(\frac{5x}{2}\right) \).

Step by step solution

01

Identifying the Formula to Use

To write the expression \( \sin(10x) + \sin(5x) \) as a product of sines and/or cosines, we can use the sum-to-product identities. The relevant identity for sum-to-product is: \( \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \).
02

Assign Values to Variables

In the formula \( \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \), assign \( A = 10x \) and \( B = 5x \).
03

Compute \( \frac{A+B}{2} \) and \( \frac{A-B}{2} \)

Calculate \( \frac{A+B}{2} = \frac{10x + 5x}{2} = \frac{15x}{2} \). Calculate \( \frac{A-B}{2} = \frac{10x - 5x}{2} = \frac{5x}{2} \).
04

Substitute into the Sum-to-Product Formula

Use the values computed in Step 3 in the sum-to-product formula: \( \sin(10x) + \sin(5x) = 2 \sin\left(\frac{15x}{2}\right) \cos\left(\frac{5x}{2}\right) \).
05

Final Expression

The expression \( \sin (10 x)+\sin (5 x) \) is rewritten as a product of sines and/or cosines: \( 2 \sin\left(\frac{15x}{2}\right) \cos\left(\frac{5x}{2}\right) \).

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Key Concepts

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

Sum-to-Product Identities
Sum-to-product identities are useful in simplifying trigonometric expressions by converting sums into products. They help in solving various trigonometric equations and enable deeper understanding by revealing underlying relationships. One commonly used identity is:
  • \( \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \)
This formula transforms the sum of sines into a product involving both sine and cosine functions.
It also highlights how different trigonometric terms are connected. To apply this identity, it's crucial to correctly substitute the values of \(A\) and \(B\) from the given problem. This ensures that the final expression is simplified accurately into a product form.
Sine and Cosine Products
The transformation of trigonometric sums into products involves changing the form to include sine and cosine functions. For the expression \( \sin(10x) + \sin(5x) \), using the identity results in:
  • \( 2 \sin\left(\frac{15x}{2}\right) \cos\left(\frac{5x}{2}\right) \)
Here, sine and cosine work together, managing complexity in the expression.
This transformation simplifies calculations, making it easier to handle integrals or other mathematical operations.
Sine and cosine products are indispensable in trigonometry for managing expressions efficiently, and they often appear in various mathematical fields beyond basic trigonometry.
Trigonometric Expressions
Trigonometric expressions often appear in mathematics and physics. Understanding them is crucial for solving problems involving angles and waves. Expressions like \( \sin(10x) + \sin(5x) \) often need simplification for practical use, such as in calculus or geometry problems.
By converting sums to products, we make the expressions easier to work with or more suitable for specific applications.
Mastering these expressions involves recognizing patterns, applying identities, and transforming them strategically. With practice, breaking down complex trigonometric forms becomes more intuitive, leading to faster problem-solving and a better grasp of mathematical principles.

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