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Chapter 7: Problem 77

Simplify. $$\cos (u+v) \cos v+\sin (u+v) \sin v$$

Short Answer

Expert verified
\(\cos u\)

Step by step solution

01

- Understand the Problem

We need to simplify the expression \(\cos (u+v) \cos v + \sin (u+v) \sin v\).
02

- Recall the Trigonometric Identity

We use the trigonometric identity for the cosine of a difference: \( \cos (a - b) = \cos a \cos b + \sin a \sin b \).
03

- Apply the Identity

Notice the given expression matches the right-hand side of the identity if we let \( a = u + v \) and \( b = v \). Hence, \(\cos (u+v) \cos v + \sin (u+v) \sin v = \cos ((u+v) - v)\).
04

- Simplify the Expression

Simplify the expression \(\cos ((u+v) - v)\) to get \(\cos (u)\).

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Key Concepts

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

Cosine Difference Identity
The cosine difference identity is a fundamental tool in trigonometry.
It states that \(\cos (a - b) = \cos a \cos b + \sin a \sin b\).
This identity is very useful when dealing with problems that involve the sum or difference of angles inside the cosine or sine functions.
In our example, the expression \(\cos (u+v) \cos v + \sin (u+v) \sin v\) matches the right-hand side of the identity.
  • Here, we identify \(a = u + v\) and \(b = v\).
By recognizing this structure, we can apply the identity to simplify the expression significantly.
This understanding helps break down complex trigonometric expressions into simpler forms involves recognizing these patterns.
Angle Simplification
Simplifying angles is a critical skill in trigonometry.
Once we apply the cosine difference identity to our expression \(\cos (u+v) \cos v + \sin (u+v) \sin v\), it transforms into \(\cos ((u+v) - v)\).
Angle simplification involves understanding how to reduce composite angles, especially when terms can cancel each other out.
  • In this case, \((u+v) - v\) simplifies to just \(u\).
This step is crucial, as it reduces the complexity of the problem, making it easier to handle and solve.
Through mastering angle simplification, you can work through many trigonometric problems more efficiently.
Trigonometric Simplification
Trigonometric simplification aims to reduce expressions to their simplest form.
In our example, we started with \(\cos (u+v) \cos v + \sin (u+v) \sin v\).
By applying the cosine difference identity and simplifying the angle, we converted this into \(\cos u\).
Steps in trigonometric simplification typically include:
  • Identifying relevant identities.
  • Applying those identities correctly.
  • Reducing angles where possible.
Understanding and applying these steps can simplify complex expressions, reducing the chance of errors and making the evaluation of trigonometric expressions more straightforward.
This process demonstrates how powerful trigonometric identities and simplification strategies can be in solving mathematical problems.

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