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Chapter 5: Problem 4

For Questions 1 through 6 , complete each sum or difference identity. $$ \cos (\theta-\phi)= $$

Short Answer

Expert verified
\( \cos(\theta - \phi) = \cos\theta \cos\phi + \sin\theta \sin\phi \)

Step by step solution

01

Recall the Cosine Difference Identity

To solve this exercise, we need to remember the cosine difference identity. The formula for \( \cos(\theta - \phi) \) is given by:\[\cos(\theta - \phi) = \cos\theta \cos\phi + \sin\theta \sin\phi\]
02

Identify the Expression Components

The given expression is \( \cos(\theta - \phi) \). We need to find the cosine of the angle difference using the identity from Step 1. This involves substituting \( \theta \) for the first angle and \( \phi \) for the second angle within the identity.
03

Substitute and Simplify

To complete the identity using the formula, replace the general components in the equation from Step 1:\[\cos(\theta - \phi) = \cos\theta \cos\phi + \sin\theta \sin\phi\]There is no further simplification required unless specific values for \( \theta \) or \( \phi \) are given.

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Key Concepts

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

Cosine Difference Identity
Understanding the cosine difference identity is crucial for solving many trigonometry problems. The identity is expressed as:\[\cos(\theta - \phi) = \cos\theta \cos\phi + \sin\theta \sin\phi\]This formula relates the cosine of the difference of two angles to the sines and cosines of the angles themselves. It is derived from the properties of the unit circle and trigonometric functions. By using this identity, complex problems involving the difference in angles become simpler.
  • The first term, \(\cos\theta \cos\phi\), represents the product of the cosines of the individual angles.
  • The second term, \(\sin\theta \sin\phi\), adds the product of their sines.
This approach is particularly helpful when exact values are known or need to be calculated, as it breaks down a potentially complicated expression into simpler components.
Angle Difference Formulas
Angle difference formulas are an essential tool in trigonometry. They include identities for sine, cosine, and tangent, helping us find the trigonometric values of angle differences. These formulas allow us to express functions like \(\sin(\theta - \phi)\) and \(\tan(\theta - \phi)\) similarly.
  • For sine: \[ \sin(\theta - \phi) = \sin\theta \cos\phi - \cos\theta \sin\phi \]
  • For cosine, as discussed: \[ \cos(\theta - \phi) = \cos\theta \cos\phi + \sin\theta \sin\phi \]
  • For tangent: \[ \tan(\theta - \phi) = \frac{\tan\theta - \tan\phi}{1 + \tan\theta \tan\phi} \]
These are powerful tools because they convert expressions involving differences into additions of products, simplifying calculations and making it easier to evaluate or integrate the trigonometric expressions.
Trigonometry Exercises
Practicing with trigonometry exercises helps solidify understanding of concepts like the cosine difference identity. These exercises often require using derived identities to solve problems involving angles. It鈥檚 beneficial to:
  • Start with identifying the type of trigonometric identity needed.
  • Substitute the given values or angles into the appropriate formula, such as the cosine difference identity.
  • Simplify the expression step-by-step to arrive at the final result.
Applying such identities in different exercises assists learners in recognizing patterns and applying the correct formulas efficiently. This practice not only enhances problem-solving skills but also deepens the understanding of underlying trigonometric principles. Regularly engaging with diverse problems is the key to mastering these concepts.

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Most popular questions from this chapter

For Questions 4 through 6, determine if the statement is true or false. \(\cos 2 A=2 \cos A\)

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