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Chapter 3: Problem 62

Simplify each expression. \(\cos (y)+\cos (-y)\)

Short Answer

Expert verified
The simplified expression is \(2\cos(y)\).

Step by step solution

01

- Recall the Cosine Function Property

Understand that the cosine function is an even function. This property means that \(\cos(-y) = \cos(y)\).
02

- Substitute the Property

Substitute \(\cos(-y)\) with \(\cos(y)\) in the original expression \(\cos(y) + \cos(-y)\) to get \(\cos(y) + \cos(y)\).
03

- Simplify the Expression

Combine like terms: \(\cos(y) + \cos(y) = 2\cos(y)\).

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Key Concepts

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

cosine function
The cosine function is one of the fundamental functions in trigonometry. It is often denoted as \(\text{cos}(x)\). The cosine function helps to relate the angles of a right triangle to the lengths of the sides of the triangle. The main features of the cosine function include:
  • It has a period of \(\text{2蟺}\), meaning that \(\text{cos}(x) = \text{cos}(x + 2蟺)\).
  • The range of the cosine function is \([-1, 1]\), meaning the function only takes values between -1 and 1.
  • The function is symmetric about the y-axis.
In our example, we are using the property of the cosine function being an even function to simplify the given expression.
even functions
An even function is a function that satisfies the property \(\text{f}(x) = \text{f}(-x)\). This means that the function is symmetric with respect to the y-axis. For trigonometric functions, the cosine function is a prime example of an even function. Because \(\text{cos}(x) = \text{cos}(-x)\), we can replace \(\text{cos}(-y)\) with \(\text{cos}(y)\).
This property is very useful in simplifying expressions, as it allows us to consolidate terms and reduce complexity. In this particular exercise, knowing that cosine is even helps us replace \(\text{cos}(-y)\) with \(\text{cos}(y)\), making the process of simplification straightforward.
simplifying expressions
Simplifying expressions is a key skill in algebra and trigonometry. The goal is to reduce an expression to its simplest form for easier computation or interpretation. Here are some common strategies for simplifying expressions:
  • Combine like terms: Identify and merge terms that are similar.
  • Use trigonometric identities: Apply known identities, such as even/odd properties, to transform the expression.
  • Factor where possible: Extract common factors to reduce the number of terms.
In our specific example, after recognizing that \(\text{cos}(-y) = \text{cos}(y)\), we combine the cosine terms:
\(\text{cos}(y) + \text{cos}(y) = 2\text{cos}(y)\).
Thus, we have successfully simplified the expression to \(2\text{cos}(y)\).

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