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Chapter 3: Problem 72

Determine whether each function is odd, even, or neither. \(f(x)=2 \sin x \cos x\)

Short Answer

Expert verified
The function \( f(x) = 2 \sin x \cos x \) is odd.

Step by step solution

01

Recall Definitions

A function is even if \(f(-x) = f(x)\), odd if \(f(-x) = -f(x)\), and neither if it does not satisfy either condition.
02

Substitute \(-x\)

Substitute \(-x\) into the function: \(f(-x) = 2 \sin(-x) \cos(-x)\).
03

Use Trigonometric Properties

Use the trigonometric identities \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \) to simplify: \( f(-x) = 2(-\sin(x))(\cos(x)) = -2 \sin(x) \cos(x)\).
04

Compare \(f(x)\) and \(f(-x)\)

Since \( f(-x) = -2 \sin(x) \cos(x) \) is equal to \( -f(x) \), the function \( f(x) \) is odd.

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Key Concepts

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

trigonometric functions
Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are crucial in various areas of mathematics and engineering.
Some of the most common trigonometric functions include:
  • Sine (\text{sin})
  • Cosine (\text{cos})
  • Tangent (\text{tan})
  • Cotangent (\text{cot})
  • Secant (\text{sec})
  • Cosecant (\text{csc})
Each function has its own properties and identities, which help in solving mathematical problems involving angles and lengths.
For example, consider the function in our exercise: \(f(x) = 2 \text{sin}(x) \text{cos}(x)\). It is a combination of the sine and cosine functions.
function properties
Understanding the properties of functions is essential to determine if they are even, odd, or neither.
  • An **even function** satisfies the property that \( f(-x) = f(x) \) for all \( x \) in its domain.
  • An **odd function** satisfies the property that \( f(-x) = -f(x) \) for all \( x \) in its domain.
  • A function is **neither even nor odd** if it doesn't satisfy either condition.
In our example, we are given **\(f(x) = 2 \text{sin}(x) \text{cos}(x)\)** and need to determine its property:
  • Step 1: Substitute \(-x\). \( f(-x) = 2 \text{sin}(-x) \text{cos}(-x) \).
  • Step 2: Use the identities \( \text{sin}(-x) = -\text{sin}(x) \) and \( \text{cos}(-x) = \text{cos}(x) \), yielding \( f(-x) = 2(-\text{sin}(x))(\text{cos}(x)) = -2 \text{sin}(x) \text{cos}(x) \).
  • Step 3: Compare with \(f(x)\). Since \( f(-x) = -f(x) \), the function is **odd**.
trigonometric identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable for which both sides of the equation are defined.
Some key identities we use frequently include:
  • Pythagorean identities: \( \text{sin}^2(x) + \text{cos}^2(x) = 1 \)
  • Angle sum and difference identities: \( \text{sin}(a \pm b) = \text{sin}(a) \text{cos}(b) \pm \text{cos}(a) \text{sin}(b) \)
  • Double angle identities: \( \text{sin}(2x) = 2 \text{sin}(x) \text{cos}(x) \)
In our case, we specifically use the odd and even identities:
  • \( \text{sin}(-x) = -\text{sin}(x) \)
  • \( \text{cos}(-x) = \text{cos}(x) \)
These identities help us in simplifying the function \( f(-x) \) and determining that \(f(x)\) is an odd function.

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Most popular questions from this chapter

Simplify \(\frac{\cos ^{3}(x)+\cos (x) \sin ^{2}(x)}{\sin (x)}\)

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State the three Pythagorean identities.
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