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Chapter 3: Problem 96

Use identities to simplify each expression. \(\frac{\sin ^{3}(-\theta)}{\sin ^{3} \theta-\sin \theta}\)

Short Answer

Expert verified
The simplified expression is \tan^{2} \theta.

Step by step solution

01

Use Odd Function Property of Sine

Recall that \(\sin(-\theta) = -\sin(\theta)\). Applying this to \sin^{3}(-\theta) gives us \-\sin^{3}(\theta)\, so the expression becomes\(\frac{-\sin^{3}(\theta)}{\sin^{3} (\theta)-\sin \theta}\).
02

Factor Out \sin \theta in the Denominator

Factor \sin \theta out of the denominator \(\sin^{3} \theta - \sin \theta\):\[ \sin^{3} \theta - \sin \theta = \sin \theta (\sin^{2} \theta - 1) \] Using the Pythagorean identity: \(\sin^{2} \theta + \cos^{2} \theta = 1\), the denominator becomes: \(\sin \theta(-\cos^{2} \theta)\).
03

Simplify Expression

Now, replace the simplified denominator back into the expression: \( \frac{-\sin^{3} \theta}{\sin \theta (-\cos^{2} \theta)}\). The \sin \theta terms cancel: \(\frac{-\sin^{2} \theta}{-\cos^{2} \theta} = \frac{\sin^{2} \theta}{\cos^{2} \theta}\).
04

Use Identity

Recall the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Applying it results in: \(\frac{\sin^{2} \theta}{\cos^{2} \theta} = \tan^{2} \theta\). Hence, the simplified expression is \tan^{2} \theta.

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Key Concepts

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

Odd Function Property
The odd function property is a fundamental concept in trigonometry. An odd function is one that satisfies the property:

\( f(-x) = -f(x) \).

For trigonometric functions, this property holds for sine and tangent. For example, the sine function exhibits odd characteristics:
\( \text{sin}(-\theta) = -\text{sin}(\theta) \).

In the given problem, this property is crucial. By using the odd property of sine, we transform \( \text{sin}^3(-\theta) \), into \(-\text{sin}^3(\theta) \).
This step simplifies the numerator, making it easier to simplify the whole expression.
Pythagorean Identity
The Pythagorean identity is another cornerstone in trigonometry:
\( \text{sin}^2(\theta) + \text{cos}^2(\theta) = 1 \).

This relation is always true for any angle \( \theta \). It's used to transform expressions involving sine and cosine. In our problem, we used this identity to simplify the denominator.

Initially, the denominator was expressed as \( \text{sin}^3(\theta) - \text{sin}(\theta) \).
By factoring it out, we got:
\( \text{sin}(\theta)(\text{sin}^2(\theta) - 1) \).
Using the Pythagorean identity, we replaced \( \text{sin}^2(\theta) - 1 \) with
\( -\text{cos}^2(\theta) \), resulting in:
\( \text{sin}(\theta)(-\text{cos}^2(\theta)) \).
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions often involves multiple steps that leverage established identities and properties. Let's break down the final steps in our problem:
  • We start with:
    \( \frac{-\text{sin}^3(\theta)}{\text{sin}(\theta)(-\text{cos}^2(\theta))} \).
  • Next, cancel out \( \text{sin}(\theta) \) from the numerator and the denominator. This gives us:
    \( \frac{-\text{sin}^2(\theta)}{-\text{cos}^2(\theta)} \), which simplifies to:

    \( \frac{\text{sin}^2(\theta)}{\text{cos}^2(\theta)} \)
  • Finally, use the identity:
    \( \tan(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} \) to transform it into:
    \( \tan^2(\theta) \).
Each of these steps is crucial for simplifying complex trigonometric expressions effectively.

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