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Chapter 6: Problem 41

Simplify each expression. $$\frac{1}{a}-\frac{\cos ^{2} x}{a}$$

Short Answer

Expert verified
\(\frac{\text{sin}^2 x}{a}\)

Step by step solution

01

Identify the common denominator

Observe that both terms in the expression have the same denominator, which is \(a\). The expression is \(\frac{1}{a}-\frac{\text{cos}^2 x}{a}\).
02

Combine the fractions

Since both fractions have the same denominator, combine them into a single fraction. The expression becomes \(\frac{1 - \text{cos}^2 x}{a}\).
03

Apply trigonometric identity

Recall the Pythagorean identity: \(1 - \text{cos}^2 x = \text{sin}^2 x\). Substitute this into the expression. The expression now is \(\frac{\text{sin}^2 x}{a}\).

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Key Concepts

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

Common Denominators
Understanding common denominators is crucial when working with fractions. In this problem, both fractions \(\frac{1}{a}\) and \(\frac{\text{cos}^2 x}{a}\) share a common denominator of \(a\). This simplifies the process of combining the fractions because you do not need to find a new common denominator.
When fractions have the same denominator, you can combine them by adding or subtracting their numerators and keeping the denominator the same. For example:
  • \( \frac{1}{a} + \frac{\text{cos}^2 x}{a} = \frac{1 + \text{cos}^2 x}{a} \)
  • \( \frac{1}{a} - \frac{\text{cos}^2 x}{a} = \frac{1 - \text{cos}^2 x}{a} \)
By understanding and identifying common denominators, you can simplify expressions easily and accurately.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides of the equality are defined. In this exercise, we use the Pythagorean identity to simplify the expression.
The Pythagorean identity states that \( \text{sin}^2 x + \text{cos}^2 x = 1 \). From this identity, we can derive other forms such as:
  • \( \text{sin}^2 x = 1 - \text{cos}^2 x \)
  • \( \text{cos}^2 x = 1 - \text{sin}^2 x \)
In this problem, we specifically use the form \( 1 - \text{cos}^2 x = \text{sin}^2 x \). This substitution helps us to simplify the given expression:
\( \frac{1 - \text{cos}^2 x}{a} = \frac{\text{sin}^2 x}{a} \)
By recognizing and applying these identities, you can transform and simplify trigonometric expressions.
Fraction Operations
Working with fractions, especially in algebra, often involves operations like addition, subtraction, multiplication, and division. In this exercise, we focused on the subtraction of fractions with common denominators.
Here鈥檚 a quick refresher on basic fraction operations with an emphasis on subtraction:
  • When subtracting fractions with the same denominator, subtract the numerators and keep the denominator the same. For example, \( \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c} \).
  • When subtracting fractions with different denominators, find a common denominator first. Then, rewrite each fraction with this common denominator before subtracting. For example, \( \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \).
In our problem, both fractions already had the common denominator, making the operation straightforward and yielding: \( \frac{1 - \text{cos}^2 x}{a} \).
Understanding how to properly perform fraction operations is key to simplifying more complex algebraic expressions.

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Most popular questions from this chapter

Find all values of \(\theta\) in the interval \(0^{\circ}, 360^{\circ}\) ) that satisfy each \right. equation. Round approximate answers to the nearest tenth of a degree. $$\sec ^{4} \theta-5 \sec ^{2} \theta+4=0$$

Find all values of \(\alpha\) in degrees that satisfy each equation. Round approximate answers to the nearest tenth of a degree. $$\sec 2 \alpha=4.5$$

Use identities to simplify each expression. Do not use a calculator. $$\frac{\tan 15^{\circ}}{1-\tan ^{2}\left(15^{\circ}\right)}$$

Solve each equation. (These equations are types that will arise in Chapter 7.) $$\frac{\sin 49.6^{\circ}}{55.1}=\frac{\sin 88.2^{\circ}}{b}$$

Solve each equation. (These equations are types that will arise in Chapter 7.) $$\frac{\sin 33.2^{\circ}}{a}=\frac{\sin 45.6^{\circ}}{13.7}$$
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