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Magazine Chapter 5: Problem 11
Multiply the numerator and denominator of the fraction by the conjugate of the denominator, and then simplify.a. \(\frac{1}{1+\sqrt{3}}\) b. \(\frac{1}{1+\cos x}\)
Short Answer
Expert verified
a. -\(\frac{1}{2} + \frac{\sqrt{3}}{2}\)
b. \(\frac{1 - \cos x}{\sin^2 x}\)
Step by step solution
01
Identify the Conjugate of the Denominator (Part a)
The given fraction is \( \frac{1}{1 + \sqrt{3}} \). The conjugate of the denominator \( 1 + \sqrt{3} \) is \( 1 - \sqrt{3} \).
02
Multiply by the Conjugate (Part a)
Multiply both the numerator and denominator by the conjugate of the denominator: \( \frac{1}{1 + \sqrt{3}} \times \frac{1 - \sqrt{3}}{1 - \sqrt{3}} \). This results in a new fraction: \( \frac{1 - \sqrt{3}}{(1 + \sqrt{3})(1 - \sqrt{3})} \).
03
Simplify the Denominator (Part a)
Use the difference of squares formula to simplify the denominator: \((1 + \sqrt{3})(1 - \sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2\).
04
Simplify the Fraction (Part a)
Write the fraction with the simplified denominator: \( \frac{1 - \sqrt{3}}{-2} = -\frac{1}{2} + \frac{\sqrt{3}}{2} \).
05
Identify the Conjugate of the Denominator (Part b)
The given fraction is \( \frac{1}{1 + \cos x} \). The conjugate of the denominator \( 1 + \cos x \) is \( 1 - \cos x \).
06
Multiply by the Conjugate (Part b)
Multiply both the numerator and denominator by the conjugate of the denominator: \( \frac{1}{1 + \cos x} \times \frac{1 - \cos x}{1 - \cos x} \). This results in a new expression: \( \frac{1 - \cos x}{(1 + \cos x)(1 - \cos x)} \).
07
Simplify the Denominator (Part b)
Using the difference of squares formula, simplify the denominator: \((1 + \cos x)(1 - \cos x) = 1^2 - (\cos x)^2 = 1 - \cos^2 x \). Recognize \(1 - \cos^2 x\) as \( \sin^2 x \) due to the Pythagorean identity.
08
Simplify the Fraction (Part b)
Write the fraction with the simplified denominator: \( \frac{1 - \cos x}{\sin^2 x} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conjugate
When you encounter a fraction like \( \frac{1}{1+\sqrt{3}} \), multiplying by the conjugate can simplify the expression. The conjugate of a binomial expression \( a + b \) is \( a - b \). It simply means switching the sign in between the terms. For the exercise above, the conjugate of the denominator \( 1 + \sqrt{3} \) is \( 1 - \sqrt{3} \).
- To eliminate irrational numbers, multiply both numerator and denominator by this conjugate.
- This helps in creating a difference of squares in the denominator, paving the way for simplification.
Difference of Squares
The difference of squares is a handy algebraic identity: \( a^2 - b^2 = (a + b)(a - b) \). It allows us to simplify expressions by converting a product of sums and differences into a single squared subtraction.
For example, in the denominator \( (1 + \sqrt{3})(1 - \sqrt{3}) \), applying the difference of squares gives \( 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 \).
For example, in the denominator \( (1 + \sqrt{3})(1 - \sqrt{3}) \), applying the difference of squares gives \( 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 \).
- This transformation is crucial because it removes the square roots, simplifying arithmetic operations.
- It also shows the close relationship between multiplication and subtraction.
Pythagorean Identity
In trigonometry, the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) is fundamental. It connects the sine and cosine functions perfectly. If you're given \( 1 - \cos^2 x \), you can immediately relate it to \( \sin^2 x \).
In the exercise, the expression \( (1 + \cos x)(1 - \cos x) \) simplifies to \( 1 - \cos^2 x \). By the Pythagorean identity:
In the exercise, the expression \( (1 + \cos x)(1 - \cos x) \) simplifies to \( 1 - \cos^2 x \). By the Pythagorean identity:
- This equals \( \sin^2 x \), which substantially simplifies the fraction \( \frac{1 - \cos x}{\sin^2 x} \).
- It highlights the interactions between cosine and sine, offering new pathways for simplification.
