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Chapter 7: Problem 40

Rationalize the denominator. $$\sqrt{\frac{1-\cos \beta}{1+\cos \beta}}$$

Short Answer

Expert verified
\[\sqrt{\frac{1-\cos \beta}{1+ \cos \beta}} = \frac{1- \cos \beta}{ \sin \beta} \]

Step by step solution

01

Identify the Expression

Identify the given expression that needs rationalizing: \[ \sqrt{\frac{1-\cos \beta}{1+\cos \beta}} \]
02

Multiply by Conjugate of the Denominator

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 + \cos \beta\) is \(1 - \cos \beta\): \[ \sqrt{\frac{1-\cos \beta}{1+\cos \beta}} \cdot \frac{1-\cos \beta}{1-\cos \beta} \]
03

Simplify the Expression

Simplify the expression by multiplying the numerators together and the denominators together: \[ \sqrt{\frac{(1-\cos \beta)^2}{(1+\cos \beta)(1-\cos \beta)}} \]
04

Apply Difference of Squares Formula

Recognize that \((1 + \cos \beta)(1 - \cos \beta)\) is a difference of squares: \[ (1 + \cos \beta)(1 - \cos \beta) = 1 - \cos^2 \beta \]
05

Use Pythagorean Identity

Apply the Pythagorean identity \(1 - \cos^2 \beta = \sin^2 \beta\): \[ \sqrt{\frac{(1-\cos \beta)^2}{\sin^2 \beta}} \]
06

Simplify the Square Root

Simplify the expression under the square root: \[ \sqrt{\frac{(1-\cos \beta)^2}{\sin^2 \beta}} = \frac{1-\cos \beta}{\sin \beta} \]

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Key Concepts

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

trigonometric identities
Trigonometric identities are mathematical equations relating to trigonometric functions. They help us simplify complex trigonometric expressions and solve equations. In our problem, understanding these identities is crucial. For example, one essential identity is the Pythagorean identity: \[ \sin^2 \beta + \cos^2 \beta = 1 \] This identity comes in handy when dealing with sine and cosine functions. Knowing these identities allows us to manipulate and simplify expressions effectively.
Pythagorean identity
The Pythagorean identity is a key trigonometric identity. It relates the sine and cosine of the same angle. Written as \[ \sin^2 \beta + \cos^2 \beta = 1 \] This identity is derived from the Pythagorean theorem applied to the unit circle. When solving trigonometric problems, you often use the identity to convert between sine and cosine.For instance, in the given problem, we simplified the denominator by recognizing that: \[ 1 - \cos^2 \beta = \sin^2 \beta \] This step was vital in rationalizing the denominator and simplifying the expression.
difference of squares
The difference of squares is another important concept in this problem. It states that: \[ a^2 - b^2 = (a+b)(a-b) \] This formula helps us factorize and simplify expressions where we have a subtraction of squares. In our exercise, we used this formula to turn \[ (1 + \cos \beta)(1 - \cos \beta) \] into \[ 1 - \cos^2 \beta. \] It鈥檚 a powerful tool in algebra and trigonometry, widely used for simplification purposes. Recognizing when and how to apply it can make solving many mathematical problems much more manageable.

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Most popular questions from this chapter

Satellite Location. \(\quad\) A satellite circles the earth in such a manner that it is \(y\) miles from the equator (north or south, height from the surface not considered) \(t\) minutes after its launch, where $$Y=5000\left[\cos \frac{\pi}{45}(t-10)\right]$$ At what times \(t\) on the interval \([0,240],\) the first \(4 \mathrm{hr}\) is the satellite 3000 mi north of the equator?

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