Question

Rationalize the numerator. $\sqrt{\frac{1+\sin y}{1-\sin y}}$

   Rationalize the numerator.
$\sqrt{\frac{1+\sin y}{1-\sin y}}$

Algebra and Trigonometry

Algebra and Trigonometry

Beecher, Judith… 1st Edition

Chapter 7, Problem 43

Instant Answer

Solved by Expert Ankit Gupta

07/02/2020

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Rationalize the numerator. $\sqrt{\frac{1+\sin y}{1-\sin y}}$

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Key Concepts

Trigonometric Identities

Trigonometric identities, such as the Pythagorean identity (sin^2? + cos^2? = 1), are often used to simplify expressions involving trigonometric functions. Recognizing these identities allows one to convert between different trigonometric forms and can be essential when rationalizing or simplifying expressions that include sine, cosine, and other trigonometric functions.

Difference of Squares

The difference of squares is a key algebraic identity which states that a^2 - b^2 = (a + b)(a - b). When an expression involves the product of conjugates, this identity can be applied to simplify the expression by directly cancelling the cross terms, which is particularly useful when rationalizing fractions with radicals.

Rationalization

Rationalization is a technique used in algebra to remove irrational numbers (typically square roots) from the numerator or denominator of a fraction. This is done by multiplying the expression by a carefully chosen form of 1 so that any radical expressions simplify or cancel, resulting in a rational expression which is often easier to work with in further computations.

Conjugate Multiplication

Conjugate multiplication involves multiplying an expression by its conjugate—a binomial formed by changing the sign between two terms—to take advantage of the difference of squares formula. This technique is particularly useful for rationalizing expressions that contain radicals, as it allows the elimination of the square root terms by transforming the product into a difference of two squares.

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