Problem 89 - Precalculus with Limits

Question

In Exercises 81-90, use the product-to-sum formulas to write the product as a sum or difference. \( \sin (x + y) \sin (x - y) \)

Step-by-Step Solution

Answer

The expression \( \sin (x + y) \sin (x - y) \) simplifies to \(0.5[ \cos(2y) - \cos(2x) ]\)

Step 1: Identify the appropriate product-to-sum formula

We should first identify the correct formula to use. The product-to-sum formulas for sine are: \[2\sin a \sin b= \cos(a-b) - \cos(a+b) \] and \[2\cos a \cos b= \cos(a-b) + \cos(a+b) \]. As we look at the expression \( \sin (x + y) \sin (x - y) \), it notably corresponds to the first formula.

Step 2: Apply the formula

Applying the formula, the expression becomes: \[\sin (x + y) \sin (x - y) = 0.5[ \cos((x+y)-(x-y)) - \cos((x+y)+(x-y)) ]\]

Step 3: Simplify the expression

Then simplify the expression inside the cosine function: \[0.5[ \cos(2y) - \cos(2x) ]\]

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is a crucial aspect of mastering trigonometry. Identities in mathematics are equations that are true for all values of the variables. Trigonometric identities are specific to angles and their corresponding trigonometric functions. These identities are used to simplify complex trigonometric expressions, solve trigonometric equations, and prove other mathematical concepts.

There are numerous identities related to the sine, cosine, tangent, and other trigonometric functions. One of the most useful categories is the product-to-sum formulas. The product-to-sum formulas are particularly handy when dealing with products of sines and cosines in trigonometry. They allow us to convert the product of two sine functions or two cosine functions into a sum or difference of two cosine functions. For example, the product-to-sum formula related to the sine function is given as: \[2\sin a \sin b = \cos(a-b) - \cos(a+b)\].

These identities not only make computation easier but also pave the way for integration in calculus and simplification of Fourier series in higher-level mathematics.

• Product-to-sum identities reduce multiplication into addition or subtraction.
• They are essential tools for simplifying and solving trigonometric expressions.
• Enable transformation of the product of trigonometric functions into a form that is easier to integrate or differentiate.

Sine Function

The sine function, denoted by \(\sin\), is one of the primary trigonometric functions and is defined as the ratio of the opposite side to the hypotenuse in a right triangle. This fundamental function is periodic with a period of \(2\pi\) radians or 360 degrees, meaning that it repeats its values in regular intervals. Typically, the values of the sine function range from -1 to 1.

Understanding the sine function is crucial for applying product-to-sum identities as seen in our example exercise where we dealt with \(\sin (x + y) \sin (x - y)\). When considering angles, the sine function can represent a multitude of phenomena such as sound waves, rotating machinery, and many other cyclic natures in the physical world.

Some key takeaways regarding the sine function include:

• It is used to model oscillatory systems.
• The graph of a sine function is a smooth wave that oscillates between -1 and 1.
• It is defined for all real numbers and its values repeat after every \(2\pi\) radians.
• Involves understanding of phases, amplitudes, and frequency when modeling real-world situations.

Algebraic Simplification

Algebraic simplification is the process of making a mathematical expression as simple as possible. This process often involves combining like terms, using distribution property, factoring, and other algebraic techniques to reduce complexity and simplify expressions into a more manageable form. Through algebraic simplification, we aim to make the expressions clearer and easier to work with, especially when evaluating, solving, or further manipulating mathematical problems.

For example, in our exercise, we've used algebraic simplification to break down the product of two sine functions into a simple expression involving cosine by applying the product-to-sum formula. Further simplification involved subtracting angles within the cosine functions and eventually yielding a clean, simplified result: \(0.5[ \cos(2y) - \cos(2x) ]\).

Key points in algebraic simplification include:

• Reduces the complexity of mathematical expressions.
• Makes it easier to evaluate or solve mathematical problems.
• Can involve a variety of methods including common factoring, expanding, and reducing fractions.
• Essential for working with equations and formulas across all areas of mathematics.

Applying algebraic simplification systematically can make even the most daunting problems tractable, laying the groundwork for further exploration and discovery in mathematics.