Problem 22 - Contemporary Precalculus

Question

Simplify the given expression. $$\sin (x-y) \cos y+\cos (x-y) \sin y$$

Step-by-Step Solution

Answer

Question: Simplify the following expression using trigonometric identities: $$\sin (x-y) \cos y+\cos (x-y) \sin y$$ Answer: The simplified expression is: $$\sin x + \cos x \sin y \cos y - \cos x \sin^2 y$$

Step 1: Apply the sum and difference angles formula for sine and cosine functions

Using the sum and difference angles formula, we can rewrite the given expression as: $$\sin (x-y) \cos y+\cos (x-y) \sin y = (\sin x \cos y - \cos x \sin y) \cos y + (\cos x \cos y + \sin x \sin y) \sin y$$

Step 2: Distribute the terms

Now, we will distribute the terms in the expression: $$= \sin x \cos^2 y - \cos x \sin y \cos y + \cos x \cos y \sin y + \sin x \sin^2 y$$

Step 3: Rearrange the terms

Next, we can rearrange the terms by grouping the like terms together: $$= \sin x(\cos^2 y + \sin^2 y) + \cos x\sin y(\cos y - \sin y)$$

Step 4: Simplify using Pythagorean identity

Recall the Pythagorean identity: \(\sin^2 a + \cos^2 a = 1\). Therefore, \(\sin^2 y + \cos^2 y = 1\). We can use this to simplify the expression: $$= \sin x(1) + \cos x\sin y(\cos y - \sin y)$$

Step 5: Final Simplification

Finally, we have: $$= \sin x + \cos x \sin y \cos y - \cos x \sin^2 y$$ So the simplified expression is: $$\sin x + \cos x \sin y \cos y - \cos x \sin^2 y$$

Key Concepts

Sine and Cosine Functions

The sine and cosine functions are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of two of its sides. Here's a simple way to remember their relationship:

• Sine (\( ext{sin}\)) of an angle is the opposite side divided by the hypotenuse.
• Cosine (\( ext{cos}\)) of an angle is the adjacent side divided by the hypotenuse.

These functions are essential tools used in various fields, from physics to engineering, because they describe wave-like patterns, motions, and are keys to solving triangles. When dealing with angles in trigonometric identities, \( ext{sin}(x-y)\) and \( ext{cos}(x-y)\) are examples where subtracting two angles affects the sine and cosine values. Understanding these functions enables better simplification and transformation of trigonometric expressions.

Pythagorean Identity

The Pythagorean identity is one of the most basic and important identities in trigonometry, reflecting a fundamental geometric truth. It states: \[\sin^2 a + \cos^2 a = 1\]This identity arises from the Pythagorean theorem applied to a unit circle. A unit circle has a radius of 1, so any point on the circle satisfies \(x^2 + y^2 = 1\), where \(x = \text{cos}(\theta)\) and \(y = \text{sin}(\theta)\).
With this identity, one can simplify expressions significantly by replacing \(\sin^2 a + \cos^2 a\) with 1, which often makes solving complex trigonometric expressions much simpler. Like in our problem, recognizing that \(\cos^2 y + \sin^2 y = 1\) was key to simplifying the expression. Keeping this identity in mind can be very useful in both theoretical and applied mathematics contexts.

Angle Sum and Difference Identities

The angle sum and difference identities allow us to calculate the sine or cosine of the sum or difference of two angles. These formulas are:

• Sine of sum/difference: \(\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\)
• Cosine of sum/difference: \(\cos(x \pm y) = \cos x \cos y \mp \sin x \sin y\)

These identities are quite powerful because they enable us to break down complex angles into simpler, more manageable parts, which can be useful in proving further identities or simplifying expressions, as we have done in the exercise provided.
By utilizing \(\sin(x - y)\) and \(\cos(x - y)\), the trigonometric expression was manipulated to simplify, showing the practical use of these identities. Mastering them can make a wide array of trigonometric equations and proofs more accessible and easily navigable.