Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Using the addition formulas and what you know about even and odd trigonometric functions, find expressions for \(\sin (x-2 y)\) and \(\cos (x-2 y)\) in terms of \(\cos x, \cos y, \sin x\) and \(\sin y\).

Step-by-Step Solution

Answer

\(\sin (x-2 y) = \sin x \cos^2 y - \sin x \sin^2 y - 2 \cos x \sin y \cos y\) and \(\cos (x-2 y) = \cos x \cos^2 y - \cos x \sin^2 y + 2 \sin x \sin y \cos y\)

Step 1: Apply the Subtraction Formulas

Start by applying the subtraction formulas for sine and cosine, which are \(\sin (x-y) = \sin x \cos y - \cos x \sin y\) and \(\cos (x-y) = \cos x \cos y + \sin x \sin y\). Using these formulas, we get \(\sin (x-2 y) = \sin x \cos 2y - \cos x \sin 2y\) and \(\cos (x-2 y) = \cos x \cos 2y + \sin x \sin 2y\).

Step 2: Apply the Double-Angle Formulas

The next step is to apply the double-angle formulas. These are used to simplify the expressions with double angles, indicated by the 2y in our equations. The double-angle formulas are \(\sin 2y = 2 \sin y \cos y\) and \(\cos 2y = \cos^2 y - \sin^2 y\). Now we substitute these expressions into the previously introduced formulas for \(\sin (x-2 y)\) and \(\cos (x-2 y)\). This gives us \(\sin (x-2 y) = \sin x (\cos^2 y - \sin^2 y) - \cos x (2 \sin y \cos y)\) and \(\cos (x-2 y) = \cos x (\cos^2 y - \sin^2 y) + \sin x (2 \sin y \cos y)\).

Step 3: Expand and Rearrange

Lastly, to get the desired expressions in terms of \(\cos x, \cos y, \sin x, \sin y\), expand and rearrange the last expressions from step 2. For \(\sin (x-2 y)\), the expanded form is \(\sin x \cos^2 y - \sin x \sin^2 y - 2 \cos x \sin y \cos y\), and for \(\cos (x-2 y)\), the expanded form is \(\cos x \cos^2 y - \cos x \sin^2 y + 2 \sin x \sin y \cos y\).

Key Concepts

Subtraction Formulas

The subtraction formulas are essential tools in trigonometry, especially when dealing with expressions like \( \sin(x - 2y) \) and \( \cos(x - 2y) \). They allow us to express these functions in terms of more manageable components. The general subtraction formulas for sine and cosine are:

• \( \sin(x-y) = \sin x \cos y - \cos x \sin y \)
• \( \cos(x-y) = \cos x \cos y + \sin x \sin y \)

Working through the exercise, when confronted with \( \sin(x - 2y) \), the subtraction formula becomes \( \sin x \cos 2y - \cos x \sin 2y \). Similarly, for \( \cos(x - 2y) \), it becomes \( \cos x \cos 2y + \sin x \sin 2y \).
These formulas break down complex expressions into simpler trigonometric terms, which are easier to manage and solve in equations. By mastering these, you can simplify and solve problems that initially seem complex.

Double-Angle Formulas

The double-angle formulas are a specific set of trigonometric identities useful when dealing with double angles like \(2y\). These formulas help to convert products of trigonometric functions into sums or differences, making them easier to handle. Here are the primary double-angle formulas:

• \( \sin 2y = 2 \sin y \cos y \)
• \( \cos 2y = \cos^2 y - \sin^2 y \)

Using these in our exercise allows us to substitute \( \sin 2y \) and \( \cos 2y \) in the subtraction formulas. After substitution, the expressions become:

• \( \sin(x-2y) = \sin x (\cos^2 y - \sin^2 y) - \cos x (2 \sin y \cos y) \)
• \( \cos(x-2y) = \cos x (\cos^2 y - \sin^2 y) + \sin x (2 \sin y \cos y) \)

These substitutions help us to further simplify and rearrange terms, bringing insights to the problem and solutions that are otherwise hidden.

Even and Odd Trigonometric Functions

Understanding even and odd trigonometric functions is crucial to simplifying expressions and solving trigonometric problems. An **even function** is symmetric about the y-axis, meaning it satisfies \( f(-x) = f(x) \). For trigonometric functions, this applies to cosine:

• \( \cos(-x) = \cos x \)

An **odd function** is symmetric about the origin, meaning it satisfies \( f(-x) = -f(x) \). For trigonometric functions, this applies to sine:

• \( \sin(-x) = -\sin x \)

These properties are especially useful in simplifying calculations and checking the accuracy of trigonometric transformations. For example, when rearranging expressions or substituting angles, knowing whether the function is even or odd can simplify your work and highlight patterns in the identities.