Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Match each expression with the correct expression to form an identity. $$\sin (x-y)= \text {_____} $$ A. \(\cos x \cos y+\sin x \sin y\) B. \(\sin x \sin y-\cos x \cos y\) C. \(\sin x \cos y+\cos x \sin y\) D. \(\sin x \cos y-\cos x \sin y\) E. \(\cos x \sin y-\sin x \cos y\) F. \(\cos x \cos y-\sin x \sin y\)

Step-by-Step Solution

Answer

Option D: \( \sin x \cos y - \cos x \sin y \)

Step 1: Recall Trigonometric Identity

We need to identify the trigonometric identity that matches with \( \sin(x-y) \). Recall that the identity for the sine of a difference is given by: \[ \sin(x-y) = \sin x \cos y - \cos x \sin y \].

Step 2: Match with Given Options

Compare the identity \( \sin(x-y) = \sin x \cos y - \cos x \sin y \) with the options provided. We are looking for an expression that exactly matches this identity.

Step 3: Identify the Correct Option

From the options given:- Option A is \( \cos x \cos y + \sin x \sin y \)- Option B is \( \sin x \sin y - \cos x \cos y \)- Option C is \( \sin x \cos y + \cos x \sin y \)- Option D is \( \sin x \cos y - \cos x \sin y \)- Option E is \( \cos x \sin y - \sin x \cos y \)- Option F is \( \cos x \cos y - \sin x \sin y \)The correct option that matches \( \sin(x-y) \) is Option D: \( \sin x \cos y - \cos x \sin y \).

Key Concepts

Sine of a difference

The concept of sine of a difference is a fundamental trigonometric identity. This identity helps in breaking down the sine function for the difference of two angles, denoted by \(x-y\). In its essence, it allows us to express \(\sin(x-y)\) in terms of the sines and cosines of the individual angles \(x\) and \(y\).

The formula is given as:

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

This identity is extremely useful for simplifying complex trigonometric expressions and solving trigonometric equations. It is worth noting that this formula is derived using particular geometric interpretations involving a unit circle, but for practical applications, memorizing the formula is often sufficient.

Using the sine of a difference formula can seem tricky at first, but with practice, it becomes an invaluable tool for tackling a variety of mathematical problems.

Trigonometric expressions

Trigonometric expressions involve combining trigonometric functions like sine, cosine, and tangent to create equations or identities. These expressions can sometimes appear complex, but understanding key identities can simplify them greatly.

Common trigonometric identities include:

• Sine and cosine identities such as \(\sin^2x + \cos^2x = 1\)
• The tangent identity \(\tan x = \frac{\sin x}{\cos x}\)
• The sine of a difference identity \(\sin(x-y) = \sin x \cos y - \cos x \sin y\)

These identities can be used to simplify trigonometric expressions by reducing the number of different types of functions present. The sine of a difference identity, for example, transforms an expression with a sine function of a difference into a sum and a product of sines and cosines of single angles, which are often easier to work with or recognize.

Therefore, mastering these identities will facilitate solving trigonometric equations, and can even make more advanced topics like calculus more approachable.

Math problem solving

In the realm of trigonometric identities, problem-solving involves recognizing patterns and substituting known identities to simplify or solve expressions. A systematic approach helps in tackling these kinds of math problems effectively.

The process often includes:

• Identifying familiar patterns or structures within an expression
• Applying appropriate trigonometric identities to simplify the expressions
• Substituting known values or expressions if applicable
• Checking the solution for consistency within the original problem context

For instance, when faced with finding a matching identity from a list, as in our exercise, the solution becomes clearer once you use the recognized identity of \(\sin(x-y) = \sin x \cos y - \cos x \sin y\) to compare with given options.

Developing these problem-solving skills in trigonometry aids students not only in academic settings but also enhances logical reasoning, which is beneficial across numerous fields.