Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Perform indicated operation and simplify the result. $$(\sin \alpha-\cos \alpha)^{2}$$

Step-by-Step Solution

Answer

The simplified result is \(1 - 2\sin \alpha \cos \alpha\).

Step 1: Write Down the Expression

The given expression is \((\sin \alpha - \cos \alpha)^2\). We need to expand this expression using the known algebraic identity for a binomial square: \((a - b)^2 = a^2 - 2ab + b^2\).

Step 2: Apply the Binomial Square Formula

Using the binomial square formula, substitute \(\sin \alpha\) for \(a\) and \(\cos \alpha\) for \(b\). The expression becomes:\[(\sin \alpha - \cos \alpha)^2 = \sin^2 \alpha - 2\sin \alpha \cos \alpha + \cos^2 \alpha\]

Step 3: Simplify Using Trigonometric Identities

Recognize that \(\sin^2 \alpha + \cos^2 \alpha = 1\) by the Pythagorean identity. Thus:\[\sin^2 \alpha + \cos^2 \alpha = 1\]Substitute this into the expression:\[1 - 2\sin \alpha \cos \alpha\]

Step 4: Express the Final Result

Now, we have simplified the expression to a single formula:\(1 - 2\sin \alpha \cos \alpha\). This is the simplest form of the given expression.

Key Concepts

Binomial Square Formula

The Binomial Square Formula is a handy tool when expanding expressions of the form \((a - b)^{2}\). It's essential to simplifying algebraic terms, particularly when dealing with trigonometric expressions like \((\sin \alpha - \cos \alpha)^{2}\).
The formula is:- \((a - b)^2 = a^2 - 2ab + b^2\)
This formula means you square the first term, subtract twice the product of the two terms, and add the square of the second term.
When applying this to our expression \((\sin \alpha - \cos \alpha)^2\), we let \(a = \sin \alpha\) and \(b = \cos \alpha\). The expansion becomes:- \(\sin^2 \alpha - 2\sin \alpha \cos \alpha + \cos^2 \alpha\)
Each part of this result plays a crucial role in further simplification.

Pythagorean Identity

The Pythagorean Identity is one of the most fundamental identities in trigonometry. It states:- \(\sin^2 \theta + \cos^2 \theta = 1\)
This identity is true for all angle measures, making it a versatile tool in simplifying expressions that involve squares of sine and cosine.
In the context of our original problem, the expanded expression contained \(\sin^2 \alpha + \cos^2 \alpha\).
By applying the Pythagorean Identity here, we substitute these terms with 1. The expression then simplifies significantly, reducing complex equations to more manageable forms.
This shows the power of the identity in reducing the number of terms and simplifying the trigonometric expressions.

Simplifying Expressions

Simplifying expressions involves reducing an equation or expression to its most straightforward or most compact form. This usually requires a good understanding of algebraic and trigonometric identities.
Let's walk through the simplification of the expanded expression:- We started with \(\sin^2 \alpha - 2\sin \alpha \cos \alpha + \cos^2 \alpha\).- Utilizing the Pythagorean Identity, substitute \(\sin^2 \alpha + \cos^2 \alpha = 1\).- Thus, the expression simplifies to \(1 - 2\sin \alpha \cos \alpha\).
This process involves recognizing and applying known identities and algebraic tricks, which can help solve and analyze trigonometric problems efficiently.
Through practice, you can identify patterns and apply the relevant identities quickly, making math problems less of a challenge and more of an engaging puzzle to solve.