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Chapter 1: Problem 41

Multiply. a. \((a-b)^{2}\) b. \((\cos \theta-\sin \theta)^{2}\)

Short Answer

Expert verified
1. Part (a): \((a-b)^2 = a^2 - 2ab + b^2\) 2. Part (b): \((\cos \theta - \sin \theta)^2 = 1 - 2\cos \theta \sin \theta\)

Step by step solution

01

Understand the Formula

We need to expand the expression using the square of a binomial formula. For any binomial \((x - y)^2\), the formula is \((x - y)^2 = x^2 - 2xy + y^2\). We will use this to expand the given expressions.
02

Apply the Formula to Part (a)

For \((a-b)^2\), we identify \(x = a\) and \(y = b\). Now apply the formula:\((a-b)^2 = a^2 - 2ab + b^2\). This gives the expanded form of \((a-b)^2\).
03

Apply the Formula to Part (b)

For \((\cos \theta - \sin \theta)^2\), identify \(x = \cos \theta\) and \(y = \sin \theta\). Apply the formula:\((\cos \theta - \sin \theta)^2 = (\cos \theta)^2 - 2(\cos \theta)(\sin \theta) + (\sin \theta)^2\).Simplify using the Pythagorean identity \((\cos \theta)^2 + (\sin \theta)^2 = 1\):\((\cos \theta)^2 - 2\cos \theta \sin \theta + (\sin \theta)^2 = 1 - 2\cos \theta \sin \theta\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square of a Binomial
The square of a binomial is a fundamental algebraic expression that most students encounter early in their math journey. When we square a binomial like \((x - y)^2\), we use a specific formula to expand it:
  • The formula is \((x - y)^2 = x^2 - 2xy + y^2\).
  • This pattern holds true for any binomial of the form \((a - b)^2\) or any letters substituted in a similar way.
To understand this better, imagine multiplying \((x-y)\) by itself. You are essentially distributing each term from the first factor to each term in the second. That鈥檚 why we end up with the terms \(x^2\), \(-2xy\), and \(y^2\). This expression not only simplifies algebraic calculations but also sets a foundation for solving complex equations easily.
Pythagorean Identity
In trigonometry, the Pythagorean identity is a remarkable relationship that connects the squares of the sine and cosine functions.
  • The identity is expressed as \((\cos \theta)^2 + (\sin \theta)^2 = 1\).
  • This tells us that the sum of the squares of cosine and sine for any angle \(\theta\) is always 1.
The Pythagorean identity is often used in various calculations and simplifications because of its reliability. It helps convert between different trigonometric expressions and can reduce complex expressions into more manageable ones, just like in our example:
  • The expression \((\cos \theta - \sin \theta)^2\) simplifies using this identity.
  • This yields \(1 - 2\cos \theta \sin \theta\), which is much simpler to handle.
Mastering the use of this identity can empower students to tackle a variety of problems with confidence.
Expanding Binomials
Expanding binomials is a common task in algebra and requires knowing certain patterns and formulas. Whenever a binomial like \((a - b)^2\) or \((x + y)^2\) appears, expansion means rewriting it as a sum or difference of terms.
  • Recognize that for any binomial \((x - y)^2\), expanding it involves identifying and calculating three terms: \(x^2, -2xy\), and \(y^2\).
  • Expansion allows these binomials to become polynomials, which are easier to work with in further algebraic manipulations.
By practicing this technique, students enhance their ability to simplify and solve equations efficiently. It鈥檚 crucial to also verify if any further simplifications can be made using another mathematical property, such as the Pythagorean identity.This dual approach of expansion and simplification makes algebraic problems less daunting and more approachable.

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Most popular questions from this chapter

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ \frac{\csc \theta}{\cot \theta}=\sec \theta $$

Find the remaining trigonometric ratios of \(\theta\) based on the given information. \(\cos \theta=\frac{2 \sqrt{13}}{13}\) and \(\theta \in \mathrm{QIV}\)

Give the reciprocal of each number. \(-\frac{2}{3}\)

Give the reciprocal of each number. \(-\frac{1}{\sqrt{2}}\)

Give the reciprocal of each number. \(-\frac{12}{13}\)
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