Problem 48 - Contemporary Precalculus

Question

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$(\sin t-\cos t)^{2}$$

Step-by-Step Solution

Answer

Question: Simplify the expression \((\sin t-\cos t)^2\). Answer: \(1 - 2(\sin t)(\cos t)\)

Step 1: Expand the expression using algebra

We start by expanding \((\sin t-\cos t)^2\) using the distributive property (also known as the FOIL method): \((\sin t-\cos t)^2 = (\sin t)^2 - 2(\sin t)(\cos t) + (\cos t)^2\)

Step 2: Use the Pythagorean identity to simplify the expression

We know that \(\sin^2 t + \cos^2 t = 1\) from the Pythagorean identity. Therefore, we can rewrite the expanded form of the expression as follows: \((\sin t)^2 - 2(\sin t)(\cos t) + (\cos t)^2 = (\sin^2 t + \cos^2 t) - 2(\sin t)(\cos t)\) Now, we substitute the Pythagorean identity, which states \(\sin^2 t + \cos^2 t = 1\): \(1 - 2(\sin t)(\cos t)\)

Step 3: Write the final simplified expression

The simplified expression for \((\sin t-\cos t)^2\) is: \(1 - 2(\sin t)(\cos t)\)

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is crucial in trigonometry and can greatly simplify many problems. This identity expresses a fundamental relationship between the sine and cosine functions of the same angle. The formula is as simple as it is powerful: \(\sin^2 t + \cos^2 t = 1\). This equation stems from the Pythagorean theorem related to the sides of a right triangle. It can be visualized as describing a right-angled triangle on the unit circle, where the hypotenuse is always 1, the opposite side is \(\sin t\), and the adjacent side is \(\cos t\).

By remembering that the sum of the squares of sine and cosine of the same angle equals one, we can easily rearrange or replace parts of trigonometric expressions, simplifying complex equations into something more manageable. In our exercise, recognizing the Pythagorean identity allowed us to replace \((\sin t)^2 + (\cos t)^2\) with 1, substantially reducing the complexity of the expression.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the involved variables where both sides of the equation are defined. These identities are useful in a variety of contexts: from simplifying algebraic expressions to solving integrals and differential equations in calculus. Some commonly used trigonometric identities include reciprocal identities, like \(\sin t = 1/\csc t\), and angle sum and difference identities, like \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\).

In our specific problem, knowing that \(\sin^2 t + \cos^2 t\) simplifies to 1 offers an immediate shortcut to simplification. It's imperative for students to familiarize themselves with the most common identities, as they are the tools that can simplify seemingly complicated expressions and solve trigonometry problems with elegance and ease. Recognizing when and how to use them is a skill developed through practice.

FOIL Method

The FOIL method stands for First, Outer, Inner, Last and is a technique used to simplify the multiplication of two binomials. It’s a handy mnemonic that reminds students of the order in which pairs of terms are multiplied. Here’s a breakdown of the process:

• First: You multiply the first terms in each parentheses.
• Outer: Then, you multiply the outermost terms in the product.
• Inner: Next, the inner terms are multiplied.
• Last: Finally, the last terms in the parentheses are multiplied together.

In the original exercise example, applying the FOIL method enabled us to expand \((\sin t - \cos t)^2\) into its components. Without this initial expansion step, utilizing the Pythagorean identity would not have been possible. Mastering the FOIL method enables students to tackle polynomial multiplication effectively, setting the stage for further simplification using identities or other algebraic principles.