Problem 5 - Contemporary Precalculus

Question

Factor the given expression. $$\cos ^{2} t-4$$

Step-by-Step Solution

Answer

Answer: The expression $$\cos^2 t - 4$$ can be factored into $$(\cos t - 2)(\cos t + 2)$$.

Step 1: Identify a and b in the given expression

In the given expression $$\cos^2 t - 4$$, we can see that $$a = \cos t$$ and $$b = 2$$, as $$\cos^2 t$$ is the square of $$\cos t$$, and $$4$$ is the square of $$2$$.

Step 2: Apply the difference of squares formula

Using the values we found for a and b, we can now apply the difference of squares formula to factor the expression: $$\cos^2 t - 4 = (\cos t - 2)(\cos t + 2)$$. So, the expression $$\cos^2 t - 4$$ can be factored into $$(\cos t - 2)(\cos t + 2)$$.

Key Concepts

Difference of Squares

The concept of the *difference of squares* is a fundamental idea in algebra that allows us to factor expressions that consist of one perfect square minus another perfect square. It is represented by the formula:

• \(a^2 - b^2 = (a - b)(a + b)\)

The key to using this formula successfully is identifying the terms \(a\) and \(b\) such that each term is a perfect square.
For instance, in the expression \( \cos^2 t - 4 \), we recognize:

• \(a = \cos t\)
• \(b = 2\)

Here's how it works: When you subtract a square, like \(4 = 2^2\), from another square, such as \( \cos^2 t \), you're left with a setup that can be easily factored using the *difference of squares* formula. This handy tool simplifies complex equations and is a stepping stone for solving more advanced problems.

Factoring Expressions

Factoring is the process of breaking down an expression into a product of simpler expressions. This technique is crucial in both simplifying expressions and solving equations.
For the expression \( \cos^2 t - 4 \), the process of factoring involves:

• Recognizing the pattern of a difference of squares.
• Rewriting the expression using the formula \((a - b)(a + b)\).

Once you identify that \( \cos^2 t - 4 \) represents \(a^2 - b^2\), you can express it as \((\cos t - 2)(\cos t + 2)\).
Factoring not only helps in simplifying expressions but also aids in solving equations by setting the factors equal to zero. By understanding this process, you can tackle a wide range of mathematical problems more efficiently.

Cosine Function

The *cosine function* is one of the fundamental trigonometric functions, denoted as \(\cos\). It relates the angle in a right triangle to the ratio of the length of the adjacent side over the hypotenuse.
In the context of functions and algebra, \( \cos^2 t \) denotes the square of the cosine function. This is simply the product of \( \cos t \) with itself.
The cosine function is periodic, with a period of \(2\pi\), meaning that it repeats its values every \(2\pi\) radians. In the context of our example \( \cos^2 t - 4 \), understanding the behavior and properties of the cosine function is pivotal.
Knowing that trigonometric identities often involve such functions enhances your ability to manipulate and simplify expressions. Mastery of the cosine function and its properties is essential for higher mathematics, as it's frequently used in calculus, physics, and engineering.