Problem 13 - Contemporary Precalculus

Question

Factor the given expression. $$\cos ^{4} t+4 \cos ^{2} t-5$$

Step-by-Step Solution

Answer

Question: Factor the given expression: $$(\cos^2{t} + 4\cos^2{t} - 5\cos^2{t})$$ Answer: $$(\cos^2{t} + 5)(-\sin^2{t})$$

Step 1: Substitution

Let's substitute $$x = \cos^2{t}$$. The expression now becomes: $$x^2 + 4x - 5$$

Step 2: Factor the quadratic equation

We want to find two numbers that multiply to -5 and add to 4. These numbers are 5 and -1. Now, we can rewrite the expression as: $$(x + 5)(x - 1)$$

Step 3: Re-substitute the original value for x

Replace $$x$$ with $$\cos^2{t}$$ back into the expression: $$(\cos^2{t} + 5)(\cos^2{t} - 1)$$

Step 4: Use trigonometric identity

Notice that $$(\cos^2{t} - 1)$$ can be simplified using the identity: $$\sin^2{t} + \cos^2{t} = 1$$ Therefore, $$\cos^2{t} - 1 = -\sin^2{t}$$ Now the factored form of the expression is: $$\boxed{(\cos^2{t} + 5)(-\sin^2{t})}$$

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that reveal inherent relationships between trigonometric functions. These identities are true for all values of the variables involved. One of the most fundamental identities is the Pythagorean identity which states that \( \sin^2{t} + \cos^2{t} = 1 \). This identity helps simplify expressions and solve equations that involve trigonometric functions.

To understand how trigonometric identities apply to our problem, consider that factoring the expression \( \cos^4 t + 4 \cos^2 t - 5 \) requires recognizing patterns and using the identities to simplify the terms. Upon re-substituting \( \cos^2{t} \) back into the factored quadratic, we can simplify \( \cos^2{t} - 1 \) further by leveraging the Pythagorean identity, transforming \( \cos^2{t} - 1 \) into \( -\sin^2{t} \) and thus obtaining a fully factored trigonometric expression.

Quadratic Equations

Quadratic equations take the form \( ax^2 + bx + c = 0 \) and are prevalent in various areas of algebra. Factoring is a powerful method for solving these equations, and it involves rewriting the quadratic as a product of two binomial expressions. The goal is to find two numbers that multiply to \( ac \) (the product of the leading coefficient and the constant term) and add up to \( b \) (the coefficient of the linear term).

In our exercise, the quadratic in question was \( x^2 + 4x - 5 \) after applying a substitution for \( \cos^2{t} \). We identified the numbers 5 and -1 as the correct pair to rewrite the quadratic as \( (x + 5)(x - 1) \). Understanding the method of factoring quadratic equations is critical, as it allows us to break down more complex expressions into simpler, more manageable components.

Substitution Method

The substitution method is a mathematical strategy used to simplify expressions or equations by replacing a variable with another expression that is easier to work with. This technique is highly useful when dealing with complex trigonometric expressions that can be transformed into more familiar algebraic forms.

In factoring trigonometric expressions, like \( \cos^4 t + 4 \cos^2 t - 5 \), we employ substitution to replace \( \cos^2{t} \) with a temporary variable, \( x \). This simplifies the expression into a recognizable quadratic equation \( x^2 + 4x - 5 \) which can then be factored using standard algebraic techniques. Once the expression is factored, we reverse the substitution, replacing \( x \) with the original trigonometric function, to obtain the final factored trigonometric expression. Utilizing the substitution method streamlines the process and bridges the gap between trigonometry and algebra.