Problem 55 - Contemporary Precalculus

Question

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$\frac{\cos ^{2} t+4 \cos t+4}{\cos t+2}$$

Step-by-Step Solution

Answer

Question: Simplify the given trigonometric expression: $$\frac{\cos ^{2} t+4 \cos t+4}{\cos t+2}$$ Answer: The simplified expression is $$\cos t + 2$$.

Step 1: Identify Factoring Pattern in the Numerator

In the numerator, we have a quadratic expression in terms of cosine: $$\cos^2 t + 4\cos t + 4$$ Observe that this expression follows the pattern $$a^2 + 2ab + b^2$$, which can be factored as $$(a+b)^2$$ We will now factor the numerator.

Step 2: Factor the Numerator

Using the pattern identified in step 1, we will factor the numerator: $$\cos^2 t + 4\cos t + 4 = (\cos t + 2)^2$$ Now, substitute this factorization into the original expression.

Step 3: Simplify the Expression

With the numerator factored, we now have: $$\frac{(\cos t + 2)^2}{\cos t + 2}$$ Since the denominator is not zero, we can simplify this expression by canceling one of the \((\cos t + 2)\) terms, leaving us with: $$\cos t + 2$$ So the simplified expression is: $$\cos t + 2$$

Key Concepts

Factoring Quadratics

Factoring quadratics is a foundational skill in algebra that involves rewriting a quadratic expression as a product of two binomials. A quadratic expression is generally of the form \(ax^2 + bx + c\). To factor it, you often look for a pattern, such as the perfect square trinomial. A perfect square trinomial follows the pattern \(a^2 + 2ab + b^2\), which can be factored into \((a + b)^2\).
For example:

• Given \(x^2 + 6x + 9\), observe that it is a perfect square trinomial. It can be rewritten as \((x + 3)^2\) because \(3^2 = 9\) and \(2 \times 3 \times x = 6x\).
• In the case of the original exercise, \(\cos^2 t + 4\cos t + 4\) matches the pattern, so it can be expressed as \((\cos t + 2)^2\).

This method simplifies complex equations, making them easier to solve. Once you master recognizing patterns, factoring quadratics becomes straightforward.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true for every value of the variables where both sides of the identity are defined. They are useful tools for simplifying expressions and solving trigonometric equations.
Some of the basic trigonometric identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Double Angle Formulas, like \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \)
• Addition and Subtraction Formulas, such as \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

In the exercise, the factorization did not require use of deep trigonometric identities, since it was primarily algebraic. However, understanding these identities can provide a deeper insight into trigonometric simplification, especially when combining different trigonometric functions or dealing with identities that might arise in different problems.

Expression Simplification

Simplifying expressions involves reducing them to their most basic form. This action makes expressions easier to interpret and resolve in algebraic equations. Typical processes include factoring, cancelling like terms, and applying mathematical identities.For instance, when simplifying fractions:

• Identify common factors in the numerator and the denominator.
• Cancel out these common factors.
• Ensure that all steps follow valid algebraic rules.

In the original exercise, after factoring the numerator as \((\cos t + 2)^2\), the expression was simplified by canceling one \((\cos t + 2)\), since it appeared in both the numerator and the denominator. This left us with \(\cos t + 2\), which is the simplest form of the expression.The essence of simplification is to make complex expressions manageable and comprehensible, allowing a clearer path to solution in various mathematical problems.