Problem 83 - Precalculus with Limits

Question

In Exercises 81 - 84, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer \( \dfrac{3}{\sec x - \tan x} \)

Step-by-Step Solution

Answer

\( \dfrac{3\cos x}{1-\sin x} \)

Step 1: Use Trigonometric Identities

We know that \( \sec x \) can be written as \( \dfrac{1}{\cos x} \) and \( \tan x \) can be written as \( \dfrac{\sin x}{\cos x} \)

Step 2: Substitute Trigonometric Identities into the Expression

We will substitute \( \sec x \) and \( \tan x \) in our original expression. So, the expression \( \dfrac{3}{\sec x - \tan x} \) now become \( \dfrac{3}{\dfrac{1}{\cos x} - \dfrac{\sin x}{\cos x}} \). This can be simplified to \( \dfrac{3}{\dfrac{1-\sin x }{\cos x}} \) by combining the fractions in the denominator.

Step 3: Simplify further

Now, to remove the fraction from our expression, we can multiply both the numerator and the denominator by \( \cos x \). That gives us \( \dfrac{3\cos x}{1-\sin x} \).

Key Concepts

Secant Function

The secant function, denoted as \( \sec x \), is one of the basic trigonometric functions. It is essentially the reciprocal of the cosine function. This means that \( \sec x \) is defined as \( \dfrac{1}{\cos x} \). The secant function is particularly useful in various mathematical contexts because it helps transform expressions involving division by cosine into simpler multiplicative forms.
Understanding \( \sec x \) is crucial for simplifying trigonometric expressions, particularly when converting fractional expressions into non-fractional ones. During simplification, recognizing that \( \sec x \) is \( \dfrac{1}{\cos x} \) allows us to work with a single common denominator when combining or rearranging terms.

Tangent Function

The tangent function, represented as \( \tan x \), is another primary trigonometric function. It can be expressed as \( \tan x = \dfrac{\sin x}{\cos x} \). This means it is the ratio of the sine function to the cosine function.
Understanding the tangent function is useful when working with expressions that need converting or simplifying. Many trigonometric identities involve \( \tan x \), making it essential to recognize its relationship with sine and cosine.

• This function is often used to write expressions purely in terms of sine and cosine, helping in simplification
• It plays a critical role in solving various trigonometric equations by using its identity form

Instead of directly manipulating \( \tan x \), expressing it using sine and cosine can provide a clearer path to simplify the expression.

Expression Simplification

Expression simplification is a fundamental process in mathematics that involves rewriting an expression in a simpler or more compact form without changing its value. Simplification is crucial to solving equations efficiently and making calculations more manageable.
In the context of trigonometric expressions, the process often involves:

• Using identities: Recognizing trigonometric identities helps rewrite expressions in a simplified form
• Finding common denominators to combine terms: As seen in the exercise, identifying common terms in the denominator (like \( \cos x \)) helps in merging fractional components
• Removing fractions: By multiplying both numerator and denominator by a common factor, you can eliminate fractions

Simplification often leads to functions that are easier to integrate, differentiate, or further manipulate in equations.

Algebraic Manipulation

Algebraic manipulation refers to the process of using algebraic methods to rewrite expressions or equations. It involves operations such as addition, subtraction, multiplication, division, and substitution to achieve a desired form of an expression.
In trigonometric contexts, algebraic manipulation often includes:

• Substituting known identities, like transforming \( \sec x \) and \( \tan x \) into expressions involving \( \sin \) and \( \cos \)
• Performing operations to simplify expressions or equations, such as factoring, expanding, or canceling terms
• Rewriting expressions to remove fractions, seen by multiplying through by a common factor

Mastery of algebraic manipulation is essential for solving complex trigonometric problems and simplifying expressions efficiently. It provides a toolkit for tackling challenges across various mathematical domains.