Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \sec ^{2} \alpha-4=0 $$

Step-by-Step Solution

Answer

The solutions are \( \alpha = \frac{\pi}{3}, -\frac{\pi}{3}, \frac{2\pi}{3}, -\frac{2\pi}{3} \) plus any full rotations (\( 2k\pi \)).

Step 1: Understand the Trigonometric Identity

The secant function, \( \sec \alpha \), is the reciprocal of the cosine function \( \cos \alpha \). Therefore, \( \sec^2 \alpha = \frac{1}{\cos^2 \alpha} \). This is an important identity to keep in mind when solving the equation.

Step 2: Rewrite the Equation

Given the equation \( \sec^2 \alpha - 4 = 0 \), start by isolating the secant term: \( \sec^2 \alpha = 4 \).

Step 3: Simplify Using Trigonometric Identity

Since \( \sec^2 \alpha = \frac{1}{\cos^2 \alpha} \), the equation \( \sec^2 \alpha = 4 \) becomes \( \frac{1}{\cos^2 \alpha} = 4 \).

Step 4: Solve for \( \cos^2 \alpha \)

Multiply both sides by \( \cos^2 \alpha \) to get 1 = 4\( \cos^2 \alpha \). Now divide both sides by 4 to find \( \cos^2 \alpha = \frac{1}{4} \).

Step 5: Find \( \cos \alpha \)

Take the square root of both sides, recalling that there are two solutions: \( \cos \alpha = \frac{1}{2} \) and \( \cos \alpha = -\frac{1}{2} \).

Step 6: Determine the Angles

The general solutions for \( \cos \alpha = \frac{1}{2} \) are \( \alpha = \frac{\pi}{3} + 2k\pi \) and \( \alpha = -\frac{\pi}{3} + 2k\pi \). For \( \cos \alpha = -\frac{1}{2} \), the solutions are \( \alpha = \frac{2\pi}{3} + 2k\pi \) and \( \alpha = -\frac{2\pi}{3} + 2k\pi \), where \( k \) is any integer.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \alpha \), is a fundamental trigonometric function. It is defined as the reciprocal of the cosine function, which means \( \sec \alpha = \frac{1}{\cos \alpha} \). This relationship is crucial for solving equations involving trigonometric identities.

• Reciprocal: Since \( \sec \alpha \) is the reciprocal, when \( \cos \alpha \) is zero, \( \sec \alpha \) is undefined.
• Application: Secant is often used in problems where division or reciprocal identities offer a simplification.

Understanding the secant function allows us to transform complex trigonometric expressions into manageable forms, simplifying the solution process for equations like \( \sec^2 \alpha - 4 = 0 \).

Cosine Function

The cosine function, \( \cos \alpha \), is one of the primary trigonometric functions that represent the ratio of the adjacent side to the hypotenuse in a right triangle. This function plays a significant role when secant, its reciprocal, is involved.

• Range: \( \cos \alpha \) can return values between -1 and 1, and by extension, the secant function returns values from 1 to infinity and -1 to negative infinity.
• Graph: The cosine function has a wave-like graph that repeats every \( 2\pi \), indicating its periodic nature.

When solving \( \sec \alpha = \frac{1}{2} \), it implies finding \( \cos \alpha \) as the inverse, which simplifies the equation since it reverts to a basic trigonometric problem.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables. In solving the equation \( \sec^2 \alpha - 4 = 0 \), understanding such identities becomes indispensable.

• Reciprocal Identities: These relate functions like secant to cosine, as seen in \( \sec \alpha = \frac{1}{\cos \alpha} \).
• Pythagorean Identities: An example is \( \sin^2 \alpha + \cos^2 \alpha = 1 \), which is utile in transforming and simplifying problems.

By using the identity \( \sec^2 \alpha = \frac{1}{\cos^2 \alpha} \), we can directly convert an equation involving secant into one involving cosine, aiding significantly in finding solutions.

General Solutions

General solutions for trigonometric equations describe all possible solutions for the given equation. These solutions are expressed in terms of an integer \( k \), representing the periodic nature of trigonometric functions.

• Periodicity: Functions like cosine have a period of \( 2\pi \), meaning they repeat their values every \( 2\pi \) intervals.
• Expression: For example, the general solution for \( \cos \alpha = \frac{1}{2} \) is \( \alpha = \frac{\pi}{3} + 2k\pi \) and \( \alpha = -\frac{\pi}{3} + 2k\pi \).

Finding the general solution allows us to encapsulate all possible angles that satisfy the equation, rather than just those falling within a specific, often limited, range.