Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \sec ^{5} \theta=4 \sec \theta $$

Step-by-Step Solution

Answer

The solutions are \( \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \text{and} \ \frac{7\pi}{4} \) in the interval \([0, 2\pi)\).

Step 1: Rewrite the Equation

Start by rewriting the equation \( \sec^5 \theta = 4 \sec \theta \) as \( \sec^5 \theta - 4 \sec \theta = 0 \). This allows us to factor the expression.

Step 2: Factor the Equation

Factor out \( \sec \theta \) from the equation: \( \sec \theta (\sec^4 \theta - 4) = 0 \). This gives two separate equations to solve: \( \sec \theta = 0 \) and \( \sec^4 \theta - 4 = 0 \).

Step 3: Solve \( \sec \theta = 0 \)

The function \( \sec \theta \) is the reciprocal of \( \cos \theta \), so \( \sec \theta = 0 \) implies \( \cos \theta = \frac{1}{0} \), which is undefined. Thus, there are no solutions for \( \sec \theta = 0 \).

Step 4: Solve \( \sec^4 \theta = 4 \)

Taking the fourth root of both sides, \( \sec^4 \theta = 4 \) implies \( \sec \theta = \pm \sqrt[4]{4} = \pm \sqrt{2} \).

Step 5: Convert \( \sec \theta \) to \( \cos \theta \)

Since \( \sec \theta = \pm \sqrt{2} \), and \( \sec \theta = \frac{1}{\cos \theta} \), rewrite as \( \cos \theta = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \).

Step 6: Find \( \theta \) for \( \cos \theta = \frac{\sqrt{2}}{2} \)

The values of \( \theta \) where \( \cos \theta = \frac{\sqrt{2}}{2} \) within \([0, 2\pi)\) are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{7\pi}{4} \).

Step 7: Find \( \theta \) for \( \cos \theta = -\frac{\sqrt{2}}{2} \)

The values of \( \theta \) where \( \cos \theta = -\frac{\sqrt{2}}{2} \) within \([0, 2\pi)\) are \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{5\pi}{4} \).

Step 8: Compile the Solutions

Combine all possible solutions from both \( \cos \theta = \frac{\sqrt{2}}{2} \) and \( \cos \theta = -\frac{\sqrt{2}}{2} \). The solutions in the interval \([0, 2\pi)\) are \( \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \text{and} \ \frac{7\pi}{4} \).

Key Concepts

Understanding the Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \). This means that whenever the cosine function is zero, the secant function becomes undefined because division by zero is not possible. In trigonometry, the secant function is important because it allows us to solve equations that involve angles in right triangles or the unit circle. When working with secant, it often simplifies the process to convert it into terms of cosine, which is more familiar and has well-known properties.

Explaining Interval Notation

Interval notation is a method used in mathematics to describe a set of numbers or range of solutions. For trigonometric equations, we often want to find solutions within a specific interval to limit the possible angles that satisfy the equation.In this exercise, the interval is \([0, 2\pi)\), which specifies all angles from 0 up to, but not including, \(2\pi\). This covers a full rotation on the unit circle, capturing all possible angles in standard position once. By defining solutions within this interval, we ensure that all relevant angles are considered without repetition.

Deriving Angle Solutions

Finding angle solutions in trigonometric equations involves determining which angles satisfy the given equation within a specified interval. For example, once we establish that \( \cos \theta = \pm \frac{\sqrt{2}}{2} \), we need to identify which angles on the unit circle correspond to these cosine values.- For \( \cos \theta = \frac{\sqrt{2}}{2} \), the angles are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{7\pi}{4} \).- For \( \cos \theta = -\frac{\sqrt{2}}{2} \), the angles are \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{5\pi}{4} \).These values fall within the interval \([0, 2\pi)\), covering all appropriate quadrants where the cosine values are positive and negative respectively.

Converting Secant to Cosine Function

Converting secant to the cosine function is a common strategy for solving equations involving \( \sec \theta \). Since \( \sec \theta = \frac{1}{\cos \theta} \), we can easily express secant equations in terms of cosine. This makes solving the equation more intuitive because cosine is a basic trigonometric function with familiar values on the unit circle.In our exercise, replacing \( \sec \theta \) with \( \frac{1}{\cos \theta} \) allowed us to set the equation \( \cos \theta = \pm \frac{1}{\sqrt{2}} \), simplifying the search for solutions to convertible cosine values like \( \pm \frac{\sqrt{2}}{2} \). This conversion is crucial because it taps into the well-defined angles associated with specific cosine values, aiding in efficiently finding angle solutions.