Problem 78 - Precalculus

Question

Find all the values of \(\theta, 0 \leq \theta \leq 2 \pi,\) for which the equation \(\cos \theta=\frac{1}{4} \sec \theta\) is true.

Step-by-Step Solution

Answer

\( \theta = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3} \).

Step 1: Rewriting the Equation

We want to find all values of \(\theta\) that satisfy \( \cos \theta = \frac{1}{4} \sec \theta \). Recall that \( \sec \theta \) is the reciprocal of \( \cos \theta \), so \( \sec \theta = \frac{1}{\cos \theta} \). Substitute \( \sec \theta \) in the equation: \( \cos \theta = \frac{1}{4} \cdot \frac{1}{\cos \theta} \).

Step 2: Clearing the Fraction

Multiply both sides of the equation by \( \cos \theta \) to eliminate the fraction: \( \cos^2 \theta = \frac{1}{4} \).

Step 3: Solving for Cosine

Take the square root of both sides of the equation: \( \cos \theta = \pm \frac{1}{2} \).

Step 4: Finding Angles for Positive Cosine

Evaluate \( \theta \) where \( \cos \theta = \frac{1}{2} \). Within the interval \([0, 2\pi]\), these values are \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).

Step 5: Finding Angles for Negative Cosine

Evaluate \( \theta \) where \( \cos \theta = -\frac{1}{2} \). Within the interval \([0, 2\pi]\), these values are \( \theta = \frac{2\pi}{3} \) and \( \theta = \frac{4\pi}{3} \).

Step 6: Compiling Solutions

List all found solutions within the interval \([0, 2\pi]\): \( \theta = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3} \).

Key Concepts

Cosine Function

The cosine function, often abbreviated as \( \cos \), is one of the fundamental functions in trigonometry. It represents the x-coordinate of a point on the unit circle as the corresponding angle \( \theta \) increases. The cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse. This provides a geometric interpretation that is useful in various calculations.

Key properties of the cosine function include:

• The range of the cosine function is from \(-1\) to \(1\).
• The cosine function is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• It has a period of \(2\pi\), meaning \( \cos(\theta + 2\pi) = \cos(\theta) \) for any angle \( \theta \).

Understanding the behavior of \( \cos \theta \) is crucial in solving equations involving trigonometric identities, especially when seeking solutions within specified intervals such as \([0, 2\pi]\).

Secant Function

The secant function, denoted as \( \sec \), is closely related to the cosine function. In fact, it is defined as the reciprocal of the cosine function. Mathematically, this relationship is expressed as \( \sec \theta = \frac{1}{\cos \theta} \). This function provides a way to express the trigonometric equation in terms of \( \cos \) when solving for angles.

Characteristics of the secant function include:

• The secant function does not exist where \( \cos \theta = 0 \) because division by zero is undefined; these points occur at \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.
• The range of \( \sec \theta \) is \(( -\infty, -1] \cup [1, \infty )\).
• Like cosine, the secant function also has a period of \(2\pi\).

Understanding the secant function's properties is essential when simplifying or transforming trigonometric equations like the one given in the exercise.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angles that satisfy a given equation, often within a specific interval. The process usually requires the application of trigonometric identities, such as transforming the equation into a simpler form using known relationships among the functions.

When solving an equation like \( \cos \theta = \frac{1}{4} \sec \theta \), the following steps are often helpful:

• Rewrite the equation: Use trigonometric identities to express all terms with common functions, like rewriting \( \sec \theta \) as \( \frac{1}{\cos \theta} \).
• Clear fractions: Multiply through by common denominators to simplify the equation further.
• Isolate the trigonometric function: Solve for \( \theta \) in terms of a familiar function, such as \( \cos^2 \theta = \frac{1}{4} \).
• Find specific solutions: Determine values for \( \theta \) by using the known angles for which these trigonometric functions give specific values (e.g., \( \cos \theta = \pm \frac{1}{2} \)).
• Consider the interval: Check that all found solutions fall within the desired interval, like \([0, 2\pi]\).

These steps ensure a comprehensive approach to solving trigonometric equations, providing a path to accurately find all solutions within the specified range.