Problem 40 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \frac{1+\cos \theta}{\cos \theta}=2 $$

Step-by-Step Solution

Answer

The solutions are \( \theta = 360k \) for any integer \( k \).

Step 1: Identify the Equation

The given trigonometric equation is \( \frac{1+\cos \theta}{\cos \theta} = 2 \). The goal is to find all \( \theta \) that satisfy this equation where \( \theta \) is measured in degrees.

Step 2: Multiply Both Sides by \( \cos \theta \)

To eliminate the fraction, multiply both sides of the equation by \( \cos \theta \), resulting in \( 1 + \cos \theta = 2 \cdot \cos \theta \).

Step 3: Isolate \( \cos \theta \)

Rearrange the equation to get all terms involving \( \cos \theta \) on one side. Subtract \( \cos \theta \) from both sides to obtain \( 1 = \cos \theta \).

Step 4: Solve for \( \theta \)

Solve \( \cos \theta = 1 \). \( \cos \theta = 1 \) is true when \( \theta = 0 + 360k \), where \( k \) is an integer. Therefore, the general solution in degrees is \( \theta = 360k \).

Step 5: Determine All Solutions

Since \( k \) can be any integer (positive, negative, or zero), the solutions are \( \theta = 0, 360, 720, -360, -720, \ldots \). These values are in increments of 360 degrees.

Key Concepts

Cosine Function

The cosine function, represented as \( \cos \theta \), is one of the primary trigonometric functions. It plays a vital role in understanding angles and triangles, especially in the unit circle representation of trigonometry.

• Definition: It measures the adjacent side to the hypotenuse in a right-angled triangle.
• Unit Circle: In a unit circle, the cosine function corresponds to the x-coordinate of a point on the circle.

For angles measured in degrees, \( \theta \), the cosine value ranges between -1 and 1. When you encounter equations like \( \frac{1+\cos \theta}{\cos \theta} = 2 \), you're analyzing how \( \cos \theta \) behaves in mathematical expressions.
The function is periodic, with a period of 360 degrees, repeating its values every 360 degrees. This characteristic is crucial for solving equations by understanding that multiple angles can yield the same cosine value.

General Solutions

In trigonometry, finding the "general solutions" of an equation means identifying all possible angles that satisfy the equation.

• Form: Trigonometric equations often produce solutions that include a term like \( \theta = a + 360k \), where \( a \) is a reference angle, and \( k \) is any integer.
• Periodic Nature: This arises from the periodic nature of trigonometric functions, such as sine and cosine, which repeat their values at regular intervals.

For the equation \( \frac{1+\cos \theta}{\cos \theta} = 2 \), solving it involves working out \( 1 = \cos \theta \). The angle \( \theta = 0 \) satisfies this condition. The cosine function achieves a value of 1 at such points, meaning solutions are expressed as \( \theta = 0 + 360k \) in degrees.
"General solutions" encompass all such results, accounting for the infinite repetition of the cosine function's periodic behavior.

Degrees Measurement

Degrees measurement is a system for measuring angles, where a full turn around a circle is 360 degrees. It's one of the most common units for angle measurement, especially in trigonometry.

• Understanding: Each degree represents \( \frac{1}{360} \) of a complete circle or rotation.
• Relevance: Used extensively for expressing angles in geometry and trigonometry.

When trigonometric problems are presented in degrees, understanding the transition between degrees and other units like radians is crucial, though most beginner problems stick with degrees.
In our trigonometric equation, finding solutions involves determining the possible angles \( \theta \) that satisfy \( \cos \theta = 1 \) in degrees. The solution, such as \( \theta = 0, 360, 720, \ldots \), showcases degrees measurement's role in representing angles as part of solving the equation.