Problem 27 - 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. $$ \cos 2 x=-1 $$

Step-by-Step Solution

Answer

The solutions are \(x = 90° + 180°n\) where \(n\) is any integer.

Step 1: Understand the Trigonometric Equation

We start with the equation \( \cos 2x = -1 \). We need to find out for which angles \(2x\) the cosine function equals \(-1\).

Step 2: Determine Key Angles for Cosine

Recall the properties of the cosine function. The cosine of an angle is \(-1\) at specific points on its unit circle representation. The most common angle where this is true is \(180° + 360°n\), where \(n\) is an integer. Therefore, \(2x = 180° + 360°n\).

Step 3: Solve for \(x\)

To solve for \(x\), we divide the equation by 2. Thus, \(x = \frac{180° + 360°n}{2} = 90° + 180°n\). This means that the solutions for \(x\) are at intervals of \(180°\), starting from \(90°\).

Step 4: General Solution

Combine the results to express the general solution for the equation. The general solution for \(x\) can be written as \(x = 90° + 180°n\), where \(n\) is any integer.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos \), is one of the primary trigonometric functions. It helps us understand the relationship between the angle of a right triangle and the lengths of its sides. Specifically, in a unit circle, the cosine of an angle \( \theta \) gives us the x-coordinate of a point on the circle. In other words, it measures how far left or right we are from the center as we travel around the circle.

The cosine function varies between -1 and 1, peaking at 1 when \( \theta = 0° \), decreasing to -1 at \( \theta = 180° \), and returning to 1 at \( \theta = 360° \). This periodic nature makes it vital for solving trigonometric equations. When solving an equation like \( \cos 2x = -1 \), we're effectively asking, "At what angles does the x-coordinate equal -1 on the unit circle?" This understanding of the cosine function provides the foundation for solving the equation.

Angle Measurement

Angles can be measured in degrees or radians, but for this discussion, we'll focus on degrees. The clockwise or counterclockwise rotation from the initial side to the terminal side forms an angle in the geometric sense.

Just as a circle is divided into 360 degrees, understanding degree measurement is fundamental to handling trigonometric problems. In our trigonometric equation \( \cos 2x = -1 \), degrees help us pinpoint specific positions on the unit circle where the cosine value hits -1.
When solving \( \cos 2x = -1 \), we observe important angles at which cosine equals -1. Here, the angle 180° is critical because it represents a half-circle rotation from the starting position. By recognizing these critical angles, we can calculate solutions effectively. Basically, each key angle gives potential solutions when they satisfy the equation's conditions.

General Solutions

In trigonometry, finding the general solution means identifying all possible values of \( x \) that satisfy the given condition. We're often dealing with periodic functions like cosine, which repeat at regular intervals.

For the equation \( \cos 2x = -1 \), identifying the angle \( 180° \) as satisfying the cosine condition is just the start. Since the cosine function repeats every 360°, we must account for multiples of this interval. This leads to the general solution for \( 2x \) as \( 180° + 360°n \), where \( n \) is any integer.
To isolate \( x \), we divide by 2, resulting in \( x = 90° + 180°n \). This general solution accounts for every possible solution by covering each repetition cycle of the cosine pattern. By using the general solution, we incorporate every instance where the conditions of the equation are met, ensuring no solution is left out. This approach is crucial in thoroughly solving trigonometric equations.