Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation in the interval \([0,2 \pi)\) algebraically. Use the table feature of a graphing utility to check your answers numerically. $$\cos x+1=-\cos x$$

Step-by-Step Solution

Answer

The solutions to the equation \(\cos x+1=-\cos x\) on the interval \([0, 2\pi)\) are \(x= 4\pi / 3\) and \(x= 5\pi / 3\)

Step 1: Rearrange the equation

To start, rearrange the given equation to isolate the cosine terms. Therefore, \(\cos x+\cos x= -1\) results in \(2\cos x = -1\).

Step 2: Solve for \(x\)

Next, isolate \(x\) by applying the inverse cosine function. Thus, \(\cos^{-1}(-1/2) = x\). However, we only obtain one solution with this, so the properties of the cosine function should be recalled. The cosine function returns the same value for angles in the first and fourth quadrants (in the unit circle interpretation), so there are two solutions in each period.

Step 3: Find all solutions on the interval \([0,2 \pi)\)

Beginning from the solutions obtained in Step 2, add multiple of \(2 \pi\) until you reach a value over \(2 \pi\). Doing this, the solutions within the interval \([0, 2\pi)\) are \(\pi + \pi / 3\) and \(2\pi - \pi / 3\), which simplifies to \(4\pi / 3\) and \(5\pi / 3\).

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 centered at the origin of a coordinate plane. This circle helps us understand the geometric interpretation of trigonometric functions. Each point on the unit circle corresponds to an angle and can be related to coordinates \( (\cos \theta, \sin \theta) \).

For cosine, specifically, the x-coordinate of a point on the circle gives us the cosine value for that angle. Thus, if you know an angle, you can easily find its cosine by looking at its corresponding x-coordinate.

Remember, angles can be measured in radians, where one full revolution around the circle is \(2\pi \). The angles \(\theta\) are often plotted starting from the positive x-axis and moving counterclockwise around the circle.

Understanding this allows us to visualize the solutions to trigonometric equations, like \(2\cos x = -1\), and find where these solutions lie on the circle.

Exploring the Cosine Function

The cosine function, denoted as \(\cos x\), is a periodic function that repeats every \(2\pi\) radians. It describes how the x-coordinate of a point on the unit circle changes as the angle increases.

Key properties of the cosine function include:

• Range: [-1, 1]
• Periodicity: It repeats every \(2\pi\).
• Symmetry: It is an even function, meaning \(\cos(-x) = \cos x\).

In this specific problem, we set \(2\cos x = -1\), leading to \(\cos x = -\frac{1}{2}\).

The cosine of \(-\frac{1}{2}\) corresponds to specific angles on the unit circle, specifically \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\), which help to solve the equation in the desired interval.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as \(\cos^{-1} x\), provide a way to find angles when we know a trigonometric value.

The notation \(\cos^{-1}\) (pronounced "arc cosine") refers to the angle whose cosine value is a given number. For our equation, \ \cos^{-1}\left(-\frac{1}{2}\right)\ corresponds to the angles \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\).

The range of \(\cos^{-1} x\) is \([0, \pi]\), but because of the cosine function's symmetry and periodicity, we can find additional solutions within any given interval by applying properties of the cosine function.

When solving an equation such as \(2\cos x = -1\), finding these inverse values is crucial to identifying all solutions within a specified range.