Problem 46 - Precalculus: Functions and Graphs

Question

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\cos x=-\frac{1}{2}$$

Step-by-Step Solution

Answer

The values of \( x \) are \( \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{8\pi}{3}, \frac{10\pi}{3} \).

Step 1: Understand the Interval and Equation

We need to find the values of \( x \) in the interval \([0, 4\pi]\) that satisfy the equation \( \cos x = -\frac{1}{2} \). This interval spans two full cycles of the cosine function.

Step 2: Recognize the Known Angles

The equation \( \cos x = -\frac{1}{2} \) occurs at specific angles on the unit circle. Recall that these angles are \( x = \frac{2\pi}{3} \) and \( x = \frac{4\pi}{3} \) within one full cycle (0 to 2\(\pi\)).

Step 3: Find Corresponding Angles in the Given Interval

Since the interval extends to \( 4\pi \), consider the second cycle by adding \( 2\pi \) to each known angle from Step 2. Thus, the angles in the second cycle are: \( x = \frac{2\pi}{3} + 2\pi = \frac{8\pi}{3} \) and \( x = \frac{4\pi}{3} + 2\pi = \frac{10\pi}{3} \).

Step 4: List All Solutions

The values of \( x \) that satisfy the equation \( \cos x = -\frac{1}{2} \) within the interval \([0, 4\pi]\) are \( x = \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{8\pi}{3}, \frac{10\pi}{3} \).

Key Concepts

Unit Circle

The Unit Circle is a fundamental concept in trigonometry that helps visualize and understand trigonometric functions like sine and cosine. It is a circle with a radius of 1 centered at the origin of the coordinate plane. Every point on the unit circle correlates to an angle and its corresponding sine and cosine values.

Here’s how it works:

• The x-coordinate of a point on the unit circle is the cosine of the angle.
• The y-coordinate is the sine of the angle.
• Angles on the unit circle are measured in radians, often represented as portions of π, such as π/3 or 2π.

For example, at an angle of 0 radians (or 0 degrees), the point on the unit circle is (1, 0). Therefore, \( \cos(0) = 1 \) and \( \sin(0) = 0 \).

Understanding the unit circle is crucial to solving trigonometric equations like \( \cos x = -\frac{1}{2} \). Knowing key angles where cosine values are recognized makes it easier to determine possible solutions.

Cosine Function

The Cosine Function, denoted as \( \cos x \), represents a wave-like graph that oscillates between -1 and 1. It is one of the primary functions in trigonometry, alongside sine and tangent.

Key characteristics of the cosine function include:

• Periodicity: The cosine function has a period of \( 2\pi \), meaning it completes one full cycle every \( 2\pi \) radians. In the interval \([0, 4\pi]\), two cycles are completed.
• Symmetry: Cosine is an even function, symmetric about the y-axis, so \( \cos(-x) = \cos(x) \).
• Critical Points: The peaks (maximum) are at 1, and the troughs (minimum) are at -1.

To solve \( \cos x = -\frac{1}{2} \), look for angles where the cosine value is -\( \frac{1}{2} \). Refer to the unit circle for angles like \( \frac{2\pi}{3} \) and \( \frac{4\pi}{3} \), then apply the periodicity to find additional solutions within the specified interval.

Interval Notation

Interval Notation is a method of writing subsets of the real number line and is especially useful for denoting domains, ranges, or solution sets of equations. In trigonometry, it helps define the specific range of angles you want to consider.

Here's a brief guide on reading interval notation:

• The "bracket" [ or ] indicates that an endpoint is included in the interval (closed interval).
• The "parenthesis" ( or ) shows that an endpoint is not included (open interval).
• For example, \([0, 4\pi]\) means the interval includes both 0 and \(4\pi\).

This notation is a concise way to express the interval in which you are trying to find solutions to equations like \( \cos x = -\frac{1}{2} \). Understanding that the interval \([0, 4\pi]\) covers two full cycles of the cosine function is crucial in ensuring all possible values of \( x \) are considered.