Problem 102 - Algebra and Trigonometry

Question

find two values of \(\theta, 0 \leq \theta<2 \pi,\) that satisfy each equation. $$ \cos \theta=-\frac{1}{2} $$

Step-by-Step Solution

Answer

The two values of \(\theta\) that satisfy the equation \(\cos \theta =-\frac{1}{2}\) within the range \(0 \leq \theta<2 \pi\) are \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\).

Step 1: Identify Quadrants for Negative Cosine Values

In unit circle, cosine function gives the x-coordinate of a point on the unit circle. Cosine is negative in the second and third quadrants. So \(\theta\) must lie in one of these two quadrants.

Step 2: Determine the Reference Angle

Since \(\cos \theta = -\frac{1}{2}\), the reference angle \(\theta'\) could be identified from the unit circle where \(\cos \theta' = \frac{1}{2}\). This happens at angles \(\theta' = \frac{\pi}{3}\) and \(\theta' = \frac{5\pi}{3}\). The positive cosine value occurs in the first and fourth quadrants.

Step 3: Adjust the Reference Angle to the Correct Quadrant

For a negative cosine value, adjust the reference angle to the second and third quadrant. In the second quadrant, \(\theta = \pi - \theta'\) giving \(\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\). In the third quadrant, \(\theta = \pi + \theta'\) giving another solution \(\theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that simplifies the understanding of trigonometric functions. It is a circle with a radius of 1 centered at the origin of the coordinate plane \(x=0, y=0\).
On this circle, each point is of the form \( (\cos \theta, \sin \theta) \), representing the cosine and sine of an angle \( \theta \), respectively. This ensures a direct relationship between trigonometric values and circle geometry.
When evaluating trigonometric equations, using the unit circle helps determine both the magnitude and the sign of the function values, making it much easier to identify solutions.
The unit circle is divided into four parts, known as quadrants, which helps in specifying the range of angles where sine and cosine values may be positive or negative.

Quadrants

The plane of the unit circle is divided into four quadrants, which provide a framework for understanding the positive and negative signs of trigonometric functions. These quadrants are:

• First Quadrant (0 to \( \frac{\pi}{2} \)): Both sine and cosine values are positive.
• Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \)): Sine is positive, cosine is negative.
• Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \)): Both sine and cosine values are negative.
• Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \)): Cosine is positive, sine is negative.

In solving trigonometric equations, recognizing which quadrant the angle \( \theta \) resides in will help determine the correct sign of the function values, such as the cosine. Since cosine represents the x-coordinate, it is negative in the second and third quadrants. Understanding this is especially useful for problems where you need to find angles that produce a negative cosine value.

Reference Angle

A reference angle is the acute angle that a terminal side of an angle makes with the x-axis. Reference angles are always between 0 and \( \frac{\pi}{2} \) radians.
They are useful for calculating the exact sine or cosine values, since the reference angle shares the same sine and cosine absolute values as the associated angle, just differing in sign depending on the quadrant.
To find a reference angle, you look for the smallest angle made from the terminal side of \( \theta \) to the x-axis. For the equation \( \cos \theta = -\frac{1}{2} \), the reference angle \( \theta' \) can be related to a positive cosine value \( \frac{1}{2} \). Thus, \( \theta' = \frac{\pi}{3} \).
The real angle \( \theta \) will then adjust based on the negative cosine value and the correct quadrants, which in this case are the second and third quadrants.

Cosine Function

The cosine function is a fundamental trigonometric function represented by \( \cos \theta \) where \( \theta \) is the angle in radians. On the unit circle, the cosine of an angle \( \theta \) represents the x-coordinate of the point on the circumference.
The cosine function is periodic, meaning it repeats its values in a regular cycle. For the unit circle, this period is \( 2\pi \) radians. This periodicity helps in solving equations because once one solution is found within a cycle, others can be determined by adding \( 2\pi \) multiples.
In the problem \( \cos \theta = -\frac{1}{2} \), it is important to recognize that the cosine value is negative in the second and third quadrants, corresponding to the solutions \( \theta = \frac{2\pi}{3} \) and \( \theta = \frac{4\pi}{3} \). Knowing the cosine's properties gives insight into how angles relate to each other on the unit circle.