Problem 17 - Algebra and Trigonometry

Question

\(17-24\) n Solve the given equation, and list six specific solutions. $$ \cos \theta=-\frac{\sqrt{3}}{2} $$

Step-by-Step Solution

Answer

Specific solutions: \( \frac{5\pi}{6}, \frac{7\pi}{6}, -\frac{7\pi}{6}, -\frac{5\pi}{6}, \frac{17\pi}{6}, \frac{19\pi}{6} \).

Step 1: Recall Basic Trigonometric Values

Recall that the cosine of an angle in a unit circle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle. \( \cos \theta = -\frac{\sqrt{3}}{2} \) means the angle is located where the adjacent side equals \( -\frac{\sqrt{3}}{2} \), a known trigonometric value.

Step 2: Identify Reference Angles

Reference angles where \( \cos \theta = \frac{\sqrt{3}}{2} \) are \( 30^{\circ} \) and \( 150^{\circ} \) (in radians, \( \frac{\pi}{6} \) and \( \frac{5\pi}{6} \)), where the value of cosine should be negative in the second and third quadrants.

Step 3: Determine angles in relevant quadrants

Given \( \cos \theta = -\frac{\sqrt{3}}{2} \), the angles in the second quadrant are \( 180^{\circ} - 30^{\circ} = 150^{\circ} \) and in the third quadrant \( 180^{\circ} + 30^{\circ} = 210^{\circ} \). Thus, \( \theta = \frac{5\pi}{6} \) and \( \theta = \frac{7\pi}{6} \).

Step 4: Express the General Solution

The solutions to \( \cos \theta = -\frac{\sqrt{3}}{2} \) are periodic, with period \( 2\pi \). The general solutions are \( \theta = \frac{5\pi}{6} + 2k\pi \) and \( \theta = \frac{7\pi}{6} + 2k\pi \) for \( k \in \mathbb{Z} \).

Step 5: List Specific Solutions

Substitute \( k = 0, 1, \) and \( -1 \) into the general solutions to list specific solutions. For \( \theta = \frac{5\pi}{6} + 2k\pi \): when \( k = 0 \), \( \theta = \frac{5\pi}{6} \); \( k = 1 \), \( \theta = \frac{17\pi}{6} \); \( k = -1 \), \( \theta = -\frac{7\pi}{6} \). Similarly for \( \theta = \frac{7\pi}{6} + 2k\pi \): when \( k = 0 \), \( \theta = \frac{7\pi}{6} \); \( k = 1 \), \( \theta = \frac{19\pi}{6} \); \( k = -1 \), \( \theta = -\frac{5\pi}{6} \).

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos \theta \), is fundamental in trigonometry. It relates the angle \( \theta \) to the x-coordinate of a point on the unit circle. Simply put, given an angle \( \theta \), the cosine function returns the horizontal distance from the origin to the point on the unit circle. This function is crucial in many fields, including physics and engineering, because it helps describe periodic phenomena like waves.
Key characteristics of the cosine function include:

• The range is between -1 and 1.
• It is an even function, meaning \( \cos(-\theta) = \cos \theta \).
• It has a period of \( 2\pi \), which means after every \( 2\pi \) interval, the function repeats itself.

Understanding cosine is essential for solving trigonometric equations like \( \cos \theta = -\frac{\sqrt{3}}{2} \). This teaches us how to find values of \( \theta \) where the cosine has specific results.

Unit Circle

The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. It is a powerful visual tool for understanding trigonometric functions. In the context of the cosine function, it illustrates how the angle \( \theta \) corresponds to a point on this circle.
When determining the cosine of an angle, you simply look at the x-coordinate of the point on the circumference where the terminal side of the angle intersects.
A few things to remember about the unit circle:

• It simplifies the process of finding trigonometric values for standard angles.
• The x-coordinate gives the cosine value, while the y-coordinate gives the sine value.
• It helps visualize the symmetry of trigonometric functions, such as \( \cos(\theta) \) being the same for angles \( \theta \) and \(-\theta \).

Using the unit circle, you can easily see why \( \cos 30^{\circ} \) and \( \cos 150^{\circ} \) are related but appear in different quadrants when solving equations.

Reference Angles

Reference angles play a pivotal role in trigonometry, especially when dealing with cosine and sine functions. The reference angle is the acute angle formed by the terminal side of the given angle \( \theta \) and the horizontal axis. It essentially translates an angle in any quadrant into a familiar context of the first quadrant.
Here are some critical points about reference angles:

• They are always positive and less than \( 90^{\circ} \).
• They help locate angles in different quadrants by using known trigonometric values of angles like \( 30^{\circ}, 45^{\circ}, \) and \( 60^{\circ} \).
• For \( \cos \theta = -\frac{\sqrt{3}}{2} \), the reference angle \( 30^{\circ} \) facilitates finding solutions by locating angles in appropriate quadrants where cosine is negative.

By understanding reference angles, solving trigonometric equations becomes more manageable as they bridge the gap between known values and unknown angles.

Quadrants

The coordinate plane is divided into four quadrants, which dictate the signs of trigonometric functions like cosine and sine. Understanding quadrants is crucial to determining where the angle \( \theta \) must lie for certain trigonometric values.
The quadrants are numbered counterclockwise:

• First Quadrant: All trig values are positive.
• Second Quadrant: Sine is positive, cosine is negative.
• Third Quadrant: Tangent is positive, but both sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, sine is negative.

In the case of \( \cos \theta = -\frac{\sqrt{3}}{2} \), the angle \( \theta \) lies in the second or third quadrant where cosine values are negative. Knowing in which quadrant a cosine function result occurs is vital for identifying the proper angles.

Periodicity of Trigonometric Functions

Periodicity is a defining feature of trigonometric functions, including cosine. It means that their values repeat at regular intervals called periods. For the cosine function, this period is \( 2\pi \). After this interval, all subsequent values repeat.
Understanding periodicity is helpful when solving trigonometric equations, particularly for finding multiple solutions:

• It allows the expression of solutions as \( \theta = \text{specific angle} + 2k\pi \), where \( k \) is an integer.
• Recognizing the repeat pattern simplifies finding additional solutions by shifting \( \theta \) by full periods.
• When \( \cos \theta = -\frac{\sqrt{3}}{2} \), its periodic nature implies that adding or subtracting \( 2\pi \) results in valid, repeating solutions.

The periodic nature of cosine ensures that once you find a solution, you can generate infinitely many, spaced consistently apart.