Problem 20 - Precalculus Mathematics for Calculus

Question

Solve the given equation, and list six specific solutions. $$\sin \theta=-\frac{\sqrt{3}}{2}$$

Step-by-Step Solution

Answer

Specific solutions are \( \frac{4\pi}{3}, \frac{10\pi}{3}, \frac{16\pi}{3}, \frac{5\pi}{3}, \frac{11\pi}{3}, \frac{17\pi}{3} \).

Step 1: Understand the Range of Sine Function

The sine function, \( \sin \theta \), has a range between -1 and 1. Given \( \sin \theta = -\frac{\sqrt{3}}{2} \), we need to identify angles where the sine function takes this value.

Step 2: Identify General Angle Solutions

The general solution for \( \sin \theta = -\frac{\sqrt{3}}{2} \) can be derived from the unit circle. This value of sine corresponds to angles in the third and fourth quadrants. Specifically, \( \theta = \frac{4\pi}{3} + 2\pi n \) and \( \theta = \frac{5\pi}{3} + 2\pi n \), where \( n \) is an integer.

Step 3: Compute Six Specific Solutions

Using the general solutions:1. For \( \theta = \frac{4\pi}{3} + 2\pi n \): - \( n = 0 \Rightarrow \theta = \frac{4\pi}{3} \) - \( n = 1 \Rightarrow \theta = \frac{10\pi}{3} \) - \( n = 2 \Rightarrow \theta = \frac{16\pi}{3} \)2. For \( \theta = \frac{5\pi}{3} + 2\pi n \): - \( n = 0 \Rightarrow \theta = \frac{5\pi}{3} \) - \( n = 1 \Rightarrow \theta = \frac{11\pi}{3} \) - \( n = 2 \Rightarrow \theta = \frac{17\pi}{3} \)

Step 4: Verify Solutions in the Interval

To verify the solutions, calculate \( \sin(\theta) \) for each angle to ensure \( \sin \theta = -\frac{\sqrt{3}}{2} \) holds true. Each angle corresponds to the sine value in the third or fourth quadrant for \( \theta \).

Key Concepts

Sine Function

The sine function, represented as \( \sin \theta \), is a fundamental concept in trigonometry. It describes the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. This function is periodic, meaning it repeats its values in regular intervals. Specifically, the sine function has a period of \( 2\pi \).

• The range of the sine function is between -1 and 1, which means it cannot produce values outside this interval.
• When dealing with trigonometric equations, we often need to find angles that satisfy \(\sin \theta = c\), where \( c \) is within the range [-1, 1].

Understanding these basics is crucial for solving equations involving the sine function, especially when finding specific angles that achieve such values.

Unit Circle

The unit circle is a key tool in understanding how trigonometric functions behave. It's a circle with a radius of one, centered at the origin of a coordinate plane. Angles on the unit circle correspond to points along the circle where trigonometric values like sine and cosine apply.

• The \( x \)-coordinate of a point on the unit circle is equal to the cosine of the angle, and the \( y \)-coordinate is equal to the sine of the angle.
• For example, if \( \sin \theta = -\frac{\sqrt{3}}{2} \), we look for points on the circle where the \( y \)-coordinate equals \(-\frac{\sqrt{3}}{2}\). This typically occurs in the third and fourth quadrants.

The unit circle helps visualize these trigonometric relationships, making it easier to determine which angles correspond to given sine values.

General Solutions

When solving trigonometric equations, deriving general solutions is a useful approach. General solutions provide a formula that can find multiple angles satisfying an equation.

• For example, \( \sin \theta = -\frac{\sqrt{3}}{2} \) has general solutions: \( \theta = \frac{4\pi}{3} + 2\pi n \) and \( \theta = \frac{5\pi}{3} + 2\pi n \), where \( n \) is an integer.
• These general solutions express an infinite number of angles, as they represent all possible rotations by full circles (\( 2\pi \) radians) around the unit circle in both positive and negative directions.

By understanding general solutions, students can calculate multiple specific angles that all satisfy the original trigonometric equation.

Specific Angles

Finding specific angles from a general solution is often required for practical applications. Specific solutions can be drawn from the general formula by assigning values to the integer \( n \).

• For the equation \( \sin \theta = -\frac{\sqrt{3}}{2} \), you can plug in values like \( n = 0, 1, 2, \ldots \) to find specific angles such as \( \theta = \frac{4\pi}{3}, \frac{10\pi}{3}, \frac{16\pi}{3} \) and so on.
• Similarly, starting with \( \theta = \frac{5\pi}{3} + 2\pi n \), you can find angles like \( \frac{5\pi}{3}, \frac{11\pi}{3}, \frac{17\pi}{3} \), etc.

These calculations help to list specific solutions that not only satisfy the trigonometric equation but are also practical for understanding rhythmic or periodic events represented in various applications.