Problem 29 - Precalculus Mathematics for Calculus

Question

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$\frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

The coterminal angles are \(\frac{11\pi}{4}\), \(\frac{19\pi}{4}\), \(-\frac{5\pi}{4}\), and \(-\frac{13\pi}{4}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but differ in the number of full rotations they have completed. In radians, two angles are coterminal if they differ by an integer multiple of \(2\pi\). Our goal is to find angles that meet this criterion with the given angle \(\frac{3\pi}{4}\).

Step 2: Finding Positive Coterminal Angles

To find positive coterminal angles, add \(2\pi\) to \(\frac{3\pi}{4}\). Repeat this process as needed:1. First positive angle: \(\frac{3\pi}{4} + 2\pi = \frac{3\pi}{4} + \frac{8\pi}{4} = \frac{11\pi}{4}\).2. Second positive angle: \(\frac{3\pi}{4} + 2(2\pi) = \frac{3\pi}{4} + \frac{16\pi}{4} = \frac{19\pi}{4}\).

Step 3: Finding Negative Coterminal Angles

To find negative coterminal angles, subtract \(2\pi\) from \(\frac{3\pi}{4}\). Repeat this process as needed:1. First negative angle: \(\frac{3\pi}{4} - 2\pi = \frac{3\pi}{4} - \frac{8\pi}{4} = -\frac{5\pi}{4}\).2. Second negative angle: \(\frac{3\pi}{4} - 2(2\pi) = \frac{3\pi}{4} - \frac{16\pi}{4} = -\frac{13\pi}{4}\).

Key Concepts

Angle Measurement

When we talk about angle measurement, it's important to understand that angles can be measured in different units like degrees or radians.

• Degrees are a more common unit outside of mathematics. A full circle is 360 degrees.
• Radians are a preferred unit in higher mathematics because they simplify many equations. A full circle is equal to \(2\pi\) radians.

The measurement of an angle dictates how far the rotation is from the initial side to the terminal side. With radians, the circumference of a circle is effectively one unit of measure, making it a more natural unit for calculus and many algebraic calculations. Understanding how to convert between degrees and radians and how to calculate coterminal angles within these units helps to solve various mathematical problems. In this context, knowing the angle measurement in radians is essential, such as given by \(\frac{3\pi}{4}\), to find coterminal angles.

Radian Measure

Radians give us a unique way to measure angles that ties directly to the concept of arc length.

• One radian is the angle created when the radius of a circle is wrapped along the circle's edge, or arc, exactly once.
• This relationship makes calculations like circumference and area of a circle simpler, as the radius is directly compared to the arc.

To work with radians, it's important to understand how they are calculated based on\(\pi\). The value \(\pi\) is crucial here, being approximately 3.14159, representing half of a circle (180 degrees) in radians. Therefore, \(\pi\) radians is equivalent to 180 degrees. This relationship is pivotal in determining coterminal angles with any given angle like \(\frac{3\pi}{4}\). To find angles coterminal to this, we simply add or subtract multiples of \(2\pi\). This process gives the angles like \(\frac{11\pi}{4}\) and \(-\frac{5\pi}{4}\) as seen in the original solution.

Standard Position

The concept of standard position is crucial for understanding trigonometry and geometry. An angle in standard position means:

• The vertex of the angle is located at the origin of the coordinate plane (0,0).
• The initial side of the angle is along the positive x-axis.
• The terminal side is the position of the angle after the rotation from the initial side.

Having this fixed starting point helps in comparing and calculating angles, especially in finding their trigonometric functions. When we work with serangels in standard position, each angle can have multiple coterminal angles. Coterminal angles can reach the terminal side via a different path, including full rotations (either positive or negative). This concept is exploited in the solution by adding or subtracting \(2\pi\) to locate equal angles, placing emphasis on angles like \(\frac{11\pi}{4}\) and \(-\frac{5\pi}{4}\).