Problem 48 - Algebra and Trigonometry

Question

Find an angle between 0 and 2p that is coterminal with the given angle. $$ 10 $$

Step-by-Step Solution

Answer

The coterminal angle is approximately 3.72 radians.

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. They are calculated by adding or subtracting full rotations (multiples of \(2\pi\)) from the given angle.

Step 2: Calculate 2π for Reference

Since the goal is to find an angle between 0 and \(2\pi\), we first need to know the approximate value of \(2\pi\). We have \(2\pi \approx 6.28\). This is the interval within which we desire our coterminal angle to fall.

Step 3: Subtract Full Rotations

Starting from the given angle \(10\), subtract \(2\pi\) continuously until the angle falls within [0, \(2\pi\)]. Perform the operation: \(10 - 2\pi = 10 - 6.28 = 3.72\).

Step 4: Verify and Conclude

Check whether the resulting angle \(3.72\) is within the desired range [0, \(2\pi\)]. Since \(3.72\) is greater than zero and less than \(6.28\), our coterminal angle is correctly within the interval.

Key Concepts

Angle Measurement

Understanding angle measurement is crucial in trigonometry. It allows us to describe and analyze various geometric properties. Angles are measured in terms of rotation, and they represent the amount of turn between two rays meeting at a common point, called the vertex. The size of an angle can be measured in different units, such as degrees or radians. Understanding these units is essential to solve problems like finding coterminal angles.

• Degrees: A full rotation around a circle is 360 degrees.
• Radians: It is another widely used unit for measuring angles, which relates directly to the radius of a circle.

Both degrees and radians are used interchangeably in trigonometry.
When working with radians, it's useful to remember that a full circle (360 degrees) equals about 6.28318 radians or precisely, \(2\pi\). This conversion is helpful when finding angles within certain ranges, such as from 0 to \(2\pi\), as required for coterminal angles.

Trigonometry

Trigonometry is the branch of mathematics that studies relationships involving lengths and angles of triangles. It is essential for analyzing periodic phenomena and solving problems involving angles and distances. When dealing with coterminal angles, trigonometry provides the necessary framework and tools to understand and compute angles within desired intervals.
Some crucial concepts in trigonometry include:

• Trigonometric Functions: Functions like sine, cosine, and tangent describe relationships between angle measures and side lengths in right triangles.
• Unit Circle: A circle of radius one, centered at the origin of a coordinate plane, used to define trigonometric functions.
• Coterminal Angles: These are angles that differ by full rotations. For example, \(\theta\) and \(\theta + 2n\pi\) are coterminal, where \(n\) is an integer.

Trigonometry provides the framework for understanding coterminal angles, helping to determine equivalent angles that share the same terminal side.

Radian Measure

Radian measure is a way to quantify angles using the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians are based on the circle's radius. This unit of measurement makes many mathematical formulas simpler and more natural.
To understand radian measure better, consider:

• Definition: One radian is the angle created when the arc length is equal to the radius of the circle.
• Relation to Pi: Since a circle's circumference is \(2\pi\times \text{radius}\), the total radian measure for a full circle is \(2\pi\).
• Conversion: To convert from degrees to radians, multiply by \(\frac{\pi}{180}\). For instance, 180 degrees equals \(\pi\) radians.

Using radians simplifies many trigonometric calculations, especially when finding coterminal angles, as it aligns more directly with the geometry of the circle.