Problem 38 - Algebra and Trigonometry

Question

\(37-42\) me Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 361^{\circ} $$

Step-by-Step Solution

Answer

The angle is \(1^{\circ}\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides, but differ by a full rotation, which is 360 degrees. This means adding or subtracting any multiple of 360 degrees will result in a coterminal angle.

Step 2: Subtract 360 Degrees

To find an angle that is coterminal with \(361^{\circ}\) and lies between \(0^{\circ}\) and \(360^{\circ}\), we subtract \(360^{\circ}\) from \(361^{\circ}\). \[ 361^{\circ} - 360^{\circ} = 1^{\circ} \]

Step 3: Verify the Angle Range

The result from subtracting \(360^{\circ}\) is \(1^{\circ}\), which falls within the specified range of \(0^{\circ}\) to \(360^{\circ}\). Therefore, \(1^{\circ}\) is a valid coterminal angle.

Key Concepts

Angle Measurement

Angle measurement is an essential concept in mathematics, particularly in trigonometry and geometry. Angles are measured in degrees, a system based on dividing a circle into 360 equal parts. Each part represents one degree. This system is widely used because of its simplicity and historical development in mathematics.

• One full rotation around a point is equal to 360 degrees.
• Angles can be smaller than 360 degrees when they don't complete a full rotation.
• Various disciplines like architecture and engineering require precise angle measurements.

Understanding the measurement of angles in degrees helps solve problems related to geometry and trigonometry, which deal with the measurement and relationships between angles and shapes.

Angle Subtraction

Angle subtraction is a basic operation used to find differences between angles or to calculate coterminal angles. When you have an angle that is more than a full circle (360°), you can subtract 360° to find an equivalent angle that lies within a single revolution.

• Subtracting 360° helps determine a coterminal angle within the desired range.
• For example, with a given angle of 361°, subtracting 360° gives 1°, which is the coterminal angle.
• This method ensures angles are within a typically used range recognizing that angles repeat every 360 degrees in circles.

This approach is crucial when you want to simplify the angle values without changing their direction in space.

Degrees in a Circle

There are 360 degrees in a circle, a fundamental concept in geometry. This structure divides a circle into 360 equal parts, each defining a degree. The reason for choosing 360 as the total number of degrees comes from ancient civilizations, understanding the simplicity of its divisibility.

• A full circle rotation measures 360°.
• This concept is useful when navigating, designing circular spaces, and understanding rotational symmetry.
• If an angle exceeds 360°, subtracting 360 helps find an equivalent angle within one circle.

Understanding the degree system allows us to easily calculate angles and navigate through problems involving rotations and circular motions.

Trigonometry Basics

Trigonometry is the branch of mathematics that studies the relationships between angles and sides of triangles. It is fundamental for understanding phenomena involving rotations and oscillations.

• Basic trigonometric functions include sine, cosine, and tangent.
• Coterminal angles are a crucial part of understanding how these functions work in rotational contexts.
• Understanding trigonometry helps in analyzing waves, architecture, and engineering tasks.

Realizing the connections between different angles, like coterminal angles, by using trigonometry deepens one's understanding of how these mathematical concepts are applicable in real-world situations.