Problem 27 - Algebra 2 and Trigonometry

Question

In \(18-27,\) for each given angle, find a coterminal angle with a measure of \(\theta\) such that \(0 \leq \theta < 360\). $$ 980^{\circ} $$

Step-by-Step Solution

Answer

A coterminal angle of 980° between 0° and 360° is 260°.

Step 1: Understand Coterminal Angles

Coterminal angles are angles that have the same initial and terminal sides. To find a coterminal angle, you can add or subtract multiples of 360° (a full rotation) from the given angle until it falls within the desired range, typically 0° to 360° for this exercise.

Step 2: Subtract 360° Until Within Range

Starting with 980°, subtract 360° repeatedly until you get an angle within the range of 0° to 360°. 1. First subtraction: \(980° - 360° = 620°\) 2. Second subtraction: \(620° - 360° = 260°\) After the second subtraction, we have 260°, which is within the required range.

Key Concepts

Angle Measurement

In geometry, understanding angle measurement is crucial because angles describe how two rays diverge from a common point, called the vertex. Angles are measured in degrees (6), which denotes the degree of rotation from one ray to another.
A complete circle is divided into 360 degrees, providing a comprehensive framework for measuring any directional change.
When measuring angles, it is essential to understand that angles can be positive or negative:

• Positive angles are measured counterclockwise from the initial side.
• Negative angles are measured clockwise.

This system allows us to determine angles in various positions and orientations. Accurately measuring and understanding angles is critical for determining coterminal angles. After all, knowing the angle in which a line turns helps align it within a specific interval, typically between 06 and 3606, which is beneficial for many practical applications.

Full Rotation

A full rotation in the context of angles refers to a complete 360-degree turn around a point or about a circle's center. Knowing that 3606 represents a full rotation is foundational in understanding how angles relate to each other.
When you visualize a circle, it is straightforward: start from any point on the circle's circumference and return to that point after tracing the entire circle's edge.
This concept is particularly helpful in calculating coterminal angles because:

• We utilize the fact that adding or subtracting 3606 (or multiples thereof) does not change an angle's ultimate direction.
• Since full rotations do not impact the angle's orientation, they allow the isolation of the minimum positive angle in a specified range.

Understanding and utilizing the notion of a full rotation is vital in many mathematical applications, especially when working with angles beyond typical bounds.

Subtracting Multiples of 360 Degrees

The process of finding coterminal angles often involves subtracting multiples of 360 degrees until the angle falls within a conventional range, such as 06 to 3606. This repetition helps simplify angles while maintaining the same endpoint:

• First, identify the given angle.
• Then, continuously subtract 3606 until the resulting angle is less than 3606 but not negative.

For example, given 9806, we subtract 3606 repeatedly:

• 9806 - 3606 results in 6206.
• 6206 - 3606 results in 2606.

Thus, 2606 is the coterminal angle in the desired range.
Using this method ensures the derived angle maintains the same geometric characteristics as the original while fitting into the standard circular angle measure. This strategy allows calculations to be simpler and makes interpreting and applying the angle line up with real-world rotational scenarios. By systematically subtracting multiples of 3606, you effectively "wrap" or "unspool" an angle to find its most intuitive rotational equivalence.