Problem 34 - Algebra and Trigonometry

Question

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ -30^{\circ}, \quad 330^{\circ} $$

Step-by-Step Solution

Answer

The angles -30° and 330° are coterminal because their difference is 360°.

Step 1: Understand Coterminal Angles

Two angles are coterminal if they share the same terminal side when drawn in standard position. This means they differ by a full rotation, which is equivalent to an angle of 360 degrees.

Step 2: Calculate the Difference

To check if two angles are coterminal, subtract one angle from the other. In this problem, calculate the difference between 330° and -30°: \[ 330^\circ - (-30^\circ) = 330^\circ + 30^\circ = 360^\circ \]

Step 3: Check Coterminal Criteria

Since the difference between the angles is 360°, which equals a full rotation, the angles -30° and 330° are coterminal.

Key Concepts

Angle Measurement

When we talk about angles, we're usually thinking about how we measure the space between two lines that meet at a point. Imagine the hands of a clock. The angle is the space between the minute and hour hand. Angle measurement is important in geometry, trigonometry, and many practical applications.

• Degrees (\(^\circ\)) are the most common units. It's what we use to measure everyday angles, from protractors to circles.
• Radians are another unit used primarily in higher-level mathematics—one complete circle equals \(2\pi\) radians.
• The complete circle in degrees is \(360^\circ\).

Understanding angle measurement helps us better grasp coterminal angles. For instance, if an angle is \(60^\circ\), and another is \(420^\circ\), the second can be thought of as just "going around" further by one full circle \((360^\circ)\). Both end at the same spot, which makes them coterminal.

Full Rotation

A full rotation is a concept that helps us understand how angles like to "go around in circles." When we say a full rotation, we mean turning around entirely to face the same direction again.

• \(360^\circ\) is one full rotation in degrees.
• In radians, this is \(2\pi\), a term used a lot in calculus and trigonometry.

In the context of coterminal angles, this concept is fundamentally about how much "extra" angle you'll need for two angles to align perfectly again.
For example, if you start at \(0^\circ\) and move to \(360^\circ\), you've made a full rotation and are back at the beginning. That's what happens with angles \(-30^\circ\) and \(330^\circ\)—they have their terminal sides overlapping because they differ by exactly one full rotation (also \(360^\circ\)).
So, understanding full rotations is crucial for things like finding coterminal angles, as it forms the basic cycle of how angles "reset."

Standard Position

The term "standard position" is used a lot when talking about angles. It's a way to draw and understand angles in a consistent and clear manner.
Here's what Standard Position means:

• The angle's vertex is always at the origin of an x-y coordinate system.
• The initial side of the angle lies along the positive x-axis.
• From this starting line, the angle opens up, either clockwise or counterclockwise.

For example, if an angle is measured counterclockwise from the positive x-axis, it's considered positive. If it's measured clockwise, it's negative. So, in our problem, \(-30^\circ\) means the angle moved 30 degrees in the clockwise direction from the x-axis, and \(330^\circ\) is 330 degrees counterclockwise. But amazingly, both end up pointing in the same direction in a standard position, showcasing their coterminal nature.
Having this standard starting point makes it easy to compare angles, calculate coterminals, and apply the same rules across different concepts in math.