Problem 38 - Precalculus Mathematics for Calculus

Question

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$50^{\circ}, \quad 340^{\circ}$$

Step-by-Step Solution

Answer

No, the angles are not coterminal. They differ by 290°, not a multiple of 360°.

Step 1: Understand Coterminal Angles

Coterminal angles share the same initial and terminal sides. In other words, they point in the same direction on the unit circle. To determine if two angles are coterminal, they must differ by a multiple of 360 degrees (a full circle).

Step 2: Calculate the Difference

Find the difference between the two angles: \(340^{\circ} - 50^{\circ} = 290^{\circ}\).

Step 3: Check for Multiples of 360 Degrees

In this case, the difference between the angles is 290°. To be coterminal, this difference should be a multiple of 360°. Since 290° is not equal to or a multiple of 360°, these two angles are not coterminal.

Key Concepts

Angle Measurement

An angle is a measure of rotation between two rays sharing a common endpoint, known as the vertex. It's important to understand that there are various units for measuring angles, including degrees and radians. Most introductory problems, such as the one here, use degrees.

Imagine a circle divided into 360 equal parts; each part represents one degree. So, a full rotation around a circle amounts to 360 degrees. When working with angles, especially coterminal angles, we concentrate on how this rotation can overlap or end at the same point despite starting differently.

To determine such overlaps, it’s helpful to look at the difference between two angles. If this difference is a multiple of 360 degrees, the angles are said to be coterminal, as they have completed whole cycles and ended up at the same place on the circle.

Standard Position

Angles in standard position help us systematically analyze their behavior and relationships on a coordinate plane. An angle is in standard position when its vertex is at the origin of a coordinate system, and its initial side is along the positive x-axis.

In this way, we can comprehensively describe an angle's position and direction, making it easier to determine relationships like coterminality. For instance, both a 50-degree and a 340-degree angle may start from the same initial side, but unless they point directly in the same direction after rotation, they're not coterminal.

The concept of standard position allows us to precisely calculate and predict the behavior of angles by providing a fixed starting point, simplifying our geometric and trigonometric calculations.

Unit Circle

The unit circle is a fundamental concept in trigonometry, serving as a powerful tool for understanding angle measures and relationships. It is a circle with a radius of one, centered at the origin on a coordinate plane.

Understanding the unit circle helps to visualize coterminal angles. Since the circle is divided into 360 degrees, angles pointing to the same position on the circle show us coterminal angles. This is because they will have the same terminal side even after multiple rotations.

• The unit circle simplifies calculations involving radians and degrees.
• It aids in visualizing sine, cosine, and other trigonometric functions effectively.
• By using the unit circle, identifying coterminal angles becomes an exercise in visualizing or computing multiples of 360 degrees.

Keeping the unit circle in mind ensures we have a clear picture of how angles relate to one another, especially when determining coterminal angles.