Problem 38 - Algebra and Trigonometry

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.

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To determine if two angles are coterminal, find the difference between the two angles and check if it's a multiple of 360°.

Step 2: Calculate the Difference Between Angles

Subtract the smaller angle from the larger angle to find the difference. Here, the difference between 340° and 50° is calculated as follows:\[340^{\circ} - 50^{\circ} = 290^{\circ}\]

Step 3: Check for Multiples of 360°

Determine if 290° is a multiple of 360°. Since 360° is a full circle, angles are coterminal if their difference is 0° or any multiple of 360°. 290° is not a multiple of 360°.

Key Concepts

Standard Position Angles

When we talk about standard position angles, we refer to angles drawn on the Cartesian coordinate plane. These angles have their vertex at the origin, which is the point \(0,0\), and one side lying along the positive x-axis.
Standard position makes it easier to compare angles by providing a common point of reference. By beginning from the positive x-axis, we can measure angles counterclockwise for positive angles and clockwise for negative angles.

• The initial side always starts at the positive x-axis.
• The terminal side rotates based on the angle's measure.

Understanding angles in standard position helps simplify the process of determining whether two angles are coterminal.

Angle Measurement

Angle measurement is essential in determining how far one side needs to be rotated to align with another in standard position. Angles can be measured in degrees, where a full circle is 360 degrees.
For most problems, including the original exercise, angles are given in degrees, although sometimes radians are used as well in higher-level mathematics.

• A right angle is 90 degrees.
• A straight angle equates to 180 degrees.
• A full rotation, or circle, is 360 degrees.

Learning the different ways to measure angles helps in identifying coterminal angles, as you'll often need to determine if the difference or sum of two angles is 360 degrees or a multiple thereof.

Multiples of 360 Degrees

Multiples of 360 degrees are key in identifying coterminal angles. Coterminal angles are those that share the same position but differ by full rotations, which are multiples of 360 degrees.
Knowing how to handle multiples of 360 degrees allows you to correctly identify coterminal angles, regardless of their numeric difference.

• If one angle exceeds the other by exactly 360 degrees, they are coterminal.
• The difference can also be negative, meaning reversing direction still results in a multiple of 360 degrees, thus signaling coterminality.

For example, if you begin with an angle measure of 340 degrees and another at 50 degrees, the difference is 290 degrees. In this case, no multiple of 360 can be obtained, confirming that these angles are not coterminal.
Remember, the ability to manipulate and understand multiples of 360 degrees is crucial for identifying coterminal angles effectively.