Problem 41 - Algebra 2

Question

Find a positive and a negative coterminal angle for the given angle. $$ -85^{\circ} $$

Step-by-Step Solution

Answer

The positive coterminal angle is 275 degrees and the negative coterminal angle is -85 degrees.

Step 1: Finding the Positive Coterminal Angle

To find a positive coterminal angle, we need to add 360 degrees to the given angle. Therefore, using the given angle of -85 degrees, the positive coterminal angle will be -85 degrees + 360 degrees = 275 degrees

Step 2: Finding the Negative Coterminal Angle

To find a negative coterminal angle, we need to subtract 360 degrees from the given angle. However, since our given angle is already negative, subtracting 360 degrees from -85 degrees would result in a larger negative angle. Hence, for a less negative coterminal angle, another 360 degrees should be subtracted from the positive coterminal angle which we just found. The negative coterminal angle is therefore 275 degrees - 360 degrees = -85 degrees

Key Concepts

Positive Angles

Angles in trigonometry are usually measured in degrees or radians on the coordinate plane. A positive angle is one that is measured counterclockwise from the initial side of the angle to the terminal side. This counterclockwise direction is considered positive by convention.

• To find a positive coterminal angle, you can add 360 degrees, as a full rotation around a circle is 360 degrees.
• This operation ensures you reach an angle that looks the same geometrically, but the numerical value will be a positive one.

For instance, if you start with an angle of -85 degrees, to find its positive coterminal counterpart, you add 360 degrees. This results in 275 degrees, a positive angle, yet it shares the same initial and terminal sides as the original angle.

Negative Angles

Negative angles, on the other hand, are measured clockwise from the initial side to the terminal side. This clockwise measurement is what makes an angle negative. Negative angles are just as valid as positive angles, but their representation differs due to the direction of measurement.

• Finding a negative coterminal angle can also involve subtracting multiples of 360 degrees from a positive angle.
• In the exercise, starting with -85 degrees, you can see that further subtracting 360 degrees results in -445 degrees, a larger negative angle.
• But, by instead subtracting from a positive coterminal angle, you achieve another valid negative coterminal angle.

This approach highlights the flexibility in representations of angles within the circle.

Angle Measurement

Angles are a fundamental concept in mathematics, essential for understanding shapes, motion, and many scientific computations. The measurement of angles can be done in degrees or radians.

• A complete circle is 360 degrees, which serves as a full rotation reference point in degree measurements.
• When measuring angles, a positive angle suggests counterclockwise rotation while a negative angle implies clockwise rotation.

Such measurements not only help in geometric representations but also play a pivotal role in trigonometric functions and calculations. Understanding how to switch between positive and negative angles can assist in solving various mathematical problems involving rotational movements.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It is essential for fields like physics, engineering, and computer science. Angles are the nucleus of trigonometric functions such as sine, cosine, and tangent.

• Trigonometric identities often rely on the periodic nature of these functions, where coterminal angles play a significant role.
• Coterminal angles have the same sine, cosine, and tangent values due to their position in a circle, even if their numerical values seem different.

Understanding trigonometry involves being comfortable with both positive and negative angles, as they provide the same trigonometric values, facilitating problem-solving across different contexts.