Problem 40 - Algebra 2

Question

Find a positive and a negative coterminal angle for the given angle. $$ 400^{\circ} $$

Step-by-Step Solution

Answer

The positive coterminal angle to \(400^{\circ}\) is \(40^{\circ}\), and the negative coterminal angle to \(400^{\circ}\) is \(-320^{\circ}\).

Step 1: Find a positive coterminal angle

To find a positive coterminal angle for 400 degrees, subtract 360 degrees from 400 degrees. This is due to the fact that adding or subtracting 360 degrees does not change the position of the angle, but it decreases or increases the value of the angle by 360 degrees.

Step 2: Calculate the positive coterminal angle

The calculation for the positive coterminal angle is then \(400^{\circ} - 360^{\circ}\), which gives \(40^{\circ}\). So the positive coterminal angle to \(400^{\circ}\) is \(40^{\circ}\).

Step 3: Find a negative coterminal angle

To find a negative coterminal angle we can subtract 360 degrees from the original angle until we get a result that is less than or equal to 0.

Step 4: Calculate the negative coterminal angle

Continuing from the positive coterminal angle of \(40^{\circ}\), we subtract 360 degrees again and get a negative coterminal angle of \(-320^{\circ}\).

Key Concepts

Angle Measurement

Angles are measured in degrees and help us understand the rotation from a fixed point. Imagine drawing a circle and rotating a line around its center. The amount of rotation is what we call an angle.

• Complete circle: 360 degrees
• Half circle: 180 degrees
• Quarter circle: 90 degrees

Angles are important in many areas, such as geometry, trigonometry, and physics. When working with angles, you might encounter terms like 'coterminal angles.' These are angles that share the same terminal side after a full rotation or several rotations are made. By adding or subtracting 360 degrees, you find coterminal angles. This will not change the direction of the line, but it alters the measured value of the angle.

Positive Angles

Positive angles are measured in the counter-clockwise direction from a fixed ray or line. When you measure an angle by rotating counter-clockwise, you are moving in the positive direction on the circle. To find a positive coterminal angle:

• Take the given angle and subtract 360 degrees until you reach a positive number less than 360.

This calculation ensures that you remain within one complete circle. For example, if you start with 400 degrees, you subtract 360 degrees to find a coterminal angle of 40 degrees. Both 400 degrees and 40 degrees point to the same direction on a full circle.

Negative Angles

Negative angles measure the rotation in the clockwise direction. When you move clockwise from a starting line, this is considered negative movement. Finding a negative coterminal angle involves subtracting multiples of 360 degrees from the angle until the result is less than or equal to zero.

• For example, starting from a known angle like 40 degrees (the positive coterminal angle of 400 degrees), you can subtract 360 degrees to get -320 degrees.

This means that -320 degrees is also a coterminal angle with the same starting and ending position on the circle.

Degrees in a Circle

One full rotation around a circle is 360 degrees. This is because a circle is divided into four quadrants of 90 degrees each. Key Points:

• 360 degrees: A complete circle
• Angles greater than 360 degrees or less than 0 degrees can be simplified within this circle.
• To simplify, you continually add or subtract 360 degrees.

Using multiples of 360 allows the identification of coterminal angles, making it easy to work with even very large or small angle measurements by finding their equivalent direction and endpoint on the circle.