Coterminal Angles: Videos & Practice Problems

Understanding angles and their relationships is crucial in geometry, particularly when dealing with coterminal angles. An angle is considered coterminal with another if they share the same terminal side when graphed on the coordinate plane. For instance, a full rotation around a circle is 360 degrees. Therefore, an angle of 390 degrees can be simplified by subtracting 360 degrees, resulting in a coterminal angle of 30 degrees. Both angles point in the same direction, demonstrating the concept of coterminality.

To find coterminal angles efficiently, one can add or subtract multiples of 360 degrees from the given angle. For example, if you have an angle of -270 degrees and need a positive coterminal angle, you can simply add 360 degrees, yielding 90 degrees. This means that -270 degrees and 90 degrees are coterminal, as they both terminate at the same point on the unit circle.

When dealing with larger angles, such as 1000 degrees, the process remains the same. By subtracting 360 degrees repeatedly until the angle falls within the range of 0 to 360 degrees, you can find the coterminal angle. In this case, subtracting 360 degrees twice from 1000 degrees results in 280 degrees, which is the coterminal angle. Thus, both 1000 degrees and 280 degrees point in the same direction on the circle.

In summary, the key to finding coterminal angles lies in recognizing that they differ by full rotations of 360 degrees. This understanding allows for quick calculations and a deeper grasp of angular relationships in geometry.

Understanding coterminal angles is essential in trigonometry, as it allows us to find angles that share the same terminal side when drawn in standard position. To determine the smallest positive angle coterminal with a given angle, we can use the concept of adding or subtracting multiples of 360 degrees, which represents a full rotation around a circle.

For example, to find the smallest positive angle coterminal with 710 degrees, we subtract 360 degrees (since 710 is greater than 360). This gives us:

710° - 360° = 350°

Since 350 degrees is less than 360 degrees, it is the smallest positive angle coterminal with 710 degrees. When sketching this angle, it is positioned just 10 degrees short of a full rotation, indicating that it lies in the fourth quadrant.

Next, consider the angle of -37 degrees. Negative angles are measured clockwise from the positive x-axis. To find the smallest positive coterminal angle, we add 360 degrees:

-37° + 360° = 323°

323 degrees is the smallest positive angle coterminal with -37 degrees. When sketching, this angle is located in the fourth quadrant, slightly less than a full rotation, and can be approximated by finding the halfway point between 270 degrees and 360 degrees.

Lastly, for the angle of -480 degrees, we first add 360 degrees:

-480° + 360° = -120°

Since -120 degrees is still negative, we add another 360 degrees:

-120° + 360° = 240°

Thus, 240 degrees is the smallest positive angle coterminal with -480 degrees. When sketching this angle, it is located in the third quadrant, closer to 270 degrees than to 180 degrees, and can be visualized by marking the angle from the positive x-axis.

In summary, finding coterminal angles involves simple arithmetic with 360 degrees, and sketching these angles requires an understanding of their positions relative to the x-axis and the quadrants of the coordinate plane.