Coterminal Angles: Videos & Practice Problems

To find coterminal angles, add or subtract multiples of 360 degrees. For example, 390 degrees is coterminal with 30 degrees, as both point in the same direction. Similarly, negative angles can be converted to positive by adding 360 degrees, such as negative 270 degrees becoming 90 degrees. For larger angles like 1,000 degrees, subtracting 360 twice yields 280 degrees, which is coterminal with 1,000 degrees. Understanding coterminal angles is essential for graphing and analyzing angular relationships in trigonometry.

Coterminal Angles

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, meaning they point in the same direction. 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. This demonstrates that both angles point in the same direction.

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. Both -270 degrees and 90 degrees point in the same direction, confirming their coterminality.

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 share the same terminal side.

In summary, the key to identifying coterminal angles lies in recognizing that they differ by full rotations of 360 degrees. This understanding allows for quick calculations and visualizations of angles on the unit circle, enhancing your grasp of angular relationships in geometry.

Example 1

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 calculation 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°

Since 323 degrees is between 0 and 360, it is the desired angle. 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 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°

Now, 240 degrees is the smallest positive angle coterminal with -480 degrees. When sketching this angle, it is found 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, to find coterminal angles, remember to add or subtract multiples of 360 degrees until you arrive at a positive angle within the range of 0 to 360 degrees. This technique is crucial for accurately sketching angles in standard position.

Here’s what students ask on this topic:

What are coterminal angles and how do you find them?

Coterminal angles are angles that share the same terminal side when drawn in standard position. To find coterminal angles, you add or subtract multiples of 360 degrees. For example, if you have an angle of 45 degrees, adding 360 degrees gives you 405 degrees, which is coterminal with 45 degrees. Similarly, subtracting 360 degrees from 45 degrees gives you -315 degrees, which is also coterminal with 45 degrees. The general formula is: $\theta \pm 360n$, where $n$ is an integer.

How do you find a positive coterminal angle for a negative angle?

To find a positive coterminal angle for a negative angle, you add 360 degrees until the result is positive. For example, if you have an angle of -150 degrees, you add 360 degrees: $-150+360=210$. Therefore, 210 degrees is a positive coterminal angle for -150 degrees. This method ensures that the angle lies within the standard range of 0 to 360 degrees.

What is the coterminal angle of 1000 degrees?

To find the coterminal angle of 1000 degrees, you subtract 360 degrees repeatedly until the result is between 0 and 360 degrees. First, subtract 360 degrees: $1000-360=640$. Then, subtract 360 degrees again: $640-360=280$. Therefore, 280 degrees is the coterminal angle of 1000 degrees.

How do you determine if two angles are coterminal?

Two angles are coterminal if they share the same terminal side when drawn in standard position. To determine if two angles are coterminal, find the difference between them and check if it is a multiple of 360 degrees. For example, to check if 45 degrees and 405 degrees are coterminal, calculate the difference: $405-45=360$. Since 360 degrees is a multiple of 360, the angles are coterminal.

What is the coterminal angle of -270 degrees?

To find the coterminal angle of -270 degrees, you add 360 degrees until the result is between 0 and 360 degrees. Add 360 degrees: $-270+360=90$. Therefore, 90 degrees is the coterminal angle of -270 degrees.