Finding Coterminal Angles for the Given Rotation

Oct 13, 2025 by talynt.org 100 views

Understanding the Basics of Angles in Trigonometry

In trigonometry, angles are a fundamental concept that plays a crucial role in solving various problems. Angles can be measured in degrees or radians, and they can be classified into different types, such as acute, obtuse, right, and straight angles. In this article, we will focus on finding coterminal, quadrant, and reference angles for a given rotation.

What are Coterminal Angles?

Coterminal angles are angles that have the same terminal side. In other words, they are angles that have the same endpoint on the unit circle. Coterminal angles can be found by adding or subtracting a full rotation (360° or 2π radians) to the given angle. For example, if we have an angle of 30°, its coterminal angles would be 30° + 360° = 390° and 30° - 360° = -330°.

Finding Coterminal Angles for the Given Rotation

The given rotation is $$\frac{64 \pi}{13}$$. To find the coterminal angle from $$0 \leq \theta \ \textless \ 2 \pi$$, we need to add or subtract a full rotation (2π radians) to the given angle. Let's start by finding the coterminal angle by adding 2π radians to the given angle:

\frac{64 \pi}{13} + 2 \pi = \frac{64 \pi}{13} + \frac{26 \pi}{13} = \frac{90 \pi}{13}

However, this angle is greater than 2π radians, so we need to subtract 2π radians from it to get the coterminal angle within the range $$0 \leq \theta \ \textless \ 2 \pi$$:

\frac{90 \pi}{13} - 2 \pi = \frac{90 \pi}{13} - \frac{26 \pi}{13} = \frac{64 \pi}{13}

However, we are looking for a coterminal angle that is not the same as the original angle. Let's try subtracting 2π radians from the given angle:

\frac{64 \pi}{13} - 2 \pi = \frac{64 \pi}{13} - \frac{26 \pi}{13} = \frac{38 \pi}{13}

This is the coterminal angle we are looking for.

What is the Quadrant of the Angle?

To determine the quadrant of the angle, we need to find the value of the angle in degrees. We can do this by multiplying the angle in radians by $$\frac{180^\circ}{\pi}$$:

\frac{64 \pi}{13} \cdot \frac{180^\circ}{\pi} = \frac{64 \cdot 180^\circ}{13} = 1153.85^\circ

Since the angle is greater than 360°, we need to subtract 360° from it to get the angle in the range $$0^\circ \leq \theta \ \textless \ 360^\circ$$:

1153.85^\circ - 360^\circ = 793.85^\circ

However, this angle is still greater than 360°, so we need to subtract another 360° from it:

793.85^\circ - 360^\circ = 433.85^\circ

This angle is in the third quadrant.

What is the Reference Angle?

The reference angle is the acute angle between the terminal side of the angle and the x-axis. To find the reference angle, we need to find the angle in the range $$0^\circ \leq \theta \ \textless \ 180^\circ$$.

Since the angle is in the third quadrant, we can find the reference angle by subtracting the angle from 180°:

180^\circ - 433.85^\circ = -253.85^\circ

However, this angle is negative, so we need to add 360° to it to get the reference angle:

-253.85^\circ + 360^\circ = 106.15^\circ

This is the reference angle.

Conclusion

In this article, we found the coterminal angle, quadrant, and reference angle for the given rotation $$\frac{64 \pi}{13}$$. We started by finding the coterminal angle by adding or subtracting a full rotation (2π radians) to the given angle. We then determined the quadrant of the angle by finding the value of the angle in degrees and subtracting 360° from it. Finally, we found the reference angle by subtracting the angle from 180° and adding 360° to it if necessary.

Key Takeaways

Coterminal angles are angles that have the same terminal side.
To find the coterminal angle, add or subtract a full rotation (2π radians) to the given angle.
The quadrant of an angle can be determined by finding the value of the angle in degrees and subtracting 360° from it.
The reference angle is the acute angle between the terminal side of the angle and the x-axis.

Final Answer

The final answer is:

Coterminal angle: $$\frac{38 \pi}{13}$$
Quadrant: third quadrant
Reference angle: $$106.15^\circ$$

Q: What is the difference between coterminal angles and equivalent angles?

A: Coterminal angles are angles that have the same terminal side, while equivalent angles are angles that have the same measure. For example, 30° and 390° are coterminal angles, but they are not equivalent angles because they have different measures.

Q: How do I find the coterminal angle of a given angle?

A: To find the coterminal angle of a given angle, you can add or subtract a full rotation (2π radians) to the given angle. For example, if you have an angle of 30°, its coterminal angle would be 30° + 360° = 390°.

Q: What is the quadrant of an angle?

A: The quadrant of an angle is the region of the coordinate plane where the angle is located. There are four quadrants: the first quadrant (0° to 90°), the second quadrant (90° to 180°), the third quadrant (180° to 270°), and the fourth quadrant (270° to 360°).

Q: How do I determine the quadrant of an angle?

A: To determine the quadrant of an angle, you can find the value of the angle in degrees and subtract 360° from it if necessary. For example, if you have an angle of 1153.85°, you would subtract 360° from it to get 793.85°, which is in the third quadrant.

Q: What is the reference angle?

A: The reference angle is the acute angle between the terminal side of the angle and the x-axis. It is used to find the sine, cosine, and tangent of an angle.

Q: How do I find the reference angle of a given angle?

A: To find the reference angle of a given angle, you can subtract the angle from 180° if it is in the second or third quadrant, or add the angle to 180° if it is in the fourth quadrant. For example, if you have an angle of 106.15°, its reference angle would be 180° - 106.15° = 73.85°.

Q: What is the relationship between coterminal angles, quadrants, and reference angles?

A: Coterminal angles are related to quadrants because they have the same terminal side. Reference angles are related to quadrants because they are used to find the sine, cosine, and tangent of an angle, which are dependent on the quadrant of the angle.

Q: How do I use coterminal angles, quadrants, and reference angles in real-world applications?

A: Coterminal angles, quadrants, and reference angles are used in various real-world applications, such as navigation, engineering, and physics. For example, in navigation, you need to find the coterminal angle of a given angle to determine the direction of a ship or a plane. In engineering, you need to find the reference angle of a given angle to determine the stress and strain on a structure.

Q: What are some common mistakes to avoid when working with coterminal angles, quadrants, and reference angles?

A: Some common mistakes to avoid when working with coterminal angles, quadrants, and reference angles include:

Not considering the quadrant of an angle when finding the coterminal angle.
Not subtracting 360° from an angle when determining the quadrant.
Not using the correct formula to find the reference angle.
Not considering the relationship between coterminal angles, quadrants, and reference angles.

Q: How can I practice working with coterminal angles, quadrants, and reference angles?

A: You can practice working with coterminal angles, quadrants, and reference angles by:

Solving problems that involve finding coterminal angles, quadrants, and reference angles.
Using online resources and calculators to check your work.
Working with a partner or tutor to review and discuss your work.
Practicing with different types of angles, such as acute, obtuse, and right angles.

Q: What are some resources available to help me learn more about coterminal angles, quadrants, and reference angles?

A: Some resources available to help you learn more about coterminal angles, quadrants, and reference angles include:

Online tutorials and videos.
Textbooks and workbooks.
Online calculators and software.
Tutoring services and online communities.
Practice problems and quizzes.

Q: How can I apply what I have learned about coterminal angles, quadrants, and reference angles to real-world situations?

A: You can apply what you have learned about coterminal angles, quadrants, and reference angles to real-world situations by:

Using the concepts to solve problems in navigation, engineering, and physics.
Applying the concepts to find the sine, cosine, and tangent of an angle.
Using the concepts to determine the direction of a ship or a plane.
Using the concepts to find the stress and strain on a structure.

Q: What are some common applications of coterminal angles, quadrants, and reference angles in real-world situations?

A: Some common applications of coterminal angles, quadrants, and reference angles in real-world situations include:

Navigation: finding the direction of a ship or a plane.
Engineering: finding the stress and strain on a structure.
Physics: finding the velocity and acceleration of an object.
Computer graphics: creating 3D models and animations.
Medical imaging: creating images of the body using MRI and CT scans.