Find The Angle Of Least Positive Measure That Is Coterminal With Each Angle.(a) 390 ∘ 390^{\circ} 39 0 ∘ (b) − 80 ∘ -80^{\circ} − 8 0 ∘ (c) 810 ∘ 810^{\circ} 81 0 ∘

Introduction

In trigonometry, angles are measured in degrees, and understanding coterminal angles is crucial for solving various problems. A coterminal angle is an angle that has the same terminal side as another angle. In this article, we will explore how to find the angle of least positive measure that is coterminal with each given angle.

What are Coterminal Angles?

Coterminal angles are angles that have the same terminal side. This means that if two angles are coterminal, they will have the same endpoint on the unit circle. For example, if we have an angle of $360^{\circ}$, we can add or subtract any multiple of $360^{\circ}$ to get a coterminal angle. In this case, $360^{\circ}$, $720^{\circ}$, and $1080^{\circ}$ are all coterminal angles.

Finding Coterminal Angles

To find the angle of least positive measure that is coterminal with a given angle, we need to add or subtract multiples of $360^{\circ}$ until we get an angle between $0^{\circ}$ and $360^{\circ}$. Let's consider the following examples:

(a) $390^{\circ}$

To find the angle of least positive measure that is coterminal with $390^{\circ}$, we need to subtract a multiple of $360^{\circ}$ from $390^{\circ}$. We can subtract $360^{\circ}$ from $390^{\circ}$ to get:

$$390^{\circ} - 360^{\circ} = 30^{\circ}$$

Therefore, the angle of least positive measure that is coterminal with $390^{\circ}$ is $30^{\circ}$.

(b) $-80^{\circ}$

To find the angle of least positive measure that is coterminal with $-80^{\circ}$, we need to add a multiple of $360^{\circ}$ to $-80^{\circ}$. We can add $360^{\circ}$ to $-80^{\circ}$ to get:

$$-80^{\circ} + 360^{\circ} = 280^{\circ}$$

However, we can also add $720^{\circ}$ to $-80^{\circ}$ to get:

$$-80^{\circ} + 720^{\circ} = 640^{\circ}$$

Since $640^{\circ}$ is greater than $360^{\circ}$, we can subtract $360^{\circ}$ from $640^{\circ}$ to get:

$$640^{\circ} - 360^{\circ} = 280^{\circ}$$

Therefore, the angle of least positive measure that is coterminal with $-80^{\circ}$ is $280^{\circ}$.

(c) $810^{\circ}$

To find the angle of least positive measure that is coterminal with $810^{\circ}$, we need to subtract a multiple of $360^{\circ}$ from $810^{\circ}$. We can subtract $360^{\circ}$ from $810^{\circ}$ to get:

$$810^{\circ} - 360^{\circ} = 450^{\circ}$$

However, we can also subtract $720^{\circ}$ from $810^{\circ}$ to get:

$$810^{\circ} - 720^{\circ} = 90^{\circ}$$

Therefore, the angle of least positive measure that is coterminal with $810^{\circ}$ is $90^{\circ}$.

Conclusion

In conclusion, finding coterminal angles is an essential concept in trigonometry. By understanding how to add or subtract multiples of $360^{\circ}$, we can find the angle of least positive measure that is coterminal with any given angle. In this article, we have explored how to find coterminal angles for the given angles $390^{\circ}$, $-80^{\circ}$, and $810^{\circ}$. We have seen that by subtracting or adding multiples of $360^{\circ}$, we can find the angle of least positive measure that is coterminal with each given angle.

Example Problems

Problem 1

Find the angle of least positive measure that is coterminal with $540^{\circ}$.

Solution

To find the angle of least positive measure that is coterminal with $540^{\circ}$, we need to subtract a multiple of $360^{\circ}$ from $540^{\circ}$. We can subtract $360^{\circ}$ from $540^{\circ}$ to get:

$$540^{\circ} - 360^{\circ} = 180^{\circ}$$

Therefore, the angle of least positive measure that is coterminal with $540^{\circ}$ is $180^{\circ}$.

Problem 2

Find the angle of least positive measure that is coterminal with $-120^{\circ}$.

Solution

To find the angle of least positive measure that is coterminal with $-120^{\circ}$, we need to add a multiple of $360^{\circ}$ to $-120^{\circ}$. We can add $360^{\circ}$ to $-120^{\circ}$ to get:

$$-120^{\circ} + 360^{\circ} = 240^{\circ}$$

However, we can also add $720^{\circ}$ to $-120^{\circ}$ to get:

$$-120^{\circ} + 720^{\circ} = 600^{\circ}$$

Since $600^{\circ}$ is greater than $360^{\circ}$, we can subtract $360^{\circ}$ from $600^{\circ}$ to get:

$$600^{\circ} - 360^{\circ} = 240^{\circ}$$

Therefore, the angle of least positive measure that is coterminal with $-120^{\circ}$ is $240^{\circ}$.

Final Thoughts

$$$$

Q: What is a coterminal angle?

A: A coterminal angle is an angle that has the same terminal side as another angle. This means that if two angles are coterminal, they will have the same endpoint on the unit circle.

Q: How do I find the angle of least positive measure that is coterminal with a given angle?

A: To find the angle of least positive measure that is coterminal with a given angle, you need to add or subtract multiples of $360^{\circ}$ until you get an angle between $0^{\circ}$ and $360^{\circ}$.

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

A: Coterminal angles are angles that have the same terminal side, while supplementary angles are angles that add up to $180^{\circ}$. For example, $30^{\circ}$ and $150^{\circ}$ are coterminal angles, while $30^{\circ}$ and $120^{\circ}$ are supplementary angles.

Q: Can I find the angle of least positive measure that is coterminal with a negative angle?

A: Yes, you can find the angle of least positive measure that is coterminal with a negative angle by adding a multiple of $360^{\circ}$ to the negative angle. For example, to find the angle of least positive measure that is coterminal with $-80^{\circ}$, you can add $360^{\circ}$ to get $280^{\circ}$.

Q: How do I know when I have found the angle of least positive measure that is coterminal with a given angle?

A: You have found the angle of least positive measure that is coterminal with a given angle when the resulting angle is between $0^{\circ}$ and $360^{\circ}$.

Q: Can I use a calculator to find the angle of least positive measure that is coterminal with a given angle?

A: Yes, you can use a calculator to find the angle of least positive measure that is coterminal with a given angle. However, it's also important to understand the concept and be able to do it manually.

Q: What are some real-world applications of coterminal angles?

A: Coterminal angles have many real-world applications, including:

• Navigation: When navigating, it's essential to know the angle of a ship or a plane relative to a reference point. Coterminal angles help to determine the correct angle.
• Engineering: In engineering, coterminal angles are used to design and build structures, such as bridges and buildings.
• Physics: In physics, coterminal angles are used to describe the motion of objects and to calculate forces and energies.

Q: Can I use coterminal angles to solve problems involving rotations?

A: Yes, you can use coterminal angles to solve problems involving rotations. For example, if a wheel rotates $720^{\circ}$, you can find the angle of least positive measure that is coterminal with $720^{\circ}$ by subtracting $360^{\circ}$ to get $360^{\circ}$.

Q: What are some common mistakes to avoid when working with coterminal angles?

A: Some common mistakes to avoid when working with coterminal angles include:

• Not understanding the concept of coterminal angles
• Not adding or subtracting the correct multiple of $360^{\circ}$
• Not checking if the resulting angle is between $0^{\circ}$ and $360^{\circ}$

Q: Can I use coterminal angles to solve problems involving trigonometry?

A: Yes, you can use coterminal angles to solve problems involving trigonometry. For example, if you need to find the sine or cosine of an angle, you can use coterminal angles to simplify the problem.

Q: What are some tips for mastering coterminal angles?

A: Some tips for mastering coterminal angles include:

• Practicing problems involving coterminal angles
• Understanding the concept of coterminal angles
• Using visual aids, such as diagrams and graphs, to help understand the concept
• Checking your work to ensure that you have found the correct angle.