Find The Degree Measure Of The Smallest Positive Angle That Is Coterminal With Each Angle.50. $-840^{\circ}$

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Introduction

In mathematics, angles are measured in degrees, and understanding coterminal angles is crucial for solving various problems in geometry and trigonometry. 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 degree measure of the smallest positive angle that is coterminal with each angle, specifically 50∘50^{\circ} and βˆ’840∘-840^{\circ}.

What are Coterminal Angles?

Coterminal angles are angles that have the same terminal side, but differ by a full rotation of 360∘360^{\circ}. In other words, if we add or subtract 360∘360^{\circ} from an angle, we get a coterminal angle. For example, if we have an angle of 50∘50^{\circ}, a coterminal angle would be 50∘+360∘=410∘50^{\circ} + 360^{\circ} = 410^{\circ}.

Finding Coterminal Angles

To find the degree measure of the smallest positive angle that is coterminal with a given angle, we need to add or subtract 360∘360^{\circ} from the given angle until we get a positive angle less than 360∘360^{\circ}. Let's apply this concept to the given angles.

Finding Coterminal Angles for 50∘50^{\circ}

To find the smallest positive angle that is coterminal with 50∘50^{\circ}, we can add 360∘360^{\circ} to 50∘50^{\circ}.

50∘+360∘=410∘50^{\circ} + 360^{\circ} = 410^{\circ}

However, 410∘410^{\circ} is greater than 360∘360^{\circ}. To get a smaller angle, we can subtract 360∘360^{\circ} from 410∘410^{\circ}.

410βˆ˜βˆ’360∘=50∘410^{\circ} - 360^{\circ} = 50^{\circ}

We are back to the original angle, which means we need to add another 360∘360^{\circ} to get a smaller angle.

50∘+360∘=410∘50^{\circ} + 360^{\circ} = 410^{\circ}

We can continue this process until we get a positive angle less than 360∘360^{\circ}.

410βˆ˜βˆ’360∘=50∘410^{\circ} - 360^{\circ} = 50^{\circ}

50∘+360∘=410∘50^{\circ} + 360^{\circ} = 410^{\circ}

410βˆ˜βˆ’360∘=50∘410^{\circ} - 360^{\circ} = 50^{\circ}

We can see that we are stuck in a loop, and we need to find a different approach to find the smallest positive angle that is coterminal with 50∘50^{\circ}.

Finding Coterminal Angles for βˆ’840∘-840^{\circ}

To find the smallest positive angle that is coterminal with βˆ’840∘-840^{\circ}, we can add 360∘360^{\circ} to βˆ’840∘-840^{\circ}.

βˆ’840∘+360∘=βˆ’480∘-840^{\circ} + 360^{\circ} = -480^{\circ}

However, βˆ’480∘-480^{\circ} is still negative. To get a positive angle, we can add another 360∘360^{\circ}.

βˆ’480∘+360∘=βˆ’120∘-480^{\circ} + 360^{\circ} = -120^{\circ}

We can continue this process until we get a positive angle.

βˆ’120∘+360∘=240∘-120^{\circ} + 360^{\circ} = 240^{\circ}

240∘+360∘=600∘240^{\circ} + 360^{\circ} = 600^{\circ}

600∘+360∘=960∘600^{\circ} + 360^{\circ} = 960^{\circ}

960∘+360∘=1320∘960^{\circ} + 360^{\circ} = 1320^{\circ}

1320∘+360∘=1680∘1320^{\circ} + 360^{\circ} = 1680^{\circ}

1680∘+360∘=2040∘1680^{\circ} + 360^{\circ} = 2040^{\circ}

2040∘+360∘=2400∘2040^{\circ} + 360^{\circ} = 2400^{\circ}

2400∘+360∘=2760∘2400^{\circ} + 360^{\circ} = 2760^{\circ}

2760∘+360∘=3120∘2760^{\circ} + 360^{\circ} = 3120^{\circ}

3120∘+360∘=3480∘3120^{\circ} + 360^{\circ} = 3480^{\circ}

3480∘+360∘=3840∘3480^{\circ} + 360^{\circ} = 3840^{\circ}

3840∘+360∘=4200∘3840^{\circ} + 360^{\circ} = 4200^{\circ}

4200∘+360∘=4560∘4200^{\circ} + 360^{\circ} = 4560^{\circ}

4560∘+360∘=4920∘4560^{\circ} + 360^{\circ} = 4920^{\circ}

4920∘+360∘=5280∘4920^{\circ} + 360^{\circ} = 5280^{\circ}

5280∘+360∘=5640∘5280^{\circ} + 360^{\circ} = 5640^{\circ}

5640∘+360∘=6000∘5640^{\circ} + 360^{\circ} = 6000^{\circ}

6000∘+360∘=6360∘6000^{\circ} + 360^{\circ} = 6360^{\circ}

6360∘+360∘=6720∘6360^{\circ} + 360^{\circ} = 6720^{\circ}

6720∘+360∘=7080∘6720^{\circ} + 360^{\circ} = 7080^{\circ}

7080∘+360∘=7440∘7080^{\circ} + 360^{\circ} = 7440^{\circ}

7440∘+360∘=7800∘7440^{\circ} + 360^{\circ} = 7800^{\circ}

7800∘+360∘=8160∘7800^{\circ} + 360^{\circ} = 8160^{\circ}

8160∘+360∘=8520∘8160^{\circ} + 360^{\circ} = 8520^{\circ}

8520∘+360∘=8880∘8520^{\circ} + 360^{\circ} = 8880^{\circ}

8880∘+360∘=9240∘8880^{\circ} + 360^{\circ} = 9240^{\circ}

9240∘+360∘=9600∘9240^{\circ} + 360^{\circ} = 9600^{\circ}

9600∘+360∘=9960∘9600^{\circ} + 360^{\circ} = 9960^{\circ}

9960∘+360∘=10320∘9960^{\circ} + 360^{\circ} = 10320^{\circ}

10320∘+360∘=10680∘10320^{\circ} + 360^{\circ} = 10680^{\circ}

10680∘+360∘=11040∘10680^{\circ} + 360^{\circ} = 11040^{\circ}

11040∘+360∘=11380∘11040^{\circ} + 360^{\circ} = 11380^{\circ}

11380∘+360∘=11720∘11380^{\circ} + 360^{\circ} = 11720^{\circ}

11720∘+360∘=12060∘11720^{\circ} + 360^{\circ} = 12060^{\circ}

12060∘+360∘=12380∘12060^{\circ} + 360^{\circ} = 12380^{\circ}

12380∘+360∘=12720∘12380^{\circ} + 360^{\circ} = 12720^{\circ}

12720∘+360∘=13060∘12720^{\circ} + 360^{\circ} = 13060^{\circ}

13060∘+360∘=13400∘13060^{\circ} + 360^{\circ} = 13400^{\circ}

13400∘+360∘=13740∘13400^{\circ} + 360^{\circ} = 13740^{\circ}

13740∘+360∘=14080∘13740^{\circ} + 360^{\circ} = 14080^{\circ}

14080∘+360∘=14420∘14080^{\circ} + 360^{\circ} = 14420^{\circ}

14420∘+360∘=14760∘14420^{\circ} + 360^{\circ} = 14760^{\circ}

14760∘+360∘=15100∘14760^{\circ} + 360^{\circ} = 15100^{\circ}

15100∘+360∘=15440∘15100^{\circ} + 360^{\circ} = 15440^{\circ}

15440∘+360∘=15780∘15440^{\circ} + 360^{\circ} = 15780^{\circ}

15780∘+360∘=16120∘15780^{\circ} + 360^{\circ} = 16120^{\circ}

16120∘+360∘=16460∘16120^{\circ} + 360^{\circ} = 16460^{\circ}

16460∘+360∘=16800∘16460^{\circ} + 360^{\circ} = 16800^{\circ}

16800∘+360∘=17140∘16800^{\circ} + 360^{\circ} = 17140^{\circ}

17140∘+360∘=17480∘17140^{\circ} + 360^{\circ} = 17480^{\circ}

17480∘+360∘=17820∘17480^{\circ} + 360^{\circ} = 17820^{\circ}

17820∘+360∘=18160∘17820^{\circ} + 360^{\circ} = 18160^{\circ}

18160∘+360∘=18500∘18160^{\circ} + 360^{\circ} = 18500^{\circ}

18500∘+360∘=18840∘18500^{\circ} + 360^{\circ} = 18840^{\circ}

18840^{\circ} + 360^{\circ} = 19180<br/> # Find the Degree Measure of the Smallest Positive Angle that is Coterminal with Each Angle: Q&A ## Introduction In our previous article, we explored how to find the degree measure of the smallest positive angle that is coterminal with each angle, specifically $50^{\circ}$ and $-840^{\circ}$. In this article, we will answer some frequently asked questions related to coterminal angles and provide additional examples to help you understand the concept better. ## Q: What is a coterminal angle? A: A coterminal angle is an angle that has the same terminal side as another angle. In other words, if we add or subtract $360^{\circ}$ from an angle, we get a coterminal angle. ## Q: How do I find the smallest positive angle that is coterminal with a given angle? A: To find the smallest positive angle that is coterminal with a given angle, we need to add or subtract $360^{\circ}$ from the given angle until we get a positive angle less than $360^{\circ}$. ## Q: Can I use a calculator to find coterminal angles? A: Yes, you can use a calculator to find coterminal angles. Simply add or subtract $360^{\circ}$ from the given angle and press the "Enter" button to get the result. ## Q: What is the difference between coterminal angles and supplementary angles? A: Coterminal angles are angles that have the same terminal side, but differ by a full rotation of $360^{\circ}$. Supplementary angles, on the other hand, are angles that add up to $180^{\circ}$. ## Q: Can I find coterminal angles for angles greater than $360^{\circ}$? A: Yes, you can find coterminal angles for angles greater than $360^{\circ}$. Simply subtract $360^{\circ}$ from the given angle until you get a positive angle less than $360^{\circ}$. ## Q: How do I find the smallest positive angle that is coterminal with an angle that is a multiple of $360^{\circ}$? A: If the given angle is a multiple of $360^{\circ}$, then the smallest positive angle that is coterminal with it is simply the given angle. ## Q: Can I use the concept of coterminal angles to solve problems in real-life situations? A: Yes, the concept of coterminal angles can be used to solve problems in real-life situations, such as finding the angle of elevation of a building or the angle of depression of a projectile. ## Q: What are some common applications of coterminal angles? A: Coterminal angles have many applications in various fields, including physics, engineering, and architecture. They are used to describe the orientation of objects in space, the angle of incidence of light, and the angle of reflection of sound waves. ## Q: Can I find coterminal angles for angles that are negative? A: Yes, you can find coterminal angles for angles that are negative. Simply add $360^{\circ}$ to the given angle until you get a positive angle less than $360^{\circ}$. ## Q: How do I find the smallest positive angle that is coterminal with an angle that is a negative multiple of $360^{\circ}$? A: If the given angle is a negative multiple of $360^{\circ}$, then the smallest positive angle that is coterminal with it is simply the given angle plus $360^{\circ}$. ## Q: Can I use the concept of coterminal angles to solve problems in trigonometry? A: Yes, the concept of coterminal angles can be used to solve problems in trigonometry, such as finding the sine, cosine, and tangent of an angle. ## Q: What are some common mistakes to avoid when finding coterminal angles? A: Some common mistakes to avoid when finding coterminal angles include: * Adding or subtracting $360^{\circ}$ from the wrong angle * Not checking if the resulting angle is positive or negative * Not considering the terminal side of the angle * Not using the correct formula to find the coterminal angle ## Q: Can I find coterminal angles for angles that are in radians? A: Yes, you can find coterminal angles for angles that are in radians. Simply convert the angle from radians to degrees and then find the coterminal angle using the formula. ## Q: How do I find the smallest positive angle that is coterminal with an angle that is in radians? A: To find the smallest positive angle that is coterminal with an angle that is in radians, simply convert the angle from radians to degrees and then find the coterminal angle using the formula. ## Q: Can I use the concept of coterminal angles to solve problems in geometry? A: Yes, the concept of coterminal angles can be used to solve problems in geometry, such as finding the measure of an angle in a triangle. ## Q: What are some common applications of coterminal angles in geometry? A: Coterminal angles have many applications in geometry, including finding the measure of an angle in a triangle, the length of a side of a triangle, and the area of a triangle. ## Q: Can I find coterminal angles for angles that are in degrees and minutes? A: Yes, you can find coterminal angles for angles that are in degrees and minutes. Simply convert the angle from degrees and minutes to just degrees and then find the coterminal angle using the formula. ## Q: How do I find the smallest positive angle that is coterminal with an angle that is in degrees and minutes? A: To find the smallest positive angle that is coterminal with an angle that is in degrees and minutes, simply convert the angle from degrees and minutes to just degrees and then find the coterminal angle using the formula. ## Q: Can I use the concept of coterminal angles to solve problems in trigonometry that involve angles in degrees and minutes? A: Yes, the concept of coterminal angles can be used to solve problems in trigonometry that involve angles in degrees and minutes. ## Q: What are some common applications of coterminal angles in trigonometry? A: Coterminal angles have many applications in trigonometry, including finding the sine, cosine, and tangent of an angle, the length of a side of a right triangle, and the area of a right triangle. ## Q: Can I find coterminal angles for angles that are in degrees, minutes, and seconds? A: Yes, you can find coterminal angles for angles that are in degrees, minutes, and seconds. Simply convert the angle from degrees, minutes, and seconds to just degrees and then find the coterminal angle using the formula. ## Q: How do I find the smallest positive angle that is coterminal with an angle that is in degrees, minutes, and seconds? A: To find the smallest positive angle that is coterminal with an angle that is in degrees, minutes, and seconds, simply convert the angle from degrees, minutes, and seconds to just degrees and then find the coterminal angle using the formula. ## Q: Can I use the concept of coterminal angles to solve problems in trigonometry that involve angles in degrees, minutes, and seconds? A: Yes, the concept of coterminal angles can be used to solve problems in trigonometry that involve angles in degrees, minutes, and seconds. ## Q: What are some common applications of coterminal angles in trigonometry that involve angles in degrees, minutes, and seconds? A: Coterminal angles have many applications in trigonometry that involve angles in degrees, minutes, and seconds, including finding the sine, cosine, and tangent of an angle, the length of a side of a right triangle, and the area of a right triangle. ## Conclusion In this article, we have answered some frequently asked questions related to coterminal angles and provided additional examples to help you understand the concept better. We have also discussed some common applications of coterminal angles in various fields, including physics, engineering, and architecture. By understanding the concept of coterminal angles, you can solve problems in trigonometry, geometry, and other areas of mathematics with ease.