Coterminal Angles In Radians & Degrees - Basic Introduction, Trigonometry

The Organic Chemistry Tutor

11 Oct 201706:19

EducationalLearning

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TLDRThe video script explains coterminal angles as angles that share the same terminal side but have different measures. It illustrates this concept with examples, such as 30 degrees and 390 degrees, which are coterminal because they point to the same position on a circle. The script teaches viewers how to find coterminal angles by adding or subtracting multiples of 360 degrees for degrees or 2π for radians. It demonstrates this process with various examples, including positive and negative angles, and converting between degrees and radians, to find angles within specific ranges, such as between 0 and 360 degrees or 0 and 2π.

Takeaways

📐 Coterminal angles are angles that point to the same terminal side but have different measures.
🔄 Coterminal angles can be found by adding or subtracting multiples of 360 degrees for angles measured in degrees, or multiples of 2π for angles measured in radians.
👉 For example, 30 degrees and 390 degrees are coterminal because they point to the same terminal side.
➕ To find a coterminal angle within a specific range, such as between 0 and 360 degrees, subtract multiples of 360 from the given angle until it falls within the desired range.
➖ Similarly, to adjust a negative angle into the range of 0 to 360 degrees, add multiples of 360 until it is positive and within the range.
🔢 The script provides an example where 430 degrees is adjusted to a coterminal angle of 70 degrees by subtracting 360 degrees.
➡️ Another example shows that a negative angle, such as -150 degrees, can be adjusted to a positive coterminal angle of 210 degrees by adding 360 degrees.
🔄 The process of finding coterminal angles is demonstrated with both positive and negative angles, as well as angles greater than 360 degrees.
📚 The script also explains how to find coterminal angles in radians, using the same principle of adding or subtracting multiples of 2π.
📈 An example with radians is given where thirteen pi over six is reduced to pi over six (equivalent to 30 degrees) by subtracting two pi (or 12 pi over six).
🔢 The final examples in the script involve adjusting negative angles in radians to positive coterminal angles within the range of 0 to 2π.

Q & A

What are coterminal angles?
-Coterminal angles are angles that share the same terminal side but have different magnitudes. They differ by full rotations, or multiples of 360 degrees.
How do you identify if two angles are coterminal?
-Two angles are coterminal if they have the same terminal side. They can be identified by checking if their difference is an integer multiple of 360 degrees.
Can you give an example of coterminal angles?
-An example of coterminal angles is 30 degrees and 390 degrees. They both point to the same terminal ray but have different magnitudes.
What happens when you add 360 degrees to an angle?
-Adding 360 degrees to an angle results in a coterminal angle that has the same terminal side but a different magnitude. It essentially represents the same angle but after one full rotation.
How do you find a coterminal angle within a specific range?
-To find a coterminal angle within a specific range, such as between 0 and 360 degrees, you can subtract or add multiples of 360 degrees to the given angle until it falls within the desired range.
What is the relationship between degrees and radians when finding coterminal angles?
-In radians, the equivalent of 360 degrees is 2π. So, to find coterminal angles in radians, you add or subtract multiples of 2π from the given angle.
Can negative angles be coterminal with positive angles?
-Yes, negative angles can be coterminal with positive angles. For example, 150 degrees and -210 degrees are coterminal because they end up at the same position on the unit circle after traveling in opposite directions.
How do you find a positive coterminal angle for a given angle greater than 360 degrees?
-To find a positive coterminal angle for an angle greater than 360 degrees, subtract multiples of 360 degrees from the given angle until the result is less than 360 degrees.
What is the process for finding a coterminal angle for a negative angle less than 0 degrees?
-For a negative angle less than 0 degrees, add multiples of 360 degrees to the angle until the result is positive and within the desired range, such as between 0 and 360 degrees.
Can you provide a step-by-step method for finding coterminal angles in radians?
-To find coterminal angles in radians, first determine if the given angle is greater than or less than 2π. If it's greater, subtract multiples of 2π until it's less than 2π. If it's less, add multiples of 2π until it's positive and within the desired range.

Outlines

00:00

📐 Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side on a circle despite having different numerical values. The script explains that these angles can be identified by their difference being a multiple of 360 degrees. For instance, 30 degrees and 390 degrees are coterminal because they point to the same direction. The concept is further clarified with examples involving both positive and negative angles, such as 150 degrees and -210 degrees. The process to find a coterminal angle involves adding or subtracting 360 degrees for angles in degrees or 2π for angles in radians. Practical examples demonstrate how to adjust given angles to find coterminal angles within a specific range, like between 0 and 360 degrees or 0 and 2π.

05:01

🔍 Calculating Coterminal Angles in Radians

This paragraph delves into the specifics of finding coterminal angles when the given angle is measured in radians. It uses the example of an angle of thirteen pi over six, which exceeds the two pi (or 360 degrees) limit. The explanation involves subtracting multiples of two pi from the given angle to find an equivalent angle within the desired range. The process is illustrated with step-by-step calculations, showing how to adjust the angle to fall between zero and two pi. The paragraph also addresses how to handle negative angles, such as negative eleven pi over four, by adding two pi until the result is positive and within the acceptable range. The key takeaway is the methodical approach to adjusting angles by adding or subtracting 360 degrees or 2π to find coterminal angles.

Mindmap

Keywords

💡Coterminal Angles

Coterminal angles are angles that share the same terminal side on a circle, despite having different numerical measures. In the context of the video, coterminal angles are crucial for understanding how angles can be equivalent despite their differing degrees or radians. For example, the script mentions that 30 degrees and 390 degrees are coterminal because they point to the same terminal ray, illustrating the concept that angles differing by full rotations (360 degrees or 2π radians) are considered coterminal.

💡Terminal Ray

A terminal ray is a line that extends from the center of a circle to a point on the circumference, defining the 'ending' part of an angle. The video script uses the concept of a terminal ray to explain coterminal angles, stating that angles like 30 degrees and 390 degrees are coterminal because they both terminate at the same ray, despite having different starting points.

💡Degrees

Degrees are a unit of measurement used to express angles. In the video, degrees are used to quantify the size of angles and to find coterminal angles. For instance, when the script discusses finding a coterminal angle for 430 degrees, it suggests subtracting 360 degrees to find the coterminal angle of 70 degrees, which falls within the desired range of 0 to 360 degrees.

💡Radians

Radians are another unit of angular measurement, often used in more advanced mathematical contexts. The script introduces radians when explaining how to find coterminal angles in this unit, stating that just as you add or subtract 360 degrees for degrees, you add or subtract 2π for radians. This is demonstrated when the script converts angles like thirteen pi over six into a coterminal angle in the range of 0 to 2π.

💡Full Rotation

A full rotation, or a complete circle, is equivalent to 360 degrees or 2π radians. The video script uses the concept of a full rotation to explain how to find coterminal angles. When an angle exceeds a full rotation, you subtract 360 degrees or 2π radians to find a coterminal angle within the standard range.

💡Add/Subtract 360/2π

The script emphasizes the process of adding or subtracting 360 degrees or 2π radians to find coterminal angles. This is a fundamental step in the process, as it allows you to adjust any given angle to find an equivalent angle within the standard 0 to 360-degree range or 0 to 2π radian range. The script provides several examples of this process, such as converting 430 degrees to 70 degrees and thirteen pi over six to pi over six.

💡Range

The range in the context of the video refers to the acceptable numerical window for angles, which is between 0 and 360 degrees or between 0 and 2π radians. The script discusses finding coterminal angles that fall within this range, which is essential for standardizing angles in mathematical and geometric contexts.

💡Negative Angles

Negative angles are discussed in the script as angles that are measured in a clockwise direction from the positive x-axis. The video explains that when dealing with negative angles, such as -150 degrees or negative 11 pi over four radians, you add 360 degrees or 2π radians to find a coterminal angle within the positive range.

💡Common Denominators

When working with fractions of pi in radians, finding a common denominator is necessary for addition or subtraction. The script mentions this when converting thirteen pi over six to a coterminal angle, where it finds a common denominator to subtract 2π (12 pi over six) from thirteen pi over six, resulting in pi over six.

💡Multiplication by Four Over Four

This term from the script refers to a method used to convert negative angles in radians into a form that can be more easily manipulated. By multiplying the numerator and denominator of a fraction by the same number, the script turns negative eleven pi over four into eight pi over four, facilitating the process of finding a coterminal angle.

💡Positive Coterminal Angle

A positive coterminal angle is an angle that is coterminal with a given angle but falls within the positive range of 0 to 360 degrees or 0 to 2π radians. The script provides examples of finding positive coterminal angles, such as converting 430 degrees to 70 degrees, which is positive and within the desired range.

Highlights

Coterminal angles are angles that land on the same position but have different measures.

For example, 30 degrees and 390 degrees are coterminal because they point to the same terminal ray.

Coterminal angles differ by a full rotation of 360 degrees.

Adding 30 degrees with 360 degrees gives the coterminal angle 390 degrees.

150 degrees and negative 210 degrees are coterminal angles that can be found by subtracting 150 by 360.

To find a coterminal angle, add or subtract 360 degrees.

In radians, coterminal angles can be found by adding or subtracting 2π.

A positive coterminal angle between 0 and 360 degrees for 430 degrees can be found by subtracting 430 by 360.

For a negative angle like -150 degrees, add 360 to find the coterminal angle 210 degrees.

To find a coterminal angle for 800 degrees, subtract 360 twice to get 80 degrees.

For a negative angle like -500 degrees, add 360 twice to get the coterminal angle 220 degrees.

For angles in radians, subtract by 2π to bring it between 0 and 2π.

The coterminal angle for thirteen pi over six is found by subtracting two pi, resulting in pi over six.

For a negative angle in radians, such as negative 11 pi over four, add two pi to find the coterminal angle.

The coterminal angle for negative 11 pi over four is five pi over four after adding two pi twice.

Transcripts

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Coterminal Angles In Radians & Degrees - Basic Introduction, Trigonometry

Takeaways

Q & A

What are coterminal angles?

How do you identify if two angles are coterminal?

Can you give an example of coterminal angles?

What happens when you add 360 degrees to an angle?

How do you find a coterminal angle within a specific range?

What is the relationship between degrees and radians when finding coterminal angles?

Can negative angles be coterminal with positive angles?

How do you find a positive coterminal angle for a given angle greater than 360 degrees?

What is the process for finding a coterminal angle for a negative angle less than 0 degrees?

Can you provide a step-by-step method for finding coterminal angles in radians?