REFERENCE ANGLES || PRE-CALCULUS

WOW MATH
1 Dec 202105:53
EducationalLearning
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TLDRThis instructional video lesson delves into the concept of reference angles, which are the positive acute angles formed by the terminal side of an angle with the x-axis. The tutorial explains how to calculate reference angles for angles in different quadrants, providing step-by-step examples. It covers angles in quadrant one (0 to 90 degrees), where the reference angle is the angle itself, quadrant two (90 to 180 degrees), where you subtract from 180, quadrant three (180 to 270 degrees), where you subtract from 180, and quadrant four (270 to 360 degrees), where you subtract from 360. The video also demonstrates how to handle negative angles by adding 360 degrees to find the reference angle. The examples given include 42 degrees, 221 degrees, 113 degrees, and -410 degrees, illustrating the process clearly. The lesson aims to educate viewers on finding reference angles in a concise and engaging manner.

Takeaways
  • πŸ“š The video lesson is about understanding reference angles in trigonometry.
  • πŸ“ A reference angle is the positive acute angle formed by the terminal side of a given angle and the x-axis.
  • πŸ”’ Reference angles are always between 0 and 90 degrees.
  • πŸ‘‰ For angles in quadrant one, the reference angle is the angle itself if it's acute.
  • βœ‚οΈ For angles in quadrant two, the reference angle is found by subtracting the given angle from 180 degrees.
  • πŸ”„ In quadrant three, there are no positive acute angles, so you subtract the given angle from 180 degrees to find the reference angle.
  • πŸ“‰ For quadrant four, the reference angle is calculated by subtracting the given angle from 360 degrees.
  • πŸ“Œ The example given for 42 degrees shows that since it's an acute angle, the reference angle is 42 degrees itself.
  • πŸ“‰ For 221 degrees, which is in quadrant three, the reference angle is calculated as 221 - 180 = 41 degrees.
  • πŸ“‰ For 113 degrees in quadrant two, the reference angle is found by 180 - 113 = 67 degrees.
  • πŸ”„ For negative angles like -410 degrees, you add 360 degrees to find the reference angle, which is |-410 + 360| = 50 degrees.
Q & A

  • What is a reference angle?

    -A reference angle is the positive acute angle formed by the terminal side of a given angle and the x-axis, ranging from 0 to 90 degrees.

  • How do you find the reference angle for an angle in quadrant one?

    -For an angle in quadrant one, which is an acute angle, the reference angle is the angle itself.

  • What is the process to find the reference angle for an angle in quadrant two?

    -For an angle in quadrant two, you subtract the given angle from 180 degrees to find the reference angle.

  • Are there positive acute angles in quadrant three?

    -No, there are no positive acute angles in quadrant three because all angles in this quadrant are obtuse.

  • How do you determine the reference angle for an angle in quadrant four?

    -For an angle in quadrant four, you subtract the given angle from 360 degrees to find the reference angle.

  • What is the reference angle of 42 degrees?

    -Since 42 degrees is an acute angle, the reference angle is 42 degrees itself.

  • How do you calculate the reference angle for 221 degrees?

    -For 221 degrees, which is in quadrant three, you subtract 180 degrees from 221 degrees to get a reference angle of 41 degrees.

  • What is the reference angle for 113 degrees?

    -For 113 degrees, which is in quadrant two, you subtract the given angle from 180 degrees, resulting in a reference angle of 67 degrees.

  • How do you find the reference angle for a negative angle like -410 degrees?

    -For a negative angle, you add 360 degrees to the given angle, then take the absolute value. For -410 degrees, it becomes 50 degrees, so the reference angle is 50 degrees.

  • What is the significance of finding reference angles in trigonometry?

    -Reference angles are significant in trigonometry because they allow for the simplification of trigonometric functions by using the sine and cosine values of the reference angle, regardless of the quadrant in which the original angle lies.

Outlines
00:00
πŸ“ Understanding Reference Angles

This paragraph introduces the concept of reference angles, which are the positive acute angles formed by the terminal side of a given angle and the x-axis. It explains that reference angles are always between 0 and 90 degrees and are used to determine the angle's location in a specific quadrant. The video provides a step-by-step method for finding reference angles for angles in different quadrants: subtracting from 180 degrees for quadrant two, and from 360 degrees for quadrant four. Examples are given for angles of 42 degrees, 221 degrees, and 113 degrees, demonstrating how to calculate their reference angles.

05:03
πŸ”’ Working with Negative Angles

The second paragraph focuses on how to deal with negative angles when finding reference angles. It explains that for negative angles, you add 360 degrees to get the absolute value before calculating the reference angle. An example is given for a negative 410-degree angle, showing that after adding 360 degrees, the angle becomes -50 degrees. The absolute value of -50 degrees is 50 degrees, which is then identified as the reference angle. The paragraph concludes with a reminder to watch the illustration provided in the video for a visual understanding of the concept.

Mindmap
Keywords
πŸ’‘Reference Angle
A reference angle is the positive acute angle that is formed by the terminal side of a given angle and the x-axis. It is a fundamental concept in trigonometry when dealing with angles outside the standard position. In the video, reference angles are used to simplify the analysis of angles in different quadrants, as they allow us to focus on the acute angle formed regardless of the angle's position.
πŸ’‘Acute Angle
An acute angle is an angle that is less than 90 degrees. It is one of the basic classifications of angles, and in the context of the video, acute angles are important because the reference angle is always an acute angle. The script mentions that if the given angle is acute, as in the case of 42 degrees, the reference angle is the angle itself.
πŸ’‘Quadrant
In the context of the video, a quadrant refers to one of the four equal parts of a Cartesian plane, each formed by the intersection of the x-axis and y-axis. The quadrants are numbered counterclockwise starting from the upper right, and they are essential in determining the reference angle of a given angle based on its position. For example, angles between 0 and 90 degrees are in quadrant one, while angles between 90 and 180 degrees are in quadrant two.
πŸ’‘Terminal Side
The terminal side of an angle is the half-line starting from the origin and passing through the point that corresponds to the angle's measure. It is used to define the position of an angle in standard position. In the video, the terminal side is mentioned in relation to forming the reference angle with the x-axis.
πŸ’‘Positive Acute Angle
A positive acute angle is an acute angle that is measured in the counterclockwise direction from the x-axis. The video emphasizes that the reference angle is always a positive acute angle, which is crucial for determining the reference angle regardless of the quadrant in which the original angle lies.
πŸ’‘Degrees
Degrees are a unit of measurement for angles, with a full circle being 360 degrees. The video uses degrees to express the measure of angles and to calculate reference angles by subtracting from 90, 180, 270, or 360 degrees depending on the quadrant of the given angle.
πŸ’‘Cartesian Plane
The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane that defines each point by a pair of coordinates, usually denoted as (x, y). It is the setting where angles are measured in relation to the x-axis and y-axis. The video discusses angles in the context of the Cartesian plane, particularly in relation to their quadrants.
πŸ’‘Absolute Value
The absolute value of a number is its non-negative value, regardless of its sign. In the video, the absolute value is used when dealing with negative angles to find their reference angle. For instance, the script mentions converting -410 degrees to its absolute value, which is 50 degrees, by adding 360 degrees to it.
πŸ’‘Subtract
Subtraction is a mathematical operation used to find the difference between two numbers. In the context of the video, subtraction is used to calculate the reference angle by subtracting the given angle from 180 or 360 degrees, depending on the quadrant in which the angle lies.
πŸ’‘Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right-angled triangles. The video is a trigonometry lesson that focuses on understanding and calculating reference angles, which are essential for solving problems involving trigonometric functions.
πŸ’‘Video Tutorial
A video tutorial is an instructional video that teaches a specific skill or concept. In this case, the video tutorial is focused on teaching the concept of reference angles in trigonometry. The script provides step-by-step instructions and examples to help viewers understand how to find reference angles for various given angles.
Highlights

Introduction to reference angles and their definition.

Explanation of the positive acute angle formed by the terminal side of a given angle and the x-axis.

Clarification that reference angles are between 0 and 90 degrees.

Method to find the reference angle in quadrant two by subtracting from 180 degrees.

Description of quadrant three having no positive acute angles and the method to find reference angles there.

Process to determine reference angles in quadrant four by subtracting from 360 degrees.

Example calculation of the reference angle for 42 degrees, which is the angle itself.

Example calculation for 221 degrees, illustrating the reference angle in quadrant three.

Explanation of how to find the reference angle for 113 degrees in quadrant two.

Demonstration of adjusting negative angles by adding 360 degrees and finding the reference angle for -410 degrees.

Final reference angle calculation for -410 degrees, resulting in 50 degrees.

Illustration of the reference angle concept with visual examples.

Encouragement to like, subscribe, and hit the bell for more video tutorials.

Closing remarks thanking viewers for watching and promoting further learning.

Transcripts

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Reference AnglesMathematicsEducationalTutorialAcute AnglesQuadrantsGeometryTrigonometryLearningVideo Lesson