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Angle Measurement by Myron Berg Dickinson State University Abstract

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Angle Measurement By Myron Berg Dickinson State University abstract • This PowerPoint deck was created for a presentation. • I discuss some ways other than degrees or radians in which angles are measured • Instead of using a radius of 1 to create radians, I discusses the results of using 1 as the circumference or area of a circle to create an alternative unit of measure. Then I examined whether these units have any advantage, including greater simplicity, when compared to the values that result from radians. Goals • Provide background information for instruction about angle measurement • Cover some strategies for helping students understand radian measure Overview • History of angle measurements • Common methods used to measure angles today • Cyclical angles • Advantages of using radians to measure angles • Helping students remember angle measurement values The Unit Circle • Expecting it to be simple – but can be a struggle Angle measurement strategies Slope Examples of grade: Grade % not evenly spaced Degrees vs Grade 50 45 40 35 30 25 20 degrees 15 10 5 0 0 20 40 60 80 100 120 grade Angles formed by repeated bisection • 256 units would be closest to 360 Compass Points use 16 divisions Wind Directions Using a scale with 0 and 360 being north Degrees Circular Reasoning to describe Circles • Why is there 360 degrees in a circle? • Because 4 right angles make a circle, and each right angle is 90° • Why is there 90 degrees in a right angle? • Because there are 360 degrees in a circle and a right angle is ¼ of a circle 360° • Reason for using 360 is unknown • 365 days in a year could be explanation • Babylonians divided circle using the angle of an equilateral triangle, then subdivided that using their sexagesimal numeric system • They inherited sexagesimal from Sumerians, who developed it around 2000 B.C. • 360 has 24 divisors (1-10 except 7) • Also occurred in ancient India • Combination of both reasons (365 days and 60 as a useful number) may be possible • Babylonians used chords for trigonometry (1/60 of a radius used as a chord forms a degree) 60 minutes were used as divisions of a degree • 60 is the smallest number divisible by 1-6, 10, 12, 15, 20, and 30. • Analog clocks and globe divisions came from Babylonians • 60 was used to divide minutes into seconds for clocks and angles 12 hours • Egyptians used sundials with duodecimal system • 12 lunar cycles per year • 12 joints on each hand excluding thumb • Hours were shorter in winter • angle bisection could produce 256 or 512 units instead of 360 • 360 provides the balance between having to use decimals and having to use 3 digit numbers. Other Angle Measurement Systems Vector angles • The sides have fixed values, even though represented by arrows. • Greeks pictured angles with infinitely long rays. Azimuth – another way North is over here Cyclical angle measurements • "We're going to turn this team around 360 degrees." - Jason Kidd, upon his drafting to the Dallas Mavericks Radians vs Degrees • Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°). • Radians are more abstract, pi is abstract • Pi contains every number pattern in it. For example, “12345” would be found in it somewhere. • If you know the radian measure of an angle, simply multiplying it by the radius gives the arc length. • Most practical (applied) uses of angles don’t care about the length. • Finding the area of a sector is about the same difficulty with radians or with degrees. • Degrees are traditionally used in navigation, surveying, and many applied disciplines. • radians are more common in mathematics and mathematical physics. Quiz 1. The illustration above is an illustration of the size of a) 1 degree b) 1 radian c) 1 percent of a full circle d) 1 percent of a right angle e) 1 gradian f) 1% grade 2. 1 radian is about _________ degrees. 3. T F The slope of the top ray (of the radian from #1) is greater than 1. Another way to illustrate a radian Important angle sizes: • Half a square can be a 45-45-90 triangle • Half an equilateral triangle can be a 30-60-90 triangle 45 degree or 30 and 60 degree angles: Oneness of radians – based upon a radius of 1 Other Alternatives using “1” in a different way and the angles they produce: Slope (Grade) of ‘1’ Circumference of ‘1’ • Radius of 1 • Circumference of 1 100 parts - 100% • cen·ti·grade (C) (sen'ti-grād) 1. Basis of an earlier temperature scale in which 100° separates the melting and boiling points of water. See: Celsius scale 2. One hundredth of a circle, equal to 3.6 degrees of the astronomic circle. http://medical-dictionary.thefreedictionary.com/centigrade (bergians) – made up name using circumference of 1. • 2휋푟 = 1 1 • 푟 = 2휋 • 푎2 + 푏2 = 푐2 1 2 • 푎2 + 푎2 = 2휋 1 • 2푎2 = 4휋2 1 • 푎2 = 8휋2 1 • 푎 = 2 2 휋 2 • 푎 = 4휋 2 1푏 • cos = 4휋 8 1 2휋 1푏 2 • cos = 8 2 1 bergian is 360 degrees • cos 0푏 = 1 1 푏 3 • cos = 12 2 1푏 2 • cos = 8 2 1푏 1 • cos = 6 2 1푏 • cos = 0 4 Area of ‘1’ • Radius of 1 • Area of 1 (Jeromians) – made up name using area of 1 • 휋푟2 = 1 1 • 푟2 = 휋 1 • 푟 = 휋 휋 • 푟 = (rationalize the denominator?) 휋 • 푎2 + 푏2 = 푐2 2 휋 • 푎2 + 푎2 = 휋 휋 • 2푎2 = 휋2 1 • 푎2 = 2휋 1 • 푎 = 2휋 2휋 • 푎 = 2휋 2휋 1푗 • cos = 2휋 8 휋 휋 1푗 2 • cos = 8 2 1 Jeromian is 360 degrees. • cos 0푗 = 1 1 푗 3 • cos = 12 2 1푗 2 • cos = 8 2 1푗 1 • cos = 6 2 1푗 • cos = 0 4 Right angle as “1” • Gradians divide a right angle into 100 units. • There are 400 gradians in a circle. Gradians (sometimes also called grade) Gradians have been removed from newer calculators Disadvantages of gradians • 30 degrees is 33.3ത gradians • 60 degrees is 66.6ത gradians • 45 degrees is 50 gradians Another possibility: Circumference = 4 (Gradians divided by 100) Myronians Gradians (Myronians) – made up name • 2휋푟 = 4 4 2 • 푟 = = 2휋 휋 • 푎2 + 푏2 = 푐2 2 2 • 푎2 + 푎2 = 휋 4 • 2푎2 = 휋2 2 • 푎2 = 휋2 2 • 푎 = 휋 2 1푀 • cos = 휋 2 2 휋 1푀 2 • cos = 1 Myronian is 90 degrees. 2 2 • cos 0푀 = 1 1푀 3 • cos = 3 2 1푀 2 • cos = 2 2 2푀 1 • cos = 3 2 • cos 1푀 = 0 Value of having the hypotenuse (radius) of 1 (Radians) • 푟 = 1 • 푎2 + 푏2 = 푐2 • 푎2 + 푎2 = 1 2 • 2푎2 = 1 1 • 푎2 = 2 2 • 푎 = 2 휋푀 2 • cos = 4 2 Degrees vs Radians in 3d Steradian Square Degree Steradians (square radians) vs Square Degrees • A steradian is the area that the portion of the sphere occupies, divided by the square of the radius of the sphere. • A total sphere is 4 pi steradians. • Astronomers often use square degrees. • Analogous to one degree being equal to π/180 radians, a square degree is equal to (π/180)2 steradians or about 1/3283 of a sphere. Square Degree • The full moon covers only about 0.2 deg2 of the sky when viewed from the surface of the Earth. The Moon is only a half degree across (i.e. a circular diameter of roughly 0.5 deg), so the moon's disk covers a circular area of: π × (0.5°/2)2, or 0.2 square degrees. • Viewed from Earth, the Sun is roughly half a degree across (the same as the full moon) and covers only 0.2 deg2 as well. • Those studying the sky calculate square degrees the same way someone on Earth figures out the area of a rectangle or a circle. If a sky object measures 1° by 2°, it’s like a box measuring 1 foot by 2 feet. The areas are 2 square degrees and 2 square feet, respectively. And while you can designate the latter 2 ft2, there’s no easy abbreviation for square degrees. Reduce memorization by looking for patterns Simplicity as a goal? • People still choose complexity over simplicity when given a choice. This is a sad fact of life, so get over it. (probably not true for students) • "There are two ways of constructing a software design; one way is to make it so simple that there are obviously no deficiencies, and the other way is to make it so complicated that there are no obvious deficiencies. The first method is far more difficult." - C. A. R. Hoare Square roots Place these in order from largest to smallest 3 _ 1 degree 4 _ 1 gradian 2 _ 1 radian 5 _ 1% grade 6 _ 1% of a circle 1 _ 1 bergian Summary • History of angle measurements • Common methods to measure angles today • Cyclical angles • Advantages of using radians to measure angles • Helping students remember angle measurement values.
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