This raises a question: how many radians are there in a circle? For a better — but still imperfect — approximation, try this. Knowing this, we can now convert between radians and degrees — just as we can convert between miles and kilometers, or Fahrenheit and Celsius. Radians become a perfectly valid, usable measure of angles. You want to know: What was wrong with degrees? Degrees are warm, friendly, familiar. Why ditch them in favor of this bizarre radian?
But Trigonometry marks a turning point in math, when the student lifts his gaze from the everyday towards larger, more distant ideas.
You begin exploring basic relationships, deep symmetries, the kinds of patterns that make the universe tick. Just like you, I learned to speak Babylonian long before I encountered radians. And for years, Babylonian remained my native tongue — to give an angle in radians first required an act of mental translation. The explanation put forward here is an excellent dissertation of the rationale for radians versus degrees. Consider trying to perform calculus, differential equations etc. The calculations would be quite cumbersome if the variable was in base 60 and the rest of the values were real numbers base By necessity the numbers would require base agreement to make any sense.
In other words, all non angular values would need to be converted to base 60 or all angular values would need to be converted to base By assuming radian measure we introduce a real valued variable base 10 that is in base agreement with all other values in the expression both numeric and angular as well as constant and variable. Essentially it is this base agreement that allows for the performance of the principles and formulas of calculus and differential equations to be performed as described in any number of textbooks, papers, etc.
Excellent Article. You have made this read interesting and informative. Thank you. Radians are a form of tyranny laid upon us. Radians are irrational, both mathematically and otherwise. Your lies to attempt to deceive me into thinking in radians was logically flawed and tedious, you have irked me. Just like solving a equation with a matrix is idiotic, so is measuring angles in radians.
Long story short, I needed a refresher on radians vs degrees. Being 3 on my Google search, along with the obvious promise of humor, here I am. These same equations work for points in other quadrants and for angles with a negative measure i. Calculations may be done in degrees or radians as long as the calculator or computer you are using is set in the proper mode.
If your calculator or computer is set for degrees, there is no need to convert to radians. If your angles are measured in degrees, use degree mode on your device. If your distance is Your examples are correct, but unduly long. There is no need to do things step by step; any calculator or computer or spreadsheet will do the work for you. That is, there is no need to change back and forth between degrees and radians. With your calculator in degree mode: and.
With your calculator in radian mode enter and. Either way, the point in polar form I hope this answers your questions. If not, write again. BTW I worked for my four years in college for a land surveyor. Work I very much enjoyed. Of course, back then we used optical transits theodolites reading the angles with a vernier scale and measured distance with a steel tape. Hello Lin, Thank you so very much for your fantastic reply. That was quite fun to do.
Baron Apparently Excel is set to work in radians. Leveling rods: I remember them. Real Of course by then our steel chains were feet, marked in decimal feet, not inches — some improvement.
You can buy real 66 foot chains on Ebay! I disagree with having the inequality with equal sign. Another approach is to graph sin x and x separately. If you then graph in degrees you will see the graphs and their slopes are quite different. Pingback: Why Radians? Reblogged this on The Maths Mann. If x is in degrees, then to differentiate a trig function you must change the degrees to radians. So with x in degrees with the argument now in radians. Then differentiate or returning the argument to degrees.
If you wanted to work entirely in degrees from the start, then the middle term of the inequality in the post would be using the formula for arc length with in degrees. Then the will work its way through the inequalities resulting in and from there into the derivative formulas. Try graphing with x in degrees and your calculator set to degree mode. In a square window that is, with equal units on both axes the graph will appear to be very flat — almost linear. Thus, you would expect the slopes derivatives to be much smaller than when working in radians.
Pingback: What are Radians? Where Do They Come From? Is this what you mean? With x in degrees you must change the argument to radians and then differentiate using the Chain Rule: This works the same way with any trig function.
That is want I meant. Hi Lin, I like the way you have explained for easy understanding. However, please expalain or give reference of the inequality you have used for the explanation. Ask Question. Asked 7 years, 7 months ago.
Active 5 years, 8 months ago. Viewed 36k times. Show 6 more comments. Active Oldest Votes. Warren Hill Warren Hill 2, 13 13 silver badges 24 24 bronze badges. Degrees are also unitless since they are a scalar multiple of a radian. Show 1 more comment. Andrew D. Hwang Andrew D. Hwang There are valid reasons but it isn't as clear-cut as it's sometimes made out to be. Show 7 more comments. Ben Grossmann Ben Grossmann k 12 12 gold badges silver badges bronze badges. This limit is computed using the squeeze theorem, and used to find the derivative.
I am claiming that if we define sin and cos to be the geometric functions on angles measured in radians, then all the nice analytic properties of sin and cos can be derived. If we chose a different unit for angles, we would get different analytic properties. Each of the properties I've given can be derived from the geometric definitions of sin and cos.
Of course you need a little calculus to make sense of this definition, but nothing circular other than, you know, the circle itself. Show 8 more comments. But how do we know how much is a radian? Add a comment. Then, connection between degree and radian become clear. Warbo Warbo 1 1 silver badge 7 7 bronze badges. Pi defines itself outside of a human's choice. If humans define the metre, second, Coulomb and kilogram, then we 'automatically' get definitions for Newtons, litres, Joules, Watts, Hertz, Bequerels, Amperes, Volts, etc.
The same happens with radians, except there are no degrees of freedom; as you say, "Pi defines itself". Aaron Meyerowitz Aaron Meyerowitz 1, 8 8 silver badges 12 12 bronze badges.
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