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. Hi Ranjay I have added an explanation of the inequality at the end of the post. Thanks for writing; you were probably not the only one who was wondering about this.
Yes, if one were to use anything other than radian measure than one would get constants popping up upon differentiating trig functions and once they appeared these constants would mestasize. For the same reason one uses e as exponential base rather than the seemly more simple choice of say This must be nipped in the bud. In elementary geometry where one is not using calculus the use of degrees or grads is perfectly OK. The radian is a linear measure of an angle, and it works best if you have your delta x and delta y in the same units.
Degrees are a made up unit. Try plotting a sine wave using radians vs degrees for the x-axis. The y-axis will be the amplitude. A linear measurement. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email.
Notify me of new posts via email. This site uses Akismet to reduce spam. Learn how your comment data is processed. To develop the derivative of the sine function you first work with this inequality At the request of a reader I have added an explanation of this inequality at the end of the post : From this inequality you determine that The middle term of the inequality is the area of a sector of a unit circle with central angles of radians.
Who needs that? Do your calculus in radians. Then the area of triangle OAB is 2. Create a free Team What is Teams? Learn more. How to know when to put calculator in radian or degree mode? Ask Question. Asked 4 years, 7 months ago. Active 3 years, 8 months ago. Viewed k times. Round to four decimals places. Ria Car Ria Car 21 1 1 gold badge 1 1 silver badge 3 3 bronze badges. It seems to me that there might be more fundamental problem than typing a value into calculator.
Show 6 more comments. Active Oldest Votes. If there is no degree symbol, then use radian mode. Add a comment. Stella Biderman Stella Biderman This makes it easy to talk about the size of an angle. Since there are degrees in a whole circle, a right angle will be one-quarter as many degrees — i. Most people learn this system so well, and at such a young age, that it becomes second nature.
What this familiarity disguises is that the number is totally arbitrary, chosen simply because the Babylonians preferred multiples of Why not divide the circle into pieces, or 5 pieces, or pieces? No more dividing the circle into some arbitrary number of units. See how it forms an angle? 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.
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