Radians vs degrees

  1. why radians are used in calculus instead of degrees?
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,682
    Staff Emeritus
    Science Advisor

    Radians have the very nice property that [tex]lim_{x->0}\frac{sin x}{x}[/tex]= 1[/tex] and [tex]lim_{x->0}{1- cos x}{x}= 1[/tex] when x is in radians. As a result the derivative of sin x is cos x and the derivative of cos x is -sin x as long as x is in radians. That's not true if x is measured in degrees. If we insist upon using degrees the corresponding derivatives would be multiplied by [tex]\frac{180}{\pi}[/tex].


    That's the easy answer. A little deeper- we don't define sine and cosine, in "higher" mathematics in terms of right triangles at all: in a right triangle would have to be between 0 and 90 degrees and we want functions to be defined as many numbers as possible. One definition widely used is this: We are given an xy-coordinate system and the unit circle (the graph of the relation x2+ y2= 1). To find sin t and cos t (for t non-negative), measure around the circumference of the circle, counter clockwise, a distance t (if t< 0, measure clockwise a distance -t). The point at which you end has coordinates, by definition, cos t and sin t. ("by definition"- in other words, whatever the coordinates are, that is how we define cos t, sin t.)

    Notice that the variable t in that definition is not measured in degrees OR radians! It is a distance, not an angle. Unfortunately, calculators are designed by engineers, not mathematicians and engineers tend to think of sine and cosine in terms of angles ("phase angles" in electromagnatism have nothing to do with angles!). "Radians" are defined so that the radian measure of an angle is the same as the length of the arc on a unit circle.
     
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