Trigonometry Radian measure and trigonometric functions on the reals The units for measuring angles tend to change depending on the context - in a geometry course, you re more likely to be measuring your angles in degrees; in a calculus course, it s more likely to be radians. Radian measure is useful for considering the trigonometric ratios as functions on the reals- in particular, it addresses some compatibility isssues. For example, the function f(x) =sinx doesn t really care if the input is in degrees or radians - it will produce the same output (on a corresponding scaled x). On the other hand, the function f(x) =x sin x is a bit problematic - the scale changes along the y as well.
More problematic is the dimension issue. A degree is essentially a fractional part of a circle [1 = 360 1 th of a circle]. Trig ratios, however, are dimensionless - the associated triangle lengths can be in meters, feet, inches, doesn t matter sin 30 = 1ft 2ft = 1m 2m = 1 2 So in real life applications involving measurement, having a trig ratio as a factor doesn t change the dimension of the answer- it s compatible with whatever quantity is being measured. However, if f(x) =x sin x is a description of some measured quantity, there s a problem if you use degrees - f(45 )=45 sin 45 = 1 2 45 The answer is in terms of a fraction of a circle. Now, under some circumstances, that may be the effect you want, but in most cases...no. So, the motivation for the use of radians is to have an angle measure that translates as a length (and is compatible with other lengths), not as a fractional part of a circle. Defining the radian Consider a circle of radius 1, a central angle θ, and an arc intercepted by θ: Say m θ = x. What is the length of the arc intercepted by θ? Saying that θ has measure x is simply saying that θ is proportionally of the intercepted arc is x 360 th of its circumference, or 360 x th of a circle. So, the length x 360 2r = x 360 2(1) = 180 x Therefore, in addition to being associated with a fraction of a circle, θ can be associated with the length of the arc it intercepts - providing another way to measure the angle. And this would be the radian measure of θ: x = 180 x rad
This form of measurement is used simply where it s more convenient - radian measure is a measure of length, compatible with other measures of length. In practice, radian measure is essentially dimensionless [ you could, for example, be working with a function of current, or charge, and the radian bit will be in whatever units you re measuring in]. It also makes a few formulas associated with circles convenient. Arc length of an intercepted arc of circle with radius r, m θ = x = 180 x rad : is conventionally expressed as L = x ( ) 360 2r = 180 x r L = rθ where θ is now the radian measure of θ. [Yes, I know, the interchangeability of the angle with its measurement is a sloppy notation convention, but it s typical.] Area of a sector of a circle with radius r, m θ = x = 180 x rad : A = x 360 r2 = 1 ( ) 2 180 x r 2 is conventionally expressed as A = 1 2 r2 θ where θ is now the radian measure of θ.
Moving along to functions... Using radian measure doesn t change the definitions of the trigonometric ratios- just imagine the triangle in a circle of radius 1. sin θ = a 1 cos θ = b 1 etc... On an xy coordinate system, you d see it like this: and can observe that the values of x and y vary with θ. So f(θ) =x =cosθ are functions on the reals. f(θ) =y =sinθ
At this point, we simply need to extend the definitions a bit more. We the trigonometric ratios defined on 0 <θ<90,now0<θ< 2, extended to take on values at 0 and 2, and extended for 90 <θ<180,now 2 <θ<; for these values sin θ =sin( θ) cos(θ) = cos( θ)...which makes sense in terms of an xy coordinate system - these values lie in the second quadrant, with x<0andy>0. So, one finishes extending: sin θ = sin(θ ) cos(θ) = cos(θ ) <θ< 3 2 3 sin θ = sin(2 θ) cos(θ) =cos(2 θ) 2 <θ<2...defines the functions as taking on the limiting values of x and y as an axis is approached:...and makes the whole thing periodic: sin0=0 cos0=1 sin 1 2 =1 cos1 2 =0 sin =0 cos = 1 sin 2 3 = 1 cos 2 3 =0 sin θ =sin(θ +2) cosθ =cos(θ +2) And there you go, trig functions on the reals. On a f(θ) vs. θ coordinate system, the values for x and y in the triangle both translate as function values on the graphs of cos θ and sin θ respectively. Of course, once you do this, the convention is to call θ by x (the new horizontal axis) and either sine or cosine by y (the new vertical axis) [ah, notation!].the demo of the triangle rolling out into the sine wave is a classic, and there s innumerable applets showing it - here s one for you: http://www.ies.co.jp/math/java/samples/graphsinx.html