2.3. CIRCULAR FUNCTIONS 51

For example, you could consider x2 +y2 = 1. Written more formally, the relation it definesis {

(x,y) ∈ R2 : x2 + y2 = 1}

Now if you give a value for x, there might be two values for y which are associated withthe given value for x. In fact y = ±√1− x2 Thus this relation would not be a function.

Recall how to graph a relation or more generally a relation. You first found lots ofordered pairs which satisfied the relation. For example (0,3),(1,5), and (−1,1) all satisfyy = 2x+3 which describes a straight line. Then you connected them with a curve.

2.3 Circular FunctionsFor a more thorough discussion of these functions along the lines given here, see my pre-calculus book published by Worldwide Center of Math. For a non geometric treatment, seemy book Single variable advanced calculus or for a different way, Pure Mathematics byHardy [19]. I much prefer methods which do not depend on plane geometry because withthis approach, many of the most difficult and unpleasant considerations become obviousand then one can use the machinery of calculus to discuss geometric significance insteadof relying so much on axioms from geometry which may or may not be well remembered.However, I am giving the traditional development of this subject here.

An angle consists of two lines emanating from a point as described in the followingpicture. How can angles be measured? This will be done by considering arcs on a circle.To see how this will be done, let θ denote an angle and place the vertex of this angle at thecenter of the circle. Next, extend its two sides till they intersect the circle. Note the anglecould be opening in any of infinitely many different directions. Thus this procedure couldyield any of infinitely many different circular arcs. Each of these arcs is said to subtendthe angle.

arc subtended by the angle

Take an angle and place its vertex (the point) at the center of a circle of radius r. Then,extending the sides of the angle if necessary till they intersect the circle, this determinesan arc on the circle which subtends the angle. If r were changed to R, this really amountsto a change of units of length. Think, for example, of keeping the numbers the same butchanging centimeters to meters in order to produce an enlarged version of the same picture.Thus the picture looks exactly the same, only larger. It is reasonable to suppose, based onthis reasoning that the way to measure the angle is to take the length of the arc subtended inwhatever units being used and divide this length by the radius measured in the same units,thus obtaining a number which is independent of the units of length used, just as the angleitself is independent of units of length. After all, it is the same angle regardless of how