52 CHAPTER 2. FUNCTIONS
far its sides are extended. This is how to define the radian measure of an angle and thedefinition is well-defined. Thus, in particular, the ratio between the circumference (length)of a circle and its radius is a constant which is independent of the radius of the circle1.Since the time of Euler in the 1700’s, this constant has been denoted by 2π . In summary, ifθ is the radian measure of an angle, the length of the arc subtended by the angle on a circleof radius r is rθ .
So how do we obtain the length of the subtended arc? For now, imagine taking a string,placing one end of it on one end of the circular arc and then wrapping the string till youreach the other end of the arc. Stretching this string out and measuring it would then giveyou the length of the arc. Later a more precise way of finding lengths of curves is given.
Definition 2.3.1 Let A be an angle. Draw a circle centered at A which intersectsboth sides of the angle. The radian measure of the angle is the length of this arc divided bythe radius of the circle.
Al
r
Thus the radian measure of A is l/r in the above. (Note that the radian measure of anangle does not depend on units of length. )There is also the wrong way of measuringangles. In this way, one degree consists of an angle which subtends an arc which goes1/360 of the way around the circle. The measure of the angle consists of the number ofdegrees which correspond to the given angle.
We avoid the wrong way of measuring angles in calculus. This is because all the theo-rems about the circular functions having to do with calculus topics involve the angle beinggiven in radians.
In any triangle, the sum of the radian measures of the angles equals π . Lets review whythis is so.
Consider the following picture.
α
α
β
γ
γ
The line at the top is chosen to be parallel to the line on the base. Then from axiomsof geometry about alternate interior angles, the diagram is correctly labeled. Now if youconsider the angle formed by a point on a straight line, then it is obvious that the circle
1In 2 Chronicles 4:2 the “molten sea” used for “washing” by the priests and found in Solomon’s temple isdescribed. It sat on 12 oxen, was round, 5 cubits high, 10 across and 30 around. This was very large if you believewhat it says in Chronicles. A cubit is thought to have been about 1.5 feet. It is remarkable how much water wascalled for in their rituals. Their sacrifices also required a great deal of wood to burn up dead animals. This templealso exceeded the efficiency of a modern meat packing plant on some special occasions, according to the Bible.
Thus, from the above, the Bible gives the value of π as 3. This is not too far off and is much less pretentiousthan the Indiana pi bill of 1897 which attempted to legislate a method for squaring the circle. A better value is3.1415926535 and presently this number is known to millions of decimal places. It was proved by Linderman in1882 that π is transcendental which implies that it is impossible to construct a square having area π using onlycompass and unmarked straight edge (squaring the circle).