8.5
Trigonometric parameterization of a circle.
There is another well known parameterization of a circle which is used in the definition of the trigonometric functions sin(t), cos(t). Suppose we take a circle of radius 1 centered at the origin:x2+ y2 = 1 A circle of radius 1 is called a unit circle.
We wish to arrange a parameter t such that given any point P on the unit circle, there is a certain value of t associated with it and given any value of the parameter t, there is a unique point P (t) associated with it.
Here is a very simple idea. Start with the point U(1, 0) on our circle. Given any real number t, walk t units along the circle in the counter clockwise direction and stop. Call the resulting point P (t). Let us see what this means.
There is a famous number called π which is defined as the ratio of the circumference of a circle to its diameter. Since our circle is a unit circle, its diameter must be 2 and hence by definition, its circumference is 2π.6
• Thus, if we take t = π/2 then we have gone a quarter circle and reach the point (0, 1); we say P (π/2) = (0, 1).
• If we go half the circle, we take t = π and P (π) = (−1, 0).
• If we further go another quarter circle then we reach the point P (3π/2) = (0, −1) and finally, if we go a complete circle, we come back to the starting point P (2π) = (1, 0).
This means that if we intend to travel a length t around the circle and t = 2π + s then the position P (t) at which we ultimately arrive will be the same as if we travel s = t − 2π. We express this symbolically as P (t) = P (s) = P (t − 2π)
Thus we conclude that to determine the position P (t) we can always throw away multiples of 2π until we get a “remainder” which lies between 0 and 2π. Formally, we can describe this as follows:
We write our
t = 2π(n) + s where 0 ≤ s < 2π and n is some integer.
Indeed, this is very much like our old division, except we have now graduated to dividing real numbers by real numbers!
6This π has a very interesting history and has occupied many talented mathematicians over the
ages. Finding more and more digits of the decimal expansion of π is a challenging mathematical problem. There are many books and articles on the subject of π including a movie with the same title! People have imagined many mysterious properties for the decimal digits of this real number, which is proven to be transcendental. This means that there is no non zero polynomial p(x) with integer coefficients such that p(π) = 0.
Then we can march s units along the unit circle counter clockwise from the point U(1, 0) and reach a point P (s) which, by definition, is also the point denoted as P (t). So for any number t we have an associated point P (t) on the unit circle. As the figure indicates there is uniquely associated to P (t) an angle made by the ray −→OP with the starting ray −→OU. We denote this angle by the symbol 6 UOP.
• This allows us to think of the number t as the measure of an angle 6 UOP;
namely the resulting angle made by the ray −→OP with the ray −→OU.
• This measure is said to be in units called “radians”. The explanation for using this term is probably as follows.
If we take a circle of radius r then the circumference of the circle, by definition, is 2πr, so the ratio distance along the arcradius is 2π. Thus the measure t can be thought of as a measure in units of “radiuses” (or should it be radii?).
• By an abuse of the notation, we may simply call t to be the angle 6 UOP , in
radians. Since it is the number t that is used in our formulas, this should not cause confusion.
However, if you are working with geometry itself, it is customary to declare t = m6 UOP , i.e. t is the measure of the angle 6 UOP .
• The possible “numerical” angles (numbers of radians) for the same locator point are infinitely many, so for a given angle of measure t we usually use the remain- der s of t after throwing away enough multiples of 2π as the radian measure of the angle. Thus, we take the liberty to say things like “This angle is y radi- ans.” when of course we know that the angle itself is a geometric figure and “y radians” is simply its measure.
Thus for the point B(0, 1) the angle of the ray −−→OB is said to be π/2 radians. • We reemphasize that we can either march off the t units directly or march off
the s units where s is the remainder of t after dividing by 2π as explained above. If we choose to march off t units directly, then a negative t requires us to march in the clockwise direction instead of the counter clockwise direction (also called anti clockwise in some books.) 7
• We formally define P (t) to be the locator point of the “angle” t (in radians).
7It is not clear why those great old mathematicians wanted “clockwise” to be the negative di-
rection, what did they have against clocks anyway? This convention of positive counter clockwise direction, however, is now firmly entrenched in tradition!
8.5. TRIGONOMETRIC PARAMETERIZATION OF A CIRCLE. 123
U
Angle UOP is arc UP P(t)
O
• A more familiar unit: degrees.
Even though this description is pleasing, it is difficult to work with, since the number π itself is such a mystery. Even a fancy calculator or a great big com- puter can only approximate the number π and hence any such calculations are basically imprecise! Then why do we still want to mention and use this number? Because, in mathematical work and especially Calculus, it is important to work with radians.
However, for our simpler application, it is more convenient to use the usual measure called degrees.
• We simply divide the circle into 360 equal parts and call the angle corresponding to each part a degree. Thus our point B(0, 1) corresponds to the angle of
360
4 = 90 degrees and it is often written as 90◦.
Here is a picture of the circle showing the degree measures for some important points. 270o 90o 180o 0o 30o 45o
• It is well worth making a simple rule to convert between radian and degree measures.
To convert radians to degrees, simply multiply by 180π .
To convert from degrees to radians, reverse the procedure and multiply by π180. We now review how the locator point P (t) is determined t is given in degrees.
We write:
t = 360n + s where 0 ≤ s < 360 and n is some integer.
Thus s is the remainder of t after division by 360 and we mark off s degrees in our circle (in the counter clockwise manner) to locate the point P (t) = P (s). Thus, for t = 765 we see that t = 360(2) + 45 so s = 45 and the corresponding point is the point at the tip of the segment making the 45 degree angle.
Note that for t = −1035, we get the same s since −1035 = 360(−3) + 45. For the reader’s convenience we present a table of the corresponding angle mea- sures of the marked points in our picture.
degrees 0 30 45 90 180 270
radians 0 π/6 π/4 π/2 π 3π/2
• In the remaining work, we shall use the degree notation◦ when we use degrees,
but when we mean to use radians, we may skip any notation or mention of units!
Definition of the Trigonometric Functions.
Given an angle t, let P (t) be its locator point. We know its x and y coordinates. Write
P (t) = (x(t), y(t)).
Then the definition of the trigonometric functions is simply: sin(t) = y(t) and cos(t) = x(t).
Before proceeding, let us calculate a bunch of these values. Some trigonometric values.
1. Find the values of cos(0), sin(0).
Answer: When the angle 6 UOP is 0, our point P is the same as U(1, 0). So
by definition,
cos(0) = 1, sin(0) = 0. 2. Find the values of cos(90◦), sin(90◦).
Answer: We already know this point P (90◦) to be (0, 1), so we get:
cos(90◦) = 0, sin(90◦) = 1.
8.5. TRIGONOMETRIC PARAMETERIZATION OF A CIRCLE. 125
cos(180◦) = −1 , sin(180◦) = 0
and
cos(270◦) = 0 , sin(270◦) = −1. 3. Find the values of cos(30◦), sin(30◦).
Answer: This needs some real work! We shall give a shortcut later.
Let us make a triangle △OMP where P is the point at 30◦ and M is the foot
of the perpendicular from P onto the segment OU.
We show a picture below where we have added the midpoint H of OP and the segment HM joining it with M.
H
P 30o
O
U M
A little geometric argument can be used to deduce that 8
d(H, O) = d(H, M) = d(H, P ) and then the triangle △HMP is equilateral. This says that
d(M, P ) = d(H, P ) = 1
2d(O, P ) = 1 2. Thus, we have given a brief idea of how to figure out that
sin(30◦) = 1
2.
8Here is a hint of the geometric argument. If we draw a circle centered at H and having OP as
diameter, then we know from our work with the circles that the point M is on this circle, since the angle between OM and P M is known to be 90◦. Therefore, all the distances are equal to the radius
How to find cos(30◦)? We can now use simple algebra using the fact that the
point P (30◦) is on the circle x2+ y2 = 1. So,
cos(30◦)2+ (1
2)
2 = 1
and it is easy to deduce that
cos(30◦)2 = 1 − 1 4 = 3 4. It follows that cos(30◦) = ±r 3 4 = ± √ 3 2 .
Which sign shall we take? Clearly, from the picture, the x-coordinate is positive, so we get
cos(30◦) =
√ 3 2 .
Bonus thought! What do we get if we take the minus sign? After a little thought, you can see that the angle of 180 − 30 = 150 degrees will have the same sin, namely 12 and its cos will be indeed −√23.
Similarly, for the angle −30◦ which is also the angle 330◦, you can see that
sin(−30◦) = −1 2 and cos(−30 ◦) = √ 3 2 . 4. Find the value of sin(13◦), cos(13◦).
Answer: There is no clever way for this calculation. All we can hope to do is to use a calculator or computer, or develop a formula which can give us better and better approximations.
A calculator will say:
sin(13◦) = 0.2249510544 and cos(13◦) = 0.9743700648.
Indeed, for most concrete values of angles, you may end up just using calculator for evaluating these functions.