Unit Circle: an Introduction
[23 Sep 2010]
I remember the first time I came across the idea of the "unit circle" and I was quite impressed. There was so much going on!
Let’s see what is involved in this very useful bit of mathematics. There are many items of math vocabulary and we can’t assume students will know all of them. It would be easy to get lost if you were uncertain of the meaning of just a few of these words. Each mathematical concept is in bold text.
We start with a point (call it O) and use it as the center of a circle, radius 1.
Next, we are going to overlay our circle with a Cartesian coordinate system (our familiar x- and y-axes), named after the French mathematician who devised it, Rene Descartes.
Next, we add a random point on the circle (0.9, 0.44) and label it P. The numbers in brackets are called the coordinates of the point and represent the distance along the x and y-axes to the point P.
We now draw the radius of the circle passing through the point P and drop the altitude from P down to where it meets the x-axis at point Q (0.9, 0), forming the triangle OPQ. We also measure the angle formed by the x-axis and the radius OP and it turns out to be 26.37°.
Line PQ has length 0.44 units and OQ has length 0.9 units.
Now, we can consider the trigonometric ratios involved in this example. The first one is the sine of the angle POQ (that is, the ratio of the length of the opposite side to the length of the hypotenuse of the right triangle).
Now, this is very neat. It means that in a unit circle, the y-value of a point on the circle we are interested in is equal to the sine of the resulting angle at the center.
Now, let’s look at the cosine case.
Once again we get a neat result. The x-value of our point on the circle is equal to the cosine of the angle at the center.
So we could now label point P as (cos 26.37°, sin 26.37°) or using our variable for the angle size in this case, P (cos θ, sin θ).
For many students, it’s a mystery how we can extend the trigonometric ratios for angles bigger than 90°, and why some of the trig ratios are positive in some quadrants and negative in others.
The unit circle helps us see why that is so. Since the point P is defined as (cos θ, sin θ), where θ is the angle subtended at the center, we can find the trigonometric ratios for angles bigger than 90°.
If we move our point P around the circle from the first to the 2nd quadrant, to the point, say (−0.5, 0.87), this is what we get:
This tells me the sine of 120° is 0.87 (the y-value) and the cosine of 120° is −0.5 (the x-value). (You can check these on your calculator.)
Note that in the first quadrant, both sine and cosine (and therefore tangent) were all positive.
In the second quadrant, sine was positive, cosine was negative (and so tangent would be negative, too).
So if sine is positive and cosine is negative, then tangent will be negative.
Extending the Unit Circle
Of course, we can extend the concepts above for any point and any circle with center (0, 0). All we need to do is change the radius and we can find the trigonometric equivalent for any point.
So for example, if the radius is 8 units, a point on the circle in the 3rd quadrant could be (−2.74, −7.52). The angle this point makes with the positive x-axis is 200°.
This time, we have:
8 sin 200° = −2.74 (negative)
8 cos 200° = −7.52 (negative)
Sine and consine are both negative in the 3rd quadrant (so tangent is positive).
The unit circle is an interesting concept that ties together several important mathematical ideas, such as Euclidean geometry, coordinate geometry, and trigonometry.