Trigonometry by the unit circle

We have defined the three most basic trigonometric functions, and we’ve looked at a few facts about these functions. Let’s do a quick review. Given our triangle, we’ve defined the functions tangent theta sine and cosine, as you know. We can do certain things, just right from the definition, pretty easily. For example, the tangent of theta is equal to the sine of theta divided by the cosine of theta. That’s clear right from the definitions that we have in front of us. Something very important is the fact that for any angle theta, sine squared theta plus cos squared theta equals 1.

Easy to see because, if you put in the definitions of sine, for example, being a over h and cosine b over h, then when you work out the sum of the squares, you get 1 by the theorem of Pythagoras. For that reason, this identity, as it’s called, the fact that sine squared plus cos squared equals 1, is sometimes called the Pythagorean identity, or sometimes the circular identity. Now the complementary angle to theta is 90 minus theta. Two angles are complementary when they add up to 90 degrees. And we have complementarity formulas. The cosine of the complement is the sine of the original angle theta, similarly for sine.

And the tangent of the complementary angle is the reciprocal of the tangent of theta. So, the complementarity formulas. Now, it turns out that, so far, our angles theta always seemed to be between 0 and 90 degrees. One can ask, what about much bigger angles? Can they be considered in trigonometry? Or what about negative angles? What would that mean, and how would we consider them? And those reasons, and others, it turns out to be extremely useful to have a way of defining the trigonometric functions that doesn’t pass by the triangles that we’ve been looking at, but by the unit circle. The unit circle means a circle of radius 1, centred at the origin of our Cartesian coordinates.

And what we’re going to do is the following. We’re going to measure off an angle theta, starting from the right-hand x-axis, and once we have theta, we’re going to go up to the circle, along the ray that theta determines, and we’re going to hit a point on the unit circle, and we’re going to define the coordinates of that point to be cos theta and sine theta. That’s the definition of cosine and sine that we’re going to go for theta. Now, you might be worried. What does that have to do with our original definition with triangles? Well, it turns out it’s going to be the same thing, when your theta is between 0 and 90 degrees, for example.

Let’s see why is the same thing. If I drop– First of all, let me note that the radius is 1, right? The unit circle. Now, if I drop a vertical down to the x-axis, then you see that a triangle is formed. And what are the sides of that triangle? Well, for the base, it’s cos theta. By definition, that’s the x-coordinate of the point at the base of the vertical. And the vertical height of the triangle is sine theta, again, by definition, the y-coordinate. So when you take, for example, for theta, the opposite over the hypotenuse, you get sine theta over 1, you get sine theta, with that triangle, the same definition as before.

So, our definition is consistent with what we had before. The advantage of this definition is that it works for any real number of theta, even a very large real number, and it works for negative numbers theta. Negative theta, by the way, simply means that you measure the angle theta going clockwise from the original position, rather than counterclockwise. So for example, if your theta is such that you wind up elsewhere on the unit circle, like this point, doesn’t matter. The definition still holds. The coordinates of that point are cos theta sine theta, by definition. So, cosine and the sine are defined. And then, how do we define tangent?

Answer– We simply define tangent to be sine over cosine, which it was before, as we had remarked. Now, of course, this definition only holds when cosine is non-zero, because you can’t divide by 0. Now, along with this definition by the unit circle, comes another way of measuring angles. It’s in radians, rather than degrees. The fundamental definition here is that one full revolution corresponds to 2 pi radians. One full revolution, 2 pi radians. So, it turns out that a radian is a bit over 57 degrees. May seem a little awkward to use the number pi there, but it turns out to be extremely useful. Now, pi over 2, by the way, then corresponds to a quarter turn.

That may seem unfair to you. You know, you turn pi over 2, you think you’d get half of the pizza pie, but you don’t. No, you get a quarter of it. Slight incoherence there in the words, but it doesn’t matter. By the way, then, in radians, the sum of the three angles of a triangle equals pi, because pi radians is 180 degrees. Henceforth, we shall always favour radians over degrees. They are always used in science, for reasons which will become apparent. OK, now here’s our new definition of cosine and sine of theta, by using the unit circle. If theta is going to be negative, we’ll work in the other direction. We’ll go clockwise, rather than counterclockwise.

So, here is minus theta, and the point determined by minus theta. As usual, the coordinates of that point will be the cosine of minus theta and the sine of minus theta. Can you see that, in terms of coordinates, the y-coordinate is the negative of what it was before, and the x-coordinate is the same? In other words, we deduce these formulas. Sine minus theta is minus sine theta, and the cosine of minus theta is the same as cos theta. To put this another way, sine theta is an odd function, and cosine is an even function. Here’s our definition.

If we do one full circle around, that’s 2 pi radians, we come back to the same point, hence the same values of cosine and sine. This reflects the fact that sine and cosine have period 2 pi. That is, the sine of theta plus or minus 2 pi is the same as the sine of theta. The same goes, of course, for cosine and for tangent. Now, what is the sign of the trigonometric functions? Where are they positive and where are they negative?

Well, that’s easy enough to see, because you see that, for example, if your angle theta is such that you wind up in the first quadrant, then the x and y-coordinates are both positive, and so the sine and cosine are positive. In the opposite, diametrically opposite quadrant, the third, both of these will be negative. And in the two remaining quadrants, one will be positive and one will be negative, depending.

Another fact about cosine and sine– Suppose that you take your angle theta, and you add to it pi radians– that’s 180 degrees– you attain the diametrically opposed point– that’s called the antipodal point, or the antipode– can you see that the sine and the cosine will both have the opposite sign to what they had initially? That is, by symmetry, you can see that the sine of theta plus pi is the same as minus sine theta, and a similar fact for cosine. So, these are the antipodality relationships for sine and cosine. Next, we’ll be looking at the graphs of all these functions.