The ancient Greek geometers only considered angles between 0° and 180°, and they considered neither the straight angle of 180° nor the degenerate angle of 0° to be angles. It’s not only useful to consider those special cases to be angles, but to also include angles between 180° and 360°, too, sometimes called “reflex angles.” With the applications of trigonometry to the subjects of calculus and differential equations, angles beyond 360° and negative angles became accepted, too.
Consider the unit circle. Denote its center (0,0) as O, and denote the point (1,0) on it as A. As a moving point B travels around the unit circle starting at A and moving in a counterclockwise direction, the angle AOB as a 0° angle and increases. When B has made it all the way around the circle and back to A, then angle AOB is a 360° angle. Of course, this is the same angle as a 0° angle, so we can identify these two angles. As B continues the second time around the circle, we get angles ranging from 360° to 720°. They’re the same angles we saw the first time around, but we have different names for them. For instance, a right angle is named as either 90° or 450°. Each time around the circle, we get another name for the angles. So 90°, 450°, 810° and 1170° all name the same angle.
If B starts at the same point A and travels in the clockwise direction, then we’ll get negative angles, or more precisely, names in negative degrees for the same angles. For instance, if you go a quarter of a circle in the clockwise direction, the angle AOB is named as –90°. Of course, it’s the same as a 270° angle.
So, in summary, any angle is named by infinitely many names, but they all differ by multiples of 360° from each other.
cos (θ + 360°) = cos θ.
Many of the modern applications of trigonometry follow from the uses of trig to calculus, especially those applications which deal directly with trigonometric functions. So, we should use radian measure when thinking of trig in terms of trig functions. In radian measure that last pair of equations read as
cos (θ + 2π) = cos θ.
sin θ = cos (π/2 – θ)
We’ve seen this before, but now we have it for any angle θ. It’s true because when you reflect the plane across the diagonal line y = x, an angle is exchanged for its complement.
We’re now ready to look at sine and cosine as functions.
Sine is an odd function, and cosine is even
cos (–θ) = cos θ.
These facts follow from the symmetry of the unit circle across the x-axis. The angle –t is the same angle as t except it’s on the other side of the x-axis. Flipping a point (x,y) to the other side of the x-axis makes it into (x,–y), so the y-coordinate is negated, that is, the sine is negated, but the x-coordinate remains the same, that is, the cosine is unchanged.
First, note that it is periodic of period 2π. Geometrically, that means that if you take the curve and slide it 2π either left or right, then the curve falls back on itself. Second, note that the graph is within one unit of the t-axis. Not much else is obvious, except where it increases and decreases. For instance, sin t grows from 0 to π/2 since the y-coordinate of the point B increases as the angle AOB increases from 0 to π/2.
Next, let’s look at the graph of cosine. Again, take the horizontal axis to be the t-axis, but now take the vertical axis to be the x-axis, and graph the equation x = cos t.
Note that it looks just like the graph of sin t except it’s translated to the left by π/2. That’s because of the identity cos t = sin (π/2 + t). Although we haven’t come across this identity before, it easily follows from ones that we have seen: cos t = cos –t = sin (π/2 – (–t)) = sin (π/2 + t).
You can also see that tangent has period π; there are also vertical asymptotes every π units to the left and right. Algebraically, this periodicity is expressed by tan (t + π) = tan t.
The graph for cotangent is very similar.
This similarity is simply because the cotangent of t is the tangent of the complementary angle π – t.
As you would expect by now, the graph of the cosecant looks much like the graph of the secant.