Now that we have been introduced to the reciprocal trig functions, secant, cosecant, and cotangent, let's consider their graphs.
Recall with sine and cosine, we saw their relationship was very closely related, especially over the period of 2π when studying the unit circle.
I always remembered sine as a sideways "S", since it starts with S.
Similarly, for cosine, I always thought of it as a "Crow" or a bird, since it looks like a bird flying in the distance and it conveniently starts with a "C".
In any case, sine and cosine are simple functions to draw. If we remember our unit circle values, sine is 0 at 0, π, and 2π, and is 1 at π/2 and 3π/2. Draw a dot for each of those points like you would for any other function, and just do your best to connect the dots with a curve. The cosine graph is the same, but where sine is 1, cos is 0, and where sine is 0, cos is 1.
What about secant and cosecant? If we never knew what sine and cosine looked like, we would never be able to make sense of them, but let's see what they look like when considering their reciprocals.
Though they look like U's, they are actually distinct parabolas heading toward infinity.
Cosine = Adjacent / Hypotenuse. When the Adjacent side is 0 on the unit circle, the angle is 90 degrees, and the hypotenuse is 1. We can divide 0 by 1, and it equals 0. What happens when you flip them? We get 1 divided by 0, which is impossible. Let's get as close to 0 as we can within reason though, and divide 1 by .000000001 - a very very small number. We get 1000000000, a very very large number. The closer and closer we get to 0, the larger and larger the number gets. So we get a vertical asymptote everywhere that cosine = 0.
It is for this reason that we can simply draw parabolas on each bend of the sine and cosine curves. I apologize for carrying on so long, the important thing to take from this is to understand why sine and cosecant are related, and cosine and secant are related.
To draw the curves (Let's graph sec(x)):
1.) Draw the reciprocal function (red). Secant relates to cosine, so pencil in the graph of cosine along 0 to 2π. We choose 0 and 2π because this is the period of the graph.
2.) Pencil in the vertical asymptotes. Wherever the graph of cosine is 0 on the y-axis, the secant will approach infinity. These are the black lines in our sample graph.
3.) Draw parabolas reflecting out of the graphs curves (shown in green). These are the periodic minimums and maximums of the graph, also known as the magnitude. The magnitude for trig functions on the unit circle are simply 1.
The accuracy of hand-drawn graphs is not particularly important, as long as you understand the values where the graph hits 0 and 1, that is what is important - most notably at 0, π/2, π, 3π/2, and 2π.
Graphs created with Desmos Graphic Calculator.
https://www.desmos.com/calculator
nice
ReplyDelete