I am math teacher and was asked the classic question, "Where am I ever going to use this in life" I usually have a real word example, but for the cosecant function I am stuck (Besides higher level math classes.) Anyone know a specific example?
I also teach math. Here is one way to respond to the importance of csc(x):
Cosecant is simply defined as: csc(x) = 1/sin(x)
So, anywhere sin(x) might occur in the “real world”, so does csc(x) since sin(x) = 1/csc(x). You could always rewrite sin(x) in terms of csc(x).
One great importance of csc(x) is that it can abbreviate expressions involving sin(x).
For example, (1/sin(x))^2 = (cos(x)/sin(x))^2 + 1 is more clearly written as
(csc(x))^2 = (cot(x))^2 + 1.
Abbreviations are very useful, they are everywhere in math. A basic example would be the number 8.
It is more convenient to write 8 instead of 1+1+1+1+1+1+1+1. But there is a trade off. Now you need to memorize that 8+1=9. That is, you need to learn more rules. This would not be the case if we just wrote it all out:
The abbreviations 8 and 9 are VERY nice, even though one needs to learn additional rules. Likewise, csc(x) is very nice, just take the time to learn the additional rules that accompany the abbreviation.
I agree that the math behind the the function is very important. As far as 1/sinx goes, I guess I could set up equation that it would be easier to use the csc, but I was hoping for a real world example that involved graphing, like using the sine or cosine function with sunrises or using the tangent and a rotating light on the wall.
The only example I can find for csc is the type of high gain antennae but thats a little to complicated for high school.
Examples for use of secant and cosecant: A few exist in Physics, mechanics at least. You might find a formula relating to friction to be more compactly written using one of those functions. How this relates to actual practice in the real world, unclear.
How about this: Two people, A and B, standing at a distance of L ft. apart.
Person A stays still. Person B starts running at velocity v in the direction perpendicular to the original line from A to B. Person A measures the angle w from the original line to the line of sight of person B’s position. Then, the distance from A to B at time t is