Why bother with the why of this?

You're busy. You barely have time to sleep. You need calculus for your degree, for your career. You just want to pass the tests and pass the course. Why take time for all the whys?

Because that's the best way to beat the tests. You'll remember a formula better when you remember the why behind it. If your answer is off by a thousand percent (but otherwise correct) you'll know. It won't feel right.

For example, why all this about sin and cos and tangent?

Because under the surface all the world comes in energy waves:

waves of radiant energy

Some of them you see, some of them you hear, some you feel as heat, and some you don't feel at all while they kill you.

They're everywhere. We wind them into a circle for the same reason you wind up your garden hose:

garden hose
garden hose

Why radians instead of degrees? Again, convenience. To make the arithmetic simpler.

Suppose your park has a big round pond a half mile across and along the way you saw a softball and bat in the grass. You left them in case someone came back looking. Now you want to send someone to see. What do you tell them?

  1. "Take binoculars and row to the fountain at the middle of the pond. Scan the shore roughly a sixth of the way around from my house, counter-clockwise. Not quite sixty degrees. Pick a landmark you can find when you walk it."

  2. "Go right a quarter mile. If it's still there you'll see it."

Radians are measured along the rim of a circle ("go right a quarter mile").

Degrees are measured from the center ("row to the fountain at the middle of the pond").

The guy who invented radians was a lot like you. He had a lot to do and radians were faster and easier. He didn't invent them to while away his time or mess you over.

Same with pi. Same with sin and cos and tangent. Same with e and ln and log. Same with integrals and second derivatives and L'Hopital and Taylor Series.

You'll remember these better when you remember why someone needed to invent them.

In math class you might get "Solve for the tangent of the following angle" or "Find the second derivative of the following function." In engineering work after college you'll get the garden hose and the pond. Your big tests in school will be closer to that engineering work. Every problem is two problems. Before you solve for the tangent, see that a tangent is needed. For that you need the why, and your calculator can't tell you that.

One more reason to take time thinking about why: To save that time on tests. If you're racing through a big important test and come up with an answer like "60 radians" you'll catch it fast, before it sinks your grade. You'll know that can't be right, and you'll know why.