For the benefit of humanity

I feel it is important, occasionally, to write Serious Posts. A post that is Important and can change the world for the better and benefit humanity.

So here goes. Something Serious. Important. That will change the world and benefit humanity.

The cosine is an even function. That is, cos(-x) = cos(x). The sine, however, is an odd function. So sin(-x) = -sin(x). If you can remember this, and remember that

sin(a+b) = sin(a)cos(b)+cos(a)sin(b),

and

cos(a+b) = cos(a)cos(b)-sin(a)sin(b),

and finally that

sin²(a) + cos²(a) = 1,

then you will be able to figure out any trig identity they throw at you. It's true. Throw any trig identity at me. Double angle formula? Well sin(2a) is just sin(a+a), and I know that! Half angle formula? cos(a) = cos(a/2+a/2) = cos²(a/2) - sin²(a/2) = 2cos²(a/2) -1. Solve for cos(a/2). Ha!

You see? Five little identities and you are totally trig proof. I love it. I told you this post would benefit humanity.

For those of you still reading, I wanted to share one of my favorite stories. This is a true story, reported by my high school math teacher. When he was a young college student, he took a math exam and failed to fully simplify one of his answers. Instead of writing the correct answer, 1/2, he left the answer as cos(pi/3). His professor docked him points for this.

My teacher was annoyed. He had done all the hard work, just hadn't simplified at the end. He complained.

"You didn't simplify," said the professor.

"Why do I need to simplify?" replied the young man who would be my teacher. "Everybody knows that cos(pi/3) is 1/2."

The professor called him on it. "Ok," he said. "Let's go outside and you choose any person off the street. If they know what cos(pi/3) is, then I will give you full credit on this problem."

They went outside. My teacher reported that he picked the smartest looking person he could find, but alas. Apparently not everyone knew that cos(pi/3) was 1/2 after all. He did not get his points back on the exam.

That is the end of the story.

But I sometimes wonder who the smart looking person would have been. And what it would be like to be approached by two people and asked the cosine of pi/3. And if I'll ever get the chance to use this with one of my students. I would like that.

Thanks to this public service trig announcement, you are now prepared to help a poor student get exam points back.