Math Is Fun Forum

Why is trigonometry taught from the perspective of trig functions being blackboxes?  It seems to be counter to what math is about.

Given some function, f(x) = some calculation involving x, we see what the function is and does.  For example, given the function, f(x) = x^2, we see that the function squares x.

But given a trig function, for example, sin(angle) = opposite/hypotenuse, we don't actually see what the function is and does.  sin() is a blackbox, and we don't see what it does to the angle.  And possibly creating much confusion here is that the notation used for trig functions can make it appear that, opposite/hypotenuse, is the operation of the function.

Over the last year I have picked up a handful of math books, looking to better my understanding of math fundamentals before going further.  One of those books which I have found to be excellent in clarifying concepts of fundamental topics is,  Yes, But Why?  Teaching for understanding in mathematics, Ed Southall.

This book is obviously aimed at teachers, but as an independent learner I have found it to be an excellent resource for getting a solid birdseye view of various topics and how things work.  I only wish that it existed long ago (it was released earlier this year), as I have wasted alot of time over the years digging through various resources, often running in circles and hitting walls or becoming bored to death.  The book begins at arithmetic and graduates through topics in algebra, geometry, and ends with fundamentals in trigonometry.

What I like about it:

- It gets right to the point of concepts without burrying the reader in a bunch of fluff or too much detail from the getgo.
- It doesn't focus on procedures over concepts.
- It makes good connections from one topic to the next, not leaving gaping holes in understanding.
- Reading it has been a series of 'Ah, ha' moments without the brickwalls of so many other resources.
- It makes good use of graphics for explaining concepts; doesn't include graphics for the sake of dressing things up.
- It isn't stodgy in style; it is very conversational in style.  It is an educational and fun read at the same time.
- It makes many mentions of where math terms come from and how they relate to the topic at hand, along with other historical bits.
- It is (along with many articles at MathisFun) what I feel in many ways, how math should be taught.

I am not associated with this book in any way.  Just sharing a resource that I have found to be exceptional among math books as an adult self-learner relearning math fundamentals.  This book, along with MathisFun, Geogebra, and exploring using a compass and rule (because working with my hands is often more satisfying) have been time well spent.  I am finding myself to be a visual learner, where numbers often are only added precision in application of concepts.

To present this as simply as possible, say I have a right triangle with sides: adjacent = 0.71; opposite = 0.71; hypotenuse = 1. The angle theta is 45 degrees. When I enter in the input box, sin(45) , I get 0.85, which is obviously not the sine of 45 degrees.

Am I doing something wrong here?  In case it is relevant, I am using Geogebra 5 with the angle unit set to degrees.

I assume that it is an abbreviation for, <something> math.

I'm relearning algebra after some years of being away from school, and I'm learning to use Geogrebra.  At the moment I'm plotting some lines to get a sense of what elementary functions look like on a grid.  I never really got a good sense of this when I was in school.

So I have:

f(x) = x , which gives a diaganol line.
f(x) = -x , which gives a mirror of the above diagonal line (Is there a math term for this?).
f(x) = 0x , which gives a horizontal line.

How would I plot a vertical line?  The closest that I have found is:

f(x) = 10^(100)x