Trigonometry: tangent finding

[louisg]

Hey, so I was working on a graphics trick, and I found this:

Tangent is undefined at pi/2.. ok, that's obvious; cos is 0, so it's undefined

But:
tan((pi/2) - (1/128)) =~ 128
tan((pi/2) - (1/64)) =~ 64
tan((pi/2) - (1/200)) =~ 200
tan((pi/2) - (1/1010)) =~ 1010

.. and so on

This is really useful, but.. why does it happen? It's been years for me since a math class, so I'm hoping one of you science people on the forum can help

[austere]

While I'm still waiting for breakfast to be prepared:

tan(pi/2 + x) = sin(pi/2 + x)/cos(pi/2 + x)
However, sin(pi/2 + x) = cos(x) and cos(pi/2 + x) = -sin(x)
i.e. tan(pi/2 + x) = - cos(x)/sin(x)

Let x be small. Then a first order approximation for cos(x) is just 1, while a first order approximation for sin(x) is x. i.e.

tan(pi/2 + x) = -1/x => tan(pi/2 - x) = 1/x => (EDIT: sub 1/x into x, i.e. x large) tan(pi/2 - 1/x) = x

Nothing to it. The real question is, how is this useful?

[RNGmaster]

It's an identity function - it can be used to find reciprocals without taking 1/x. And it's also freaking cool.
Identity functions can be used for all sorts of proofs, so yeah.

[austere]

This is an approximation not a trigonometric identity. An identity function maps every value in its range to the same value.

-- So, yeah.

[louisg]

Nothing to it. The real question is, how is this useful?

Hopefully this makes sense (been recovering from a cold)...

I was playing with mode 7 style perspective.. assuming that each row of the screen is a different viewing angle (ala raycasting), tana = z/h (where z=distance "into" the screen and h is the height the camera is above the ground) => h*tana = z. From there, I needed to figure out the maximum viewing distance. If angle=0 is straight down at the ground and 90-degrees is the horizon, the maximum distance viewable would be just-under-90-degrees. If I know by how much, that same value can also double as the maximum viewing distance without any (or too much) additional calculation.

[austere]

I see. You could just use cotan, btw.

[louisg]

I see. You could just use cotan, btw.

Yeah, I don't think I'll have to do it dynamically. It just relates directly to the table size. For example, if you have a tangent table with 256 entries, then it's handy to know that the highest value in the table under the horizon (90 degrees) is 256!