Timber Framers Guild Forums General General Forum Questions degrees/angles

If a 12 pitch is 45 degrees why is a 6 pitch 26.565 degrees instead of 22.5 degrees? The tangent is .5, that is half the tangent of a 12 pitch, but not half the angle. Tim

Tim, the realtionship between the tangent of an angle and the angle itself is non-linear.

Angles are ratios; the measure of an angle in radians is the arc length/radius. In other words we are measuring the arc lengths of parts of a circle of unit radius. The tangent of the angle is the rise/run.

Since we are not measuring the same quantities to begin with, halves or other fractions of the original won't be equal either.

Trigonometric Functions of a Unit Circle

This is off topic but ... is there something wrong with the "Edit" function or what? I wanted to add something to my post about ten seconds after submitting it but it seems that my time has already expired.

It's not possible to bisect (or take any other proportion) of both the angle and the tangent ...

Rise Run.pdf

Oops! There's a typo in the link in the last post. Here it is again:

Rise Run.pdf

Tim, it's because you are working with a right angled triangle, not a triangle with equal angles at either end of the short side. Below is a sketch that may help, the original 12:12 triangle is in yellow

yes, the edit button would be ever so handy.....

The Forum EDIT function is disabled until software/server upgrade.

Mark, I checked the numbers 5" is close to bisecting the angle, 4.97" would be closer but the answer is irrational.

Tim, if you could graph the tangent function from 0 to 90 degrees the result will be a curved line from 0 on to the infinite, meaning that the slope ratio has no discrete value at 90 degrees.

On a more analytic note:

arctan x = x – (x^3)/3 + (x^5)/5 – (x^7)/7 + (x^9)/9 – ...

If we take any portion of x, the tangent of the angle, we cannot be taking the same portion of the angle (the value returned by the formula).

Or, looking at the Formula for the Tangent of an Angle

Again, dividing the angle, x, by any number does not produce the same fraction of the number produced by the formula (the tangent of the angle).

I think the geometric interpretations are the easiest to understand. Bisect or take any part of the angle and the geometry says it's impossible to take the same fraction of the tangent. Bisect the tangent and we cannot bisect the angle.