Picture of what I am looking for
I have a formula where if I have a circle sitting in an saddle angled at 10 degrees and I know the distance from the bottom of the saddle to the top of the circle it will result in the diameter of the circle. The formula looks to be some form of relation ship to the size of the circle and the 10 degree angle. Distance * (1+1/cos10)/2 Is anyone able to tell me what this relationship is called so that I can look it up and figure out how someone came up with this. No matter how big the circle is, the saddle is always tangent to the circle.
Mathematics involves thousands upon thousands of formulae, proofs, concepts, algorithms, etc. Naming them all would be pointless. We only name the ones that we need to reference a lot, and so need a short-hand way to refer to them. And even among those (still many thousands), any one mathematician is unlikely to know more than a few hundred of the names. Contrary to the impression left upon many students, mathematics is not about memorizing lists of formulas.
What we do instead is learn tool sets for deriving the formulas we may need. This one is solved by a little geometry and trigonometry. Add the lines shown.
The angle at $A$ is right, so $ABC$ is a right triangle. The angle in that triangle at $B$ is $90^\circ - \theta$, and the hypotenuse is $h$, so $$\sin(90^\circ - \theta) = \frac rh$$ But $\sin(90^\circ - \theta) = \cos \theta$, so with a little rearranging, we can rewrite this as $$h = \frac r{\cos \theta}$$
Now $$x = r+ h = r + \frac r{\cos \theta} = r\left(1+ \frac 1{\cos \theta}\right)\\r = \dfrac x{1+\dfrac 1{\cos \theta}}$$
Since you are calculating the diameter, which is $2r$, this is $$2r = \dfrac {2x}{1+\dfrac 1{\cos \theta}}= \dfrac x{\dfrac{1+\dfrac 1{\cos \theta}}2}$$
In your formula, someone has dropped the inversion of the cosine in the denominator. But that is not what the math is telling me. They also divided by $2$ in the denominator instead of simply multiplying the numerator by $2$. Seems more complicated to me, but hey. The form in your denominator is also $\left(\cos \frac\theta 2\right)^2$, which might have some bearing on how whoever came up with this formula did so.
But my derivation shows that it isn't correct. It has $\cos \theta$ where it should be $\frac 1{\cos \theta}$. You could simplify my formula to $$ \text{diameter} = \dfrac{2\cos\theta}{1 + \cos\theta}x$$ but that is still not the same.