I don't have a particularly good reason to want to do this, and I'm just asking out of curiosity.
I am looking for the coordinates of point $\pmb B$, a point on the circumference of a circle.
If I know the following:
- The equation of the circle (the coordinates of the center $\pmb O$, and its radius $\pmb R$)
- The coordinates of a point on the circumference, point $\pmb A$
- The central angle $\widehat{AOB}$ (I know the angle might be useless without trigonometry)
- The length of the arc $\newcommand{arc}[1]{\stackrel{\Large\frown}{#1}}\arc{AB}$
- Points $\pmb A$ and $\pmb B$ are both in the upper half of the circle, and the angle $\widehat{AOB}$ is less than $\frac{\pi}{4}$ radians (for this specific example)
Is it possible to find the ccoordinates of $\pmb B$ without using any trigonometric functions (sin,cos,tan,etc)?
I'm not sure how to create an image for this, otherwise I would've included one for clarity.
Edits: I realize that the arc length can be calculated from the central angle and the radius, so point 4 is redundant.
Someone else pointed out that there would be two possible points given the angle and arc length. Would it be possible to find either one or both without using trig?
Best Answer
The trig functions provide a dictionary between the arc measurement of an arbitrary$^\dagger$ point on a circle (or similarity class of a right triangle described by an angle) and the rectangular coordinates of the point on the circle (or ratio of side lengths of the triangle). Nothing more, nothing less.
You have one type of information—namely, the radius and angle—and you're asking about the other. Any possible tool to accomplish this is equivalent to using the trig functions (possibly obfuscated a bit).
$^\dagger\!$ Caveat: for "special" angles, the values of the trig functions (i.e., ratios of sides of a triangle with those angles or coordinates of a rotated point) can be found using the Pythagorean theorem and algebra, but they are like dust among the continuum of possible angles.