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I got the coordinates of the center of a circle $(a,b)$ as well as one other point $(x, y)$. From those I can derive the radius by applying square root to the result of following formula.

$$ (x-a)^2 + (y-b)^2 = r^2 $$

This should allow me to compute the angle but I have only this formula:

$$ x = a + r\cos t$$ $$ y = b + r\sin t$$

How can I compute $t$?

I am a bit lost in solving this system of equations. Any help?

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1 Answer 1

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You know $x, y, a, b$ and $r$. Then $$ \cos(t) = \frac{x-a}{r} = c_{1} \quad \text{and} \quad \sin(t) = \frac{y-b}{r} = c_{2} $$ Note that $c_{1}$ and $c_{2}$ are just two numbers that you compute based on what you know. How many solutions does each of the above trigonometric equations have in $t \in [0, 2\pi)$?

Of course, in the end, the right $t$ should satisfy both equations..

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  • $\begingroup$ So once found c1 or c2 I should be able to derive the angle? Is there a formula to derive t from the cos or sin value? $\endgroup$
    – mm24
    Dec 21, 2014 at 10:33
  • $\begingroup$ Not really. For certain angles people usually know the cosine and sine values: for example if $t=30^{o}$ (or $t=\pi/6$ in radians), then we know $\cos{t} = \sqrt{3}/2$ and $\sin{t} = {1}/{2}$. So, inversely if $c_{2}=1/2$ in our example, then we know that $t=30^{o}$ or $t=30^{o} + 90^{0}=150^{o}$. Take a look at the unit circle for such angles: en.wikipedia.org/wiki/Trigonometric_functions#mediaviewer/… If $c_{1}$ and $c_{2}$ are arbitrary, you may use a scientific calculator to compute the inverse cos/sin (arccos/arcsin). $\endgroup$
    – megas
    Dec 21, 2014 at 16:34

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