To find the measure of an angle knowing the coordinates of the endpoint (x,y) you use the next:
[tex]b-y_0=r\sin \theta[/tex]Solve the equation to θ
[tex]\begin{gathered} \frac{b-y_0}{r}=\sin \theta \\ \\ \sin ^{-1}(\frac{b-y_0}{r})=\theta \end{gathered}[/tex]b is the coordinate in y of endpoint
y0 is the coordinate in y where is the center of the angle
r is the radius of the circle
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In this case for θ1: Endpoint B (3.05, 1.91)
b= 1.91
y0= 0 (angle in in (0,0)
r= 3.6 (The distance from the origin to point A in x is equal to the radius)
[tex]\begin{gathered} \theta_1=sin^{-1}(\frac{1.91-0}{3.6}) \\ \\ \theta_2=32.04º=0.55\text{rad} \end{gathered}[/tex]The radian measure of ∠AOB is 0.55. θ1 =0.55rad-----------------------------
As you know the length of the arc BC (13.824 units long) you use the next equation to find the measure of the angle)
[tex]S=r\cdot\theta[/tex]Solve the equation for θ:
[tex]\theta=\frac{S}{r}[/tex]S is the length of the arc
r is the radius
In this case for θ2:
S= 13.824 units
r= 3.6
[tex]\begin{gathered} \theta_2=\frac{13.824}{3.6} \\ \\ \theta_2=3.84\text{rad} \end{gathered}[/tex]The radian measure of ∠BOC is 3.84. θ2=3.84rad---------------------------------
To find the coordinates of point C you add angles θ1 and θ2:
[tex]\theta_1+\theta_2=0.55rad+3.84rad=4.39\text{rad}[/tex]Use the next formulas to find coordinates of a endpoint with center of angle in (0,0):
[tex]\begin{gathered} x=r\cdot\cos \theta \\ y=r\cdot\sin \theta \end{gathered}[/tex]In this case for point C:
θ = 4.39rad
r= 3.6
[tex]\begin{gathered} x=3.6\cdot\cos 4.39\text{rad} \\ x=-1.140 \\ \\ y=3.6\cdot\sin 4.39\text{rad} \\ y=-3.41 \end{gathered}[/tex]The coordinates of point C are (-1.140 , -3.41)x= -1.140y= -3.41