Formula for calculating third point from two points and angle - math

Question

I am trying to find angle of yellow line from center of circle. I know circle radius, red and blue point coordinates and angle between red and yellow lines.
What kind of formula should I use?

Answer 1

Think the situation is as follows:
Points R and B are given, as well as angle θ. What is asked for is angle φ.
I place a coordinate system on the center of the circle and expressed R in polar coordinates
d = sqrt( (x_R-x_B)^2 + (y_R-y_B)^2 )
ψ = atan2( (y_R-y_B), (x_R-x_B) )
Then use the law of cosines to find l
l = sqrt(r^2 + d^2 -2*r*d*cos(θ))
Now to find φ and ψ we use the following two equations
d*cos(ψ) = r*cos(φ)-l*cos(θ-φ)
d*sin(ψ) = r*sin(φ)+l*sin(θ-φ)
This is where I am stuck now.

Answer 2

Assuming the following are known:
side BC = a
radius CA = b
angle ∡BAC = α
It follows from the law of sines that sin ∡ABC = AC sin ∡BAC / BC = b sin α / a. The right-hand side is a known quantity, so the equation can be solved for ∡ABC then the third angle of △ABC is ∡BCA = π - α - ∡ABC. This gives the angle between the yellow line and the known segment BC.

Answer 3

Yes, it is possible.
Assuming the vector length is the same, you could define the first point as the origin (0,0).
Then describe the first vector in polar coordinates, (r, \displaystyle \thetaθ), where \displaystyle r = \sqrt{x_{1 }^{2} + y_{1 }^{2}}r=
x
1
2
​
+y
1
2
​
​
and \displaystyle \thetaθ = \displaystyle arctan(y_{1 }/x_{1 })arctan(y
1
​
/x
1
​
).
Then create the second vector by adding k degrees to \displaystyle \thetaθ: \displaystyle \theta_{2 }θ
2
​
= \displaystyle \thetaθ + k
Then convert from polar coordinates back to rectangular coordinates--if you need them that way:
\displaystyle x_{2 } = r cos(\theta_{2 })x
2
​
=rcos(θ
2
​
) and
\displaystyle y_{2 } = r sin(\theta_{2 })y
2
​
=rsin(θ
2
​
)