Trig question

[markraz]

Hi, I have 2 circles. I only know the center points and radii of both circles and I know the theta of the yellow line. I need to find the XY value of point A on the larger circle. I can find the X value of this point but how do I find the Y value using trig?

I am able to find this xy using point using algebra via: https://www.analyzemath.com/CircleEq/circle_line_intersection.html

but I want to use trigonometry. Is there an easy way to do this with some kind of trig function? Thanks in advance

 

[skeeter]

[MATH]r [/MATH] = small circle radius

[MATH]R[/MATH] = large circle radius

[MATH]\theta[/MATH] = angle between the two radii

let the circles be centered at the origin

x-coordinate of the red vertical line is [MATH]x = -r\sin{\theta}[/MATH]
Large circle equation ...

[MATH]x^2 + y^2 = R^2 \implies y = \sqrt{R^2 - x^2} = \sqrt{R^2 - r^2 \sin^2{\theta} }[/MATH]

 

[markraz]

I

[MATH]r [/MATH] = small circle radius

[MATH]R[/MATH] = large circle radius

[MATH]\theta[/MATH] = angle between the two radii

let the circles be centered at the origin

x-coordinate of the red vertical line is [MATH]x = -r\sin{\theta}[/MATH]
Large circle equation ...

[MATH]x^2 + y^2 = R^2 \implies y = \sqrt{R^2 - x^2} = \sqrt{R^2 - r^2 \sin^2{\theta} }[/MATH]

I can't thank you enough, appreciate it