How can I calculate the tilt of the Earth using trigonometry?

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The Earth has a tilt of approximately 23.439281°, influencing seasonal changes as it orbits the sun. A preliminary MatLab program has been developed to visualize a 3D model of the Earth, including equatorial rings at tilt maxima and minima. The user seeks assistance with the trigonometric equations needed to calculate the 3D loop connecting these tilt points through the origin. Specific equations for theta ranging from 0 to 2π and incorporating the Earth's obliquity are required. Suggestions for these calculations are welcomed to enhance the model.
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Earth has a tilt to it about 23.439281° to -23.439281° from the Tropic of Cancer to the Tropic of Capricorn. The tilt faces the sun at differrent points from 23.439281° to -23.439281° creating the seasons etc. I have written a preliminary MatLab program showing the 3D Earth Equitorial Model plus some rings at the equator and at the tilt maxima and minima. What I need is some help with the trigonomic equation of the 3D loop ring that connects the two tilt maxima and minima thru the origin. Here is an example of what I have. Center loop:

x = r * cos(theta) * sin(phi)
y = r * sin(theta) * sin(phi)
z = r * cos(phi)

where r = 6378100, phi = 90 and theta = 0 - 360
 

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I have had nothing but trouble finding a set of equations to calculate the Tilt of the Earth trigonometrically for a theta (0 to 2pi) and a given Obliquity = 23.439281° (for Earth). Please disregard the MatLab program above. Any suggestions would be much appreciated.
 
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