How to determine surface temperatures at specific points

I have a planet around 11242 km in diameter. It is very similar to Mars, with most surface water trapped at the poles. For this, I want the polar temperature below 0 C, but the equatorial temperature to be consistently above 50 C; this way, I can have the water stay at the poles and not evaporate or melt. Assuming that the star's radius is 1.21 solar radii and 5992 K, how can I figure out the temperature of a region at the poles of the planet, as opposed to the temperature of the equator, so I can accurately determine the distance at which my planet needs to orbit? A simple equation would help.

I have thought of using an average surface temperature calculator such as this one to determine the temperature of a small asteroid orbiting at the distance at which the poles are from the sun and another asteroid orbiting at the distance of the equator and using those as my temperature values, but I don't know how to compensate for heat transport.

Note: the planet only has one atmospheric cell.

UPDATE:

On Medium, at this article, it states that the relationship between latitude and temperature is roughly sinusoidal, so I suppose the question now is: how do I find the extremes of the graph?

posted over 4 years ago

System‭

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