Question 717941
The sun, as it travels from dawn (horizontal) to noon (vertical) travels through an angle of 90 degrees and it takes 6 hours to do this. Assuming that the angle of the sun changes at a fixed rate will it goes from dawn to noon then the ratio of the angle you found over 90 will be the same as the ratio of how many hours past 6 am it is and 6 (the total time from dawn to noon). So
{{{58/90 = x/6}}} 
where x is the number of hours past dawn when the angle is 58 degrees. Now we solve for x. Multiplying by 6:
{{{58/15 = x}}} 
Since 1/15 of an hour is 4 minutes, x is 58*4 or 232 minutes. This is 3 hours and 52 minutes. Remembering that is how much time past dawn, the time of day when the angle is 58 degrees would be 6 hours + 3 hours and 52 minutes or 9:52 am.