Why the length of day and the solar angle are (almost) constant around the solstice and vary greatly around the equinox ?

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On the solstice, the changing length of day changes which direction it is going. It goes from getting longer every day to shorter every day for the summer solstice. Because its changing direction, it is slowing down as it approaches the change point and then going the other direction

The rate of change will continue to grow until the equinox which is when the length of the day changes by the most per day, and will then begin to slow until it reaches the next solstice and changes direction

All this is because the Earth is tilted and the sun progresses up and down the surface as a sine wave. The highest rate of change for a sine wave is in the middle, and there is no change at the tops of the peaks and bottoms of the valleys

The sun follows the pattern of a pendulum. The solstice are at the top where the pendulum is barely moving. The equinox is at the bottom where it is moving the fastest.

In the 30 days around the spring equinox the sun goes from the bottom of the sky to the top.