I encountered a book discussion that defines the Sun's declination as 23.5 degrees (inclination of Earth's equator to the ecliptic) times the sine of the Sun's longitude. Does this make sense? How do i visualize it?
Yes, it is correct. For better precision: Declination =23.5*sin(longitude) This is true only if you don't take into account the eccentricity of Earth's orbit and other movement. It isn't easy to understand why, but I'll try ti explain it. The different declination of the sun during the year is generated by the different angle of inclination of the Earth (the north pole point always in the same direction, so during the earth revolution the poles don't point always in the sun direction and the protection of the inclination on the sun-earth plane change during the year) So, what wee see of the sun is the projection of the circular movement of the earth on a line of sight, so an harmonic motion.
Yes, see graph. At the March equinox, the Sun's ecliptic longitude $\lambda_\odot$ is zero, and the Sun's declination $\delta_\odot$ is zero and increasing. At the June solstice, $\lambda_\odot$ is 90$^\circ$ and $\delta_\odot$ is at a maximum, 23.5$^\circ$ north of the equator. At the September equinox, $\lambda_\odot$ is 180$^\circ$ and $\delta_\odot$ is zero and decreasing. At the December solstice, $\lambda_\odot$ is 270$^\circ$ and $\delta_\odot$ is at a minimum of -23.5$^\circ$.