Chapter 7.1, Problem 83AE

In Problems 79-84, use the following discussion. The formula $D=24\left[1-\frac{{\cos }^{-1}\left(\tan i\tan \theta \right)}{\pi }\right]$ can be used to approximate the number of hours of daylight D when the declination of the Sun is $i^{\circ }$ at a location ${\theta }^{\circ }$ north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle i between the equatorial plane and any ray of light from the Sun. The latitude of a location is the angle $\theta$ between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula, ${\cos }^{-1}(\tan i\tan \theta )$ must be expressed in radians.

Approximate the number of hours of daylight at the Equator ( $0^{\circ }$ north latitude) for the following dates:

Summer solstice $\left(i={23.5}^{\circ }\right)$

Vernal equinox $i=0^{\circ }$

July 4 $i=0^{\circ }\left(i={22}^{\circ }48\text{'}\right)$

What do you conclude about the number of hours of daylight throughout the year for a location at the Equator?