Common Core State Standards

The background to … F-TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Mankind is on a perpetual scientific and intellectual quest, to answer the fundamental question:

What is this universe we find ourselves in?

Our need to understand our existence and our place and role in the universe, and the nature of the universe itself, has propelled grand scientific, psychological, theological, social, and creative musings since the dawn of time. The study of astronomy was one of the earliest fields of scientific pursuit.

Imagine a human back at the dawn of time, sitting on the ground, observing the universe around her. She notices the Sun, the Moon, and the stars, and their motion. Each body seems to move in arcs across the day or night sky. It is natural to wonder what these objects are, how high or far away they are, what their influence on us might be, and so on. The mathematics to begin understanding the heavenly motions dates back to the ancient Babylonians (ca. 2000 bce), if not earlier.

Let's address one particular natural question: How high is the Sun?

Each day, the Sun rises in the east, moves across the day sky in a large arc, sets in the west, and then returns to rise in the east again the next day (on average twelve hours later). It seems natural to suspect that the Sun stays in motion during the night, moving perhaps below us on the other side of the ground. Can we determine the height of the Sun at any desired time of day?

Unfortunately it is not possible to climb up to the Sun, drop a rope back down to the ground, and measure its length. From our vantage point, here on the ground, there is only one measurement we can make: the measure of the angle of elevation at which we observe the Sun.

Can we determine the height of the Sun from one angle measurement?

Scholars of ancient times were fully aware that the Sun does not move on a perfectly circular arc across the day sky.