3 - The Basics of “Circle-ometry”

Summary

Common Core State Standards

More 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.

Movement towards …

F-TF.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/ 3, π/ 4 and π/ 6, and use the unit circle to express the values of sine, cosine, and tangent for π – x, π + x, and 2π – x in terms of their values for x, where x is any real number.

F-TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions

Let's imagine a point—the Sun, say—moving counterclockwise about a circle centered at the origin, always rising in the east and setting in the west.

A problem: What radius circle do we consider?

Just to get the mathematics going, let's work with a circle of radius of 1 unit. For the Sun moving about the Earth (from our perspective) this is one very large unit: the number of miles from the Earth to the Sun. But let's call it one unit nonetheless.

Comment. Astronomers call the mean distance of the Earth from the Sun one astronomical unit, so we are right in line with the astronomical origins of this subject doing this!

If the Sun has risen x degrees from the positive horizontal axis (east), we define

sin (x) , read “sine of x,” is the height of the Sun at that angle of elevation.

cos (x) , read as “cosine of x,” is the “overness” of the Sun at that angle of elevation.

That's it!

Comment. Indian scholars did not think to set a convention of using a circle of radius 1. In theirwork the value of sine for a given angle of elevation also depended on the radius of the circle being examined.

We'll see later on how to change the radius of the circle in our considerations too.