Common Core State Standards

Still towards…G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

We introduced sin(x) as the actual physical height of the Sun traversing a circle of radius 1 observed at an angle of elevation of x degrees.

For a circle of a different radius r, the physical height of the Sun observed at an angle of elevation of x degrees is now

height = r sin(x).

The meaning of sine has now subtly changed. Solving for sin(x) we get

and sine is now a ratio of lengths, not an actual length in and of itself!

Comment. Another shift of thinking also occurred soon after this. In the mid 1700s, the Swiss mathematician Leonhard Euler noted that sin(x) plays the role of a function: to each angle x is assigned a number between –1 and 1. (The analogous idea holds for cos(x) too.) Euler was the first to articulate the notion of a function and seeing sine and cosine as functions provided a new mindset for thinking about them: he could graph these trigonometric functions, compose them, ask for their function inverses, and the like.

“CIRCLE-OMETRY” BECOMES TRIGONOMETRY

In the mid-1500s the scholar Jaochim Rheticus turned the study of circleometry into a study of right triangles. His approach is the one used today in practically all introductory texts to the subject.

Rheticus realized that in the most general case of a circle of arbitrary radius r, sine and cosine each represent a ratio of lengths in diagrams and are not themselves physical lengths (except in the case r = 1, perhaps).

We see this if we isolate the right triangle we see in this diagram. For example, sin(x)is the ratio of the side of length r sin(x) (that is, the side opposite the angle x) to r, the length of the hypotenuse of the right triangle. We can thus focus on a study of right-triangle sides and their ratios.