As with all math, the genesis of trigonometry was trying to understand the natural world around us. Trigonometric functions are periodic, which means they repeat themselves over and over again on on a regular interval. We see other periodic things around us every day. Days get longer then shorter. The seasons change. The tide goes in an out. It turns out, actually, that all of these things have to do with the single most observable object in our world, the sun. If we needed math to explain the world around us, then it makes sense that the ancient Babylonian civilization in the Mesopotamian region (modern day Iraq) saw the sun and tried to quantify certain things about. Some of the very basic things we use in trigonometry today are believed to come from these attempts at explaining the natural world.
When it comes to the duration of daylight each day throughout the year, it turns out that a trigonometric function (all of which are periodic) models the data pretty much perfectly (high R^2 values for a regression indicate very strong correlations between the data and the "line of best fit" that we generated). If we multiplied the inputs of the negative cosine function by a number smaller than one, we stretched out it's default period (something we will learn about later) to make it closer to a year (our x-axis in this example represents days). By adding 14.612 to the inputs, we shift everything to the left by 14.612 days to make our day that had the least daylight correspond to the actual day of the year that has the least sunlight (the Winter solstice). The amplitude of our cosine function, which by default is 1 (something we'll also learn later), is 2.08871, which is basically how more and how much less daylight can we have throughout the year compared to the average (which is represented by the vertical shift of 12.1189).
Although much of what we talk about in the study of Trigonometry involves angles and triangles, it is worth looking at the origin of some of the common measures we use when discussing these things, and doing so involves looking first at circles. The Sumerian and Babylonian of the Mesopotamian region are believed to the first to use and quantify angles, and their sexagesimal number system (base 60) is why we have 360° instead of some other number. Though an angle itself is just a measurement, we can standardize their use in the study of trigonometry by drawing them in a standard position. An angle in standard position has its initial side on the positive x-axis, formed by a ray extending from the origin, and its terminal side formed from a ray extending from the origin. Coterminal angles are angles drawn in the standard position that have the same terminal side. The sum of the absolute value of coterminal angles will always be divisible by 360°.
The Unit Circle might be the most feared mathematical topic. This is completely unfair to the Unit Circle itself. The Unit Circle is just a circle centered at the origin with a radius of 1. That's it. By drawing angles in standard position within the unit circle, we can make some very handy simplifications that form the foundation behind most of the study of trigonometry. We use these simplified angles and side lengths to generate relationships between the side of a right triangle and the angle in standard position. Where the Unit Circle gets its bad rap is when we introduce several common angles, the corresponding side lengths of the triangles that we can draw within the unit circle, and maths teachers' nasty tendency to require that students memorize all of these values. If we can instead just start to visualize the unit circle, consider where x values are positive/negative, where y values are positive/negative, and familiarize ourselves with certain common angles, we can minimize the amount of memorization required and make the entire topic less intimidating.
To be clear, the only thing required of you this week is to read/watch the information above, and to complete the form at the link directly above this sentence. You will have new material that includes practice problems and a mini quiz available starting Monday, 4/6.