Here's what's left for the year:
This week we will be looking at applications of some of the trigonometric concepts we've learned in the last month and a half. We will be focusing primarily around two different types of applications: modeling periodic events and calculating heights and distances using right angle trigonometry. With the foundation we've built, neither of these should be particularly challenging but there is a little bit of new vocabulary and some procedure things that might be confusing at first.
One quite practical thing we can do with trigonometry is find lines of best fit for periodic phenomena. We've discussed in the past how math is used to model data and predict future events, and trigonometric functions, specifically the sine and cosine function, are used frequently for this purpose. The process of fitting a sine or cosine curve to data is known as sinusoidal regression.
Computers and calculators can perform regressions on data and generally get pretty fantastic results, but for periodic functions it is definitely possible for us to perform the same analysis by hand and get within spitting distance of that accuracy. I have not covered how to perform a sinusoidal regression on a Texas Instruments graphing calculator, but I do have a video that shows that process on a related topic.
In order to write any trigonometric function, we only need to know four things: The amplitude, the period, the phase shift, and the vertical shift. By visualizing what the data represents, we can pull all four of those things from the raw data itself and generate a function.
By generating a function that represents the data, we can then reasonably accurately predict future events. In the example of the Bay of Fundy, if the Canadian government even uses a function to predict generate those tide predictions, it is a more complicated one than the one we came up with. The reason is that tides are effected by more than just the moon, so the intensity of the tides vary throughout the year (the sun also has an effect on tides). In any case, the idea holds (and certainly applies to ferris wheels!)
There are often times in the real world where you may want to measure distances or heights that are inaccessible. For example, if you need to know how tall a building is but don't have access to any sort of records and don't have access to the building itself, you can measure known horizontal distances and record some angles to find the height with a fair degree of accuracy. This may seem like a far-fetched scenario that only math teachers could dream up, but it's not! Civil engineers have developed measuring equipment that can record angles with a great level precision and then use those angles in conjunction with measured horizontal distances to map elevation changes. Technology has advanced to a point where that particular equipment is basically irrelevant now, but this kind of application is still a very real-world situation.
If the world was nicely divided up into right triangles, we could just measure everything. The thing is... we really can use right triangles to simplify different measurements in most every scenario. In fact, more involved trigonometric concepts like the Law of Sines and the Law of Cosines (they're not that involved) are fairly simply derived using simple right angle trigonometry.
Two terms we often hear when discussing these types of applications are "Angle of Elevation" and "Angle of Depression". Sometimes these are called "Angle of Inclination" and "Angle of Declination", but they are always measured off of the horizontal. These kinds of problems come up all the time on standardized tests, so make sure you know how to do them.
Notice how we tend to end up using tangent more often than sine or cosine when solving problems involving angles of elevation and depression. Maybe we can say that since sine and cosine are used more for regression, that this is tangent's time to shine? In these types of problems, it is also possible to find angles of elevation or depression when given the necessary heights/lengths (triangle side lengths), but doing so requires using something called inverse trigonometric functions, which we haven't talked about yet. This is a fairly simple concept (made complicated in a full fat trigonometry class) and using it solve these kinds of problems is just a matter of pressing an extra button on your calculator (2nd or shift and then the trig function you're after).
When you're ready, go to Microsoft Forms and complete the quiz under the Assignments tab called "Trigonometry Quiz". Please do this by Friday, May 15th at 11:59PM. If you are able to print the pdf that is attached to the assignment, then do that and fill it out. Otherwise, you can solve the problems on a separate sheet of paper. Submit your work in Microsoft Teams and make sure to clearly label your work for each question. By then you will also have your final exam ready to take (which does not involve doing any calculations whatsoever so don't stress out about that), so make sure to budget your time accordingly.