Chapter 5.3 – Sum and Difference Identities

Objectives (click/tap to expand)

- Use a sum or difference identity to find an exact value without a calculator
- Simplify expressions
- Use identities and transformations to match graphs and equations
- Use identities to convert sum/difference of multiple sinusoids into a single sinusoid

All Trigonometric Identities

Here is a formula sheet for all the trig identities we will learn this year, for your convenience.

The first page has the Fundamental Identities. Think of this sheet as a set of "training wheels" that you will eventually need to learn to do without. These are the formulas that I keep insisting need to make sense -- they shall not simply be memorized. You will be expected to know all of them for the assessments.

The second page has the Advanced Identities. You will eventually need to know how to use all of them, but I make no claims that these should be self-evident in any way. So, I will provide a sheet of only the Advanced Identities for your assessments.

5-3 Sum and Difference Identities

Watch before Day #47 lesson

Introduction to the Sum & Difference trigonometric identities which allow us to evaluate sin(A+B), sin(A-B), cos(A+B), cos(A-B), etc. In class we'll derive these formulas from our basic trig skills, but if interested, here's another approach showing how the cos(A-B) identity comes as a result of just some basic geometry and trigonometry knowledge.

You're welcome (Day #48, no video)

Parallel Parking Article in New York Times Magazine

The mathematics of parallel parking had me stumped for about a week until I realized that the same "Sum and Difference Identities" that we're studying right now would enable me to solve for the variable that I wanted to isolate. Once I realized this, I wondered why I didn't see this earlier (See, that happens to me too).

How this ended up in the NYT magazine is still a bit of a mystery to me, but it was an interesting experience and a nice example of how these formulas do more than just enable us to find the trig functions of a few more angles without a calculator.

Here's one of the original articles on Professor Blackburn's version of the Parallel Parking formula.

Here are interactive GeoGebra constructions of my model as well as Professor Blackburn's original model.

My model reflects pulling forward after the initial car backup, ending some allowable distance from the curb. It still includes a lot of simplifying assumptions, but I'd claim it captures a crucial maneuver in parallel parking.

5-3 Sum of Sinusoids

Watch before Day #49 lesson

When given a function that is the sum (or difference) of two sinusoids, such as f(x) = 8sin(x) + 4cos(x), how can we rewrite this as a single sinusoid using our Sum/Difference formulas?

Assessment on 5-1 and 5-3

Day #50, no video

Here is exactly what the next assessment will look like (actual problems have been blurred, of course). Content and wording reflects what you see in your homework. Any questions on the content and wording need to be addressed before the assessment block.

Go to next page, Chapter 5.4.

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