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The Pythagorean Identity:
sin2X + cos2X = 1
The trig identity that you should never forget: sin2X + cos2X = 1. In this video I show you where it comes from, how to use it in proofs, and how to spot proofs where it might come in handy.
Sum & Difference Formulas, and Even-Odd Properties
This video contains the remainder of the "popular" sum and difference formulas that almost every trig teacher seems to go over for sine, cosine & tangent. If you have to memorize these, I tell you tips for that as well, but most teachers will give them to you. Also, what do "even" and "odd" have to do with trig?
Half-Angle Formulas for Sine, Cosine & Tangent
"Half-angle formula" and "double-angle formula" sound pretty similar, so you'd think they'd be equally important. Nope! But I give the half-angle formulas their own video anyway because they seem to generate the most confusion vis-a-vis which angle to pick for θ and θ/2.
Double-Angle Formulas:
sin2X = 2sinXcosX &
cos2X=cos2X-sin2X
Of all the formulas in the Trig Identities chapter, the double-angle formulas are the only ones you'll ever see again in Calculus. In this video we'll take a look at the double-angle formulas for sine and cosine and work a few examples. And I throw a proof in there, just in case you're in honors and have an aggro teacher.
Tricky Proofs - Honors Only! (free)
This video features some brutal examples which show how the Big Three proof tools can get you through even the diciest proofs. It also shows that even folks who are pretty good at trig don't necessarily see how to solve a proof when we begin; you just have to keep messing around until it gets there! If you're in honors, these trig proof examples are for you!
Trig Proofs With Complex Fractions
Trig proofs get a lot harder when they don't have exponents in them, since you can't use the Pythagorean Identities on them. Instead, you've got to use the tricks I show you in this video to turn denominators like (1 + sinX) and into expressions like (1 - sin2) where we CAN use the Pythagorean Identities. (If you still remember Algebra 2, you'll recognize this "conjugate trick" as a way we rationalized complex fractions and roots.)
Proofs Using The "Other Two" Pythagorean Identities: tan2X+1 = sec2X &
1+cot2X = csc2X
These two identities show up all the time in trig proofs, but they're really easy to get mixed up (wait, does tan go with sec or csc?). So, in this video I show you a great trick to memorize them so you can write them down at the top of your quiz or test (a practice I highly recommend). I also show you how to spot trig proof problems where they'll come in handy, and work a few examples.
These problems are frustrating for students because they pretty much all look the same, yet you can lose big points if you get one little shift or label wrong. Plus there's the annoying vocab: Amplitude, "b", phase shift, vertical shift. No worries, the videos in this chapter will sort it all out using explanations and techniques my students have found helpful, and I'll point out common errors to avoid.
Proofs strike fear in many a heart because most students were traumatized by proofs in Geometry. But have no fear. In this chapter I'll show you how to easily memorize the key identities you'll need, as well as take you through the Big Three techniques that will help you spot and solve most proofs.
"General Solutions" vs "Specific Solutions"(without exponents)
This video will answer a host of questions: What does [0,2pi] mean? Why "2npi" or "360n"? This is where you have to pay careful attention to whether they're asking for a general solution where you have to account for infinite co-terminal angles (thus the "n"), or just limit yourself to the unit circle other interval.
Solving Trig Equations on your Calculator
When trig teachers let you use a calculator for something, nine times out of nine it's a trap, and this is no exception! See, when you solve a trig equation on your calculator, your only choice is the inverse trig buttons. But guess what: those will limit you to only a couple quadrants! Teachers know this, hence the trap. In this video, I'll show you how to deal with that and find the answers your calculator doesn't want to give you.
Solving Squared Trig Equations -- No Factoring
Putting an exponent on the sine or cosine makes trig equations twice as hard. In this video we'll work the slightly easier type where there is ONLY a squared trig function -- nothing to the first power -- which allows us to square root both sides: cos2X-1=0, sin2X-2=0, and one with no solution! In the next video on the trig equations page, we'll factor.
Factoring - Let's review!
If you're already a factoring master, feel free to skip this video. But if you haven't factored in a while you might want to watch it, since factoring trig equations is going to be twice as difficult as factoring with just x. For even more factoring practice, especially if you have a tough teacher, check out the Algebra 2 factoring page.
Solving Really Hard Trig Equations (with factoring!)
If you're lucky, maybe your teacher won't even make you do these, especially if you're not in honors. But for most kids, this will be the worst factoring problems you'll ever see. Stuff like sin2x-2sin+1=0, cos2x+2cosx-3=0, and 3tan2x-4tanx-5=0. (Notice I didn't say worst trig equations, because I'll reserve that honor for equations with a number in front of the x!)
Trig equations are problems where you're solving for X or Theta but they're hidden behind a trig function, like "2sinX-1=0" or "tan2X-1=0". Lots of vocab in this one -- specific solutions, general solutions, 2npi, 360n -- and lots of factoring to do too. Easy to get confused between these and inverse trig functions!
What if there's a 2 or 3 in front of Theta/X?
These are the worst -- cos3X-1=0, sin2X-2=0 -- and they're typically for honors students only. Believe me, I hate these as much as students do, and it took me a long time to figure out a good way to explain them. Luckily, that was a while back, so you get the fruits of my experience on this one! Still no fun, but if you drop your 360n or 2pi(n) in at the right time, it becomes a plug-and-chug affair.
Solving Basic Trig Equations (without exponents)
This first video just covers how to solve basic trig equations with no exponents and nothing in front of the variable, such as 2cosX-1=0, sinX-2=0, and tanX=2. T The trick with these is to forget all that stuff you just had to memorize for inverse trig functions about which quadrants go with what: the whole unit circle is fair game!
Also called Arcsin, Arccos, Arctan, etc., these are problems like sin-1(1) where they give you the sin/cos/tan of an angle and you're supposed to give the angle in the correct quadrant. Key for solving trig equations, I explain how to do these with the unit circle or a calculator.
Intro to Inverse Sine, Cosine & Tangent
Inverse trig functions are the exact opposite of the unit circle stuff you've seen up to this point. Before, it was always "what's the sine of X angle." Well, now it's going to be "give me the angle whose sine is X." And it turns out that's a bit tougher, especially since you've got to know which quadrants you're allowed to use, and it's different for each function!