The Physics Philes, lesson 21: Double Down on Double Angles
In which indentities are defined, trigonometric equations are solved, and the door to calculus is opened.
Ermahgerd, you guys. This is the last post before I delve head first into calculus. This is the last post where anything is familiar to me at all! There are fun times to come.
But first, let’s finish off this trig stuff. There are two topics I need to review: identities and trigonometric equations.
An equation is an identity if that equation is always true, regardless of the input. One such set of identities are the Pythagorean identities. These identities created the Pythagorean theorem (please don’t as me how). You remember that theorem, right? Say it with me!
The sum of the legs of a right triangle is equal to the square of the hypotenuse!
But that is neither here nor there. What are the identities?! There are three:
First, let’s input into cos^2 x + sin^2 x = 1.
See? It equals 1, just like the identity said it should. We can use these Pythagorean identities to simplify problems, as well. Let me show you how. Using a Pythagorean identity, I’ll simplify cos^2 x / sin x + sin x.
Hmm…that numerator looks familiar. Oh! It looks like the first Pythagorean identity! So we can just replace the numerator with 1 and we’ll be all set. Right?
Wrong! Remember last week when I went over all those trigonometric functions? Which one does this look like? That’s right! Cosecant! So our initial equation can ultimately be simplified to…
It’s pretty easy once you know how, huh? But there are more identities we need to talk about. It’s time for double-angle formulas. This allows us to trigonometric equations containing double-angles using single-angle expressions. If you’re like me and had no idea what double angles are, they are sin 2x or cos 2θ. Double-angle formulas basically replace the coefficient 2. Here are the formulas:
There are actually three equations you can replace cos 2x with. More to remember, but it also gives you more flexibility, I guess? Oh well. Let’s do a quick problem to demonstrate the double-angle formula’s abilities.
Factor and simplify cos^4 θ + sin^4 θ.
Oh man. Do you remember way back when when we did factoring? In this situation, we’d use the pattern (a + b)(a – b) to get:
Anything about this equation look familiar? The first set of parentheses is the first Pythagorean identity. So we know the first parenthesis is equal to 1. The second parentheses is one of the cosine double-angle formulas, we know that the second parentheses is equal to cos 2x. Which means that:
(1)(cos 2x) = cos 2x
Ta da! Simplified! And see what we did? We used a Pythagorean identityand a double-angle formula! I love when things come together like that, don’t you?
There’s just one more thing in trigonometry I need to review before I move on to calculus. I need to learn how to solve trigonometric equations. I think the best way to do this is to jump right into an example.
Solve the equation cos 2x – cos x = 0 on the interval [0, 2Π).
OK, before we do anything else, what is up with that [0, 2Π) nonsense? That is interval notation. The interval [0, 2Π) means that x is less than or equal to 0 and less than 2Π. Or, in math speak 0 ≤ x < 2Π. It indicates the lower and higher boundaries of the interval. The bracket indicated that the boundary is included, and the parenthesis indicates that the boundary is not included. If infinity is the boundary (example: x ≥ 7, or [7, ∞)) then parenthesis is used.
I know, I know. It’s a little hard for me to wrap my brain around, too. But let’s power through, shall we?
So we’re solving cos 2x – cos x = 0 on the interval [0, 2Π). First, we need to eliminate all the double-angle formulas so all the terms left are single formulas:
To find the answer, we need to determine when cosine is equal to -1/2 and when cosine is equal to 1. We determine this by looking at a unit circle chart. So we know that the answers – in the interval of [0, 2Π) – are:
So there we go! I’m done with trig! At least done with what the particular book I’m using thinks I should know before venturing into calculus. But I miss physics. So from here on out I’ll be inserting some physics in between my math posts. Physics is just more fun, no?
Stay tuned next week as I Muppet flail my way through limits!
Featured image credit: jimmiehomeschoolmom