Small questions can lead to big things happening. Asking your friend, "How was your day today?" can lead to a 20-minute rant about the evils of capitalism, and how their manager, in particular, is the most twisted despot this side of Sauron.
Here's another question that seems simple enough: what happens if we try to add an angle to itself? That seems simple, but it leads us straight to the double-angle identities.
sin (α + α) = sin α cos α + cos α sin α = 2sin α cos α
Or, to have an actual double angle involved in this:
sin (2α) = 2sin α cos α
That's it, seriously. Easiest proof ever, right? Cosine is the same way:
cos (2α) = cos (α + α) = cos α cos α – sin α sin α
= cos2 α – sin2 α
Oh hey, squared trig functions. Remember that trig2 α is the exact same as (trig α)2. But they aren't the same as trig (α2). Got it?
An easy string of discoveries in a row can't last forever, can it? If anyone is going to ruin this, it's going to be tangent. Tangent is like a professional party ruiner.
Tangent must be on break or something, though, because the double-angle identity for tangent is easy too.
We're not complaining.
Sample Problem
Find the exact value of sin (120°).
We know all about sine and cosine of 60°, so let's use those and our shiny and new double-angle formula for sine to get sin (120°):
sin (120) = sin (2(60)) = 2sin (60) cos (60)
Notice that it this actually the same value as sin 60°. What could be behind such a coincidence? As always, our first thought is aliens. Our second thought is that the reference angle for 120° is 60°, and that sine is positive in Quadrant II.
Maybe that does explain it. If aliens were involved, they must have been super sneaky, and super useless. They didn't even do anything. How lazy.