Breaking Down a Cool Sin(3x) Identity

Proof with trig identities

Image made by me, background image from Commons (public domain)

There’s a a cool Twitter bot called infseriesbot. The other day they posted this:

These are some neat identities for sin(3x), cos(3x), and tan(3x)!

I’m going to quickly break down why the sin(3x) identity is true, step-by-step, pointing out the identities I use along the way.

Angle-Sum Identities

We’ll start with these two trig identities:

Using these two identities, we can calculate sin(π/3 + x) and sin(π/3 - x):

But we can simplify these by finding sin(π/3) and cos(π/3). The angle “π/3” is 60 degrees. We can find sin(π/3) and cos(π/3) with a 30–60–90 triangle:

Image from Commons (public domain)

Basically, sin(π/3)=(√3/2), and cos(π/3)=(1/2). So now we have the following:

Difference of Squares: (a+b)(a-b)

You’ll notice the 2 formulas so far are very similar.

If we define a=(√3/2)cos(x) and b=(1/2)sin(x), then the following is true:

We have this handy identity from algebra:

Apply the identity and simplify and we get:

Multiply both sides by 4 to cancel out those (1/4) fractions:

Multiply by sin(x)

In order to get the left-hand-side of our original identity, all we need to do now is multiply by sin(x):

The cos²(x) Identity

Now it’s time to introduce another identity you’ve probably seen before, sin²(x)+cos²(x) = 1.

Rearranging the terms a bit, we also know cos²(x) = 1-sin²(x). Let’s replace cos²(x) with (1-sin²(x)) in our equation:

Now we have this equation:

Proof That sin(3x) = 3sin(x)-4sin³(x)

Now we need to prove that 3sin(x)-4sin³(x)=sin(3x).

We could prove this by turning 3sin(x)-4sin³(x) into sin(3x). But I think it’s easier to turn sin(3x) into 3sin(x)-4sin³(x) instead.

Start with this:

Angle-Sum Identity (Again)

Here’s a trig identity we used earlier:

Apply the identity:

Double Angle Identities

Here are two other trig identities (these are the last new identities we’ll need):

Apply the identities, then simplify:

Like we showed before, cos²(x) = 1-sin²(x). Replace cos²(x) in the equation and simplify:

Putting Everything Together

We proved two separate equations. First, we showed that:

Second, we showed that:

Putting those two equations together, we get the original identity we wanted to prove:

And that’s the end of our proof!

At the beginning, you saw the bot’s identities for cos(3x) and tan(3x) too. The proofs are similar, but we’re at the end of this post, so… I guess those are left as exercises to the reader!