*Jeff Calareso*Show bio

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Lesson Transcript

Instructor:
*Jeff Calareso*
Show bio

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

In trigonometry, a double angle is twice as large as a common angle. Learn how to calculate the double angle formula, explore trigonometry twins for sine, cosine, and tangent, and apply this information in practice problems using the double angle formula.
Updated: 09/28/2021

Let's say you're angle *x*. You're a happy angle. Life is good. You've got your home in a right triangle. And you have your straightforward sine, cosine and tangent friends, and when you get together with them, you know your values.

With sine, you're the opposite over the hypotenuse. With cosine, you're the adjacent over the hypotenuse. And with tangent, you're the opposite over the adjacent. SOH CAH TOA, nice and simple.

Then, one day, you find out you have a long-lost twin. Suddenly, you're not just *x*, you're 2*x*. Your world no longer makes sense. Most relevant to this lesson, your sine, cosine and tangent friends are confused. Fortunately, there are formulas for each of your friends to help. The **double angle formulas** are formulas that define the relationship between a trigonometric value and the double of the original angle. So take your twin and your old trigonometry friends and let's figure this out.

Let's start with the double angle formula for sine. It's **sin(2 x) = 2sin(x) cos(x)**. Wait, how can that be? Why couldn't we just find the sine of

Let's test our double angle formula with the same angle. So sin(2*x*) is sin60. For our double angle formula to be true, sin60 = 2sin(30) cos(30). Sin60, again, is (root 3) / 2. Sin30 is still ½, and the cosine of 30: (root 3) / 2. So we have 2 * 1/2 * (root 3) / 2. These twos cancel and we're left with (root 3) / 2 = (root 3) / 2. That'll put sine at ease.

So, where does sin(2*x*) = 2sin(*x*) cos(*x*) come from? It comes from the summation formula for sine. If we didn't have 2*x* but had *x* and *y*, then sin(*x* + *y*) = sin(*x*) cos(*y*) + sin(*y*) cos(*x*). If those *y*s become *x*s, you have sin(*x*) cos(*x*) + sin(*x*) cos(*x*), which is our double angle formula: 2sin(*x*) cos(*x*).

Next, let's look at the formula for cosine. This is also derived from the summation formula. If you recall, cos(*x* + *y*) = cos(*x*) cos(*y*) - sin(*x*) sin(*y*). Let's turn those *y*s to *x*s; cos(2*x*) = cos(*x*) cos(*x*) - sin(*x*) sin(*x*). This simplifies to **cos(2 x) = cos^2(x) - sin^2(x)**. And that's our cosine double angle formula!

Should we test this one, too? Your cosine friend says that just because it works for sine doesn't mean it'll work with cosine. They're not twins, too, you know. Let's try it with *x* = 30 again. So cos(2*x*) is cos(60). cos(60) is 1/2.

Then we have cos^2(30) - sin^2(30). cos(30) is (root 3)/2. If we square that, we get 3/4. Sin(30) is 1/2. If we square that, we get 1/4. So we have 1/2 = 3/4 - 1/4. 3/4 - 1/4 is 2/4, or 1/2. So we did it!

What would your buddies sine and cosine be without tangent? It's like Moe and Larry without Curly, Harry and Ron without Hermione, Snap and Crackle without Pop. Rice Krispies that just snap and crackle? That would never work.

Anyway, let's talk tangent. Again, what's the tangent summation formula?

- tan(
*x*+*y*) = (tan(*x*) + tan(*y*)) / (1 - tan(*x*) tan(*y*))

Oh, tangent, you're the Walter to the Dude and Donny - so complicated. Fortunately, tangent gets simpler as a double angle formula. Let's magically make our *y*s into *x*s again, and voila: tan(2*x*) = (tan(*x*) + tan(*x*)) / (1 - tan(*x*) tan(*x*)). We can simplify that to get **tan(2 x) = 2tan(x) / (1 - tan^2(x))**.

What do we do with double angle formulas? We test them. Should we use *x* = 30 again? Well, knowing what we know about tangent (that it's the opposite over adjacent), let's use *x* = 60. That'll make our math a little simpler. So that looks like this: tan(120) = 2tan(60) / (1 - tan^2(60)). The tangent of 120 is -(root 3). The tangent of 60 is (root 3). So we have -(root 3) = (2(root 3)) / (1 - (root 3)^2). We square (root 3) to get 3. 1 - 3 is -2. So we have 2(root 3) / -2, which simplifies to just -(root 3). Looks like we did it again!

Now we have our three formulas. To review, it's:

- sin(2
*x*) = 2sin(*x*) cos(x) - cos(2
*x*) = cos^2(*x*) - sin^2(*x*) - tan(2
*x*) = 2tan(*x*) / (1 - tan^2(*x*)).

We made your trigonometry friends feel better, but how else might we use these? You might see a problem that looks like this: find the value of cos(2*x*) if sin(*x*) = 3/5 and *x* is in Quadrant II.

This seems daunting, but our problem is packed with useful info. Let's start with that last bit. We're in Quadrant II. If we draw ourselves a quick *x*- and *y*-axis, we know the angle is somewhere over here. Our triangle looks like this. If sin(*x*) is 3/5, and sine is opposite over hypotenuse, we can label our opposite side and hypotenuse. Guess what? This is a 3-4-5 triangle! (If it wasn't, or if we forget about 3-4-5 triangles, we can always use the Pythagorean Theorem.) So this other side is 4. Since we're in Quadrant II, it's -4. So, cos(*x*) is -4/5.

Now, to our formula!

- cos(2
*x*) = cos^2(*x*) - sin^2(*x*). - cos(2
*x*) = (-4/5)^2 - (3/5)^2. - -4/5 squared = 16/25.
- 3/5 squared = 9/25.
- 16/25 - 9/25 = 7/25.

So cos(2*x*) is 7/25. That's our answer!

Let's practice one more. Find sin(2*x*) if tan(*x*) = -2/3 and *x* is between 270 and 360 degrees.

Again, let's start with locating our angle. Between 270 and 360 degrees is Quadrant IV. Since tangent is opposite over adjacent, we can label our side here -2 and here 3.

- What's the hypotenuse? 2^2 + 3^2 is 4 + 9, or 13. So the hypotenuse is root 13.
- That means sin(
*x*) = -2 / (root 13). - Cos(
*x*) will be 3 / (root 13). - What's sin(2
*x*)? 2sin(*x*) cos(*x*). So, that's 2 * (-2 / (root 13)) * 3 / (root 13). - Start with the numerators. 2 * -2 * 3 is -12.
- root 13 * root 13 is just 13.

So -12/13 is our answer! As you can see, if you know your formulas, all you need to do is draw a picture and then plug in your values. From there, it's just arithmetic. That should set long-lost twins at ease.

In summary, we defined, tested and practiced using double angle formulas. A double angle formula defines the relationship between trigonometric functions and the double of an angle.

We first looked at the formula for sine, which is **sin(2 x) = 2sin(x) cos(x)**.Then we looked at cosine, which is

After this lesson is done, you should be able to:

- Understand the trigonometric functions of the double angle formula
- Explain the formula of sine
- Identify the formula for cosine
- Recognize how you solve the formula for tangent

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