sin(3x)

Introduction

Hey there! In this section, we're going to explore how to figure out the sin(3x) equation using a cool geometric method. This way of looking at things helps us understand the equation really well, so we won't forget it easily.

To get started, We would first need to break the "3x" triangle into three triangles having x angles, and place them one on top of the other, Such that the total angle is equal to (3x).

Identify the sides.

In this task you have to find out how long the sides are.

Here's a tip: when the longest side(Hypotenuse) is 1 unit long, sin and cos tell us the lengths of the other sides. Move the labels for the sides to the right spots and then hit submit.

Constructions

Looking at the pictures, you'll see that sin(3x) goes straight up, but second and third sin(x) goes sideways. To figure out the vertical part of those sin(x), we need to do a special construction. You can play with the "Construct" and "Deconstruct" buttons to see how this construction works

Identify the angles.

Once we've built the entire figure, the next step is to figure out the angles inside the triangles. Drag and drop the angles where they fit and then hit submit.

sin(x) side resized

Okay, now we're going to get started with resizing the sine components to fit perfectly inside the sin(3x) side. We will start with the bottom sin(x).

Try moving the sin(x) side around and notice how it gets shorter. To figure out by how much, we're going to think about two triangles that look alike—one smaller and inside the other. See, when you change the sin(x) side, its new length is the regular sin(x) length times the length of the inside triangle's slanted side(Hypotenuse).

Figuring out components of sin(x)

Remember how the second and third sin(x) is tilted? Well, we're interested in just the part that is vertical. In the below figure transform the sin(x) sides to show the cos components of the sin(x) sides.

sin(2x)

The below button based animation shows us how the First and the Second sin(x) components transform to make the sin(2x) side.

Resized sin(2x)

The the below figure, the red line represents sin(2x) side, click and drag it to an appropriate position so that it's lenght is only represented by terms dependent on the x angle. Since, sin(2x) is made up of the transformed parts of two sin(X) sides, the two parts have to be moved independently of one another.

Bringing It All Together - The Last Equation

Now that we have the right parts for all the three sin(x) sides, let's put everything in one place and see how the equation works.Click the buttons below and watch how each of the sin(x) sides change.

Going from 3sin(x) to sin(3x)

After figuring out the transformed length of each sin(x) side, we can see in the below image how the sum of the three transformed sin(x) equals the length of sin(3x).

Simplification

Finally, we want to simplify the above equation so that it consists of only sin(x) terms and we will have the final equation.