cos(x+y)
Introduction
Hey there! In this section, we're going to explore how to figure out the cos(x+y) 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 "x+y" triangle into two triangles having x and y angles, and place them one on top of the other, Such that the total angle is equal to (x+y).
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 cos(x+y) is horzontal, but sin(y) goes sideways. To figure out the horizontal part of sin(y), 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.
Similar Triangles.
Okay, now we're going to talk about something called "similar triangles." It's like when two triangles are related to each other in a special way. Let's see how it works with the interactive thing below. Try moving the sin(x) side around and notice how cos(x) 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, The cos(x) side's length is the regular cos(x) length times the length of the inside triangle's slanted side(Hypotenuse).
Changing the cos(x) side.
Now, let's use the similar triangles idea in the picture below. We want to find the right length for the changed cos(x) side. This is important for understanding the sin(x+y) equation. In the animation, move the cos(x) side to the right spot and hit submit. Remember, there's some multiplying going on.
Figuring out sin(y) side parts
Remember how sin(y) is tilted? Well, we're interested in just the part that is horizontal. The simualtion below will show you how that part is formed when we use sin(x) and cos(x) ratios.
Bringing It All Together - The Last Equation
Now that we have the right parts for sin(x) and sin(y) sides, let's put everything in one place and see how the equation works. Click the buttons below and watch how the sides change.