Math with Construction Paper

Chris explained the proof for the sum of angles identities (cosine and sine) to me at breakfast. He used colors, because I get lost when there are more than two triangles overlapping, even when all the points are labeled. Also, colors are pretty and pleasant and non-threatening, which helps me get through math that would otherwise intimidate me.

We decided it would be awesome to post a picture of the proof, but Chris wanted a cleaner version than the one he'd used at breakfast. So he and I made a proof out of construction paper pieces (one piece for each of the four triangles that matter for the proof).

Here it is! I'm going to try and explain what is happening in each picture as I go. I think it's surprisingly pretty and comprehensible! Some of the pictures ended up oriented incorrectly (oops!!!).

Start with two right triangles, with the second (yellow) triangle's base equal to the length of the first (orange) triangle's hypotenuse.

The triangle that we use to find the Sum of Angles (the point of the whole exercise) is white here. The base runs along the bottom/adjacent side of the orange triangle, and the opposite side is a vertical line dropped from the highest point of the yellow triangle. This makes the corner of the white triangle which covers the corners of the orange and yellow triangles equal to the Sum of (their) Angles.

The last triangle we need is the green one. It is created from having a line perpendicular to the white triangle split the yellow triangle. I'm not sure if I'm describing it very well, but hopefully the picture is fairly clear.

Alright! Now that we know the general layout of our four triangles, let's start actually defining the lengths of the sides! An important note: all of the triangles we are working with a RIGHT triangles, so we know ahead of time that one of the angles is equal to 90 degrees.

We start with the yellow triangle, and say that the hypotenuse equals 1. We name the angle we're interested in beta (I'll use "b" when I'm typing, but the symbol in the picture is a Greek letter). Using info from the Unit Circle, (and at least for me, the mnemonic "Soh Cah Toa"), we label the adjacent side as having a length equal to cos(b) and the opposite side as having a length equal to sin(b).

Next we label the white triangle. We know that hypotenuse is 1, the same as the yellow triangle. The angle we're interested in is alpha plus beta, or a+b. Again, using the Unit Circle derived information (or my cheating Soh Cah Toa method) we label the adjacent side as having length cos(a+b) and the opposite side having length sin(a+b). But actually calculating cos or sin of two angles added together is a pain. So a formula explaining how to manage it would be clever.

The orange triangle comes next (for me, at least). We call the interesting angle alpha, or a. Unlike the nice hypotenuse of the yellow triangle, the orange triangle is not 1. Laying it next to the yellow triangle, we can see that it turns out to equal cos(b).

Once we know the hypotenuse of the orange triangle, we can state the lengths of the other two sides. First we say that cos(some angle)=(length of adjacent)/(length of hypotenuse), and with some algebra we get (length of adjacent)=(length of hypotenuse)cos(some angle). Plugging in the info we already know about our hypotenuse and our angle, we get (length of adjacent)=cos(b)cos(a). Then we use the same process for figuring out the length of the opposite side. Written out, it looks like: sin(some angle)=(length of opposite)/(length of hypotenuse) and with some algebra we get (length of opposite)=(length of hypotenuse)sin(some angle). Again, plugging in the information we already know, it turns out that (length of opposite)=sin(a)cos(b). Ta-da! That's our orange triangle!

Our white triangle shows us that the length of cos(a+b), or the length of the adjacent side of the white triangle, is less than the length of cos(b)cos(a), or the length of the adjacent side of the orange triangle. Also, the length of sin(a+b), or the length of the opposite side of the white triangle, is greater than the length of cos(b)sin(a), or the length of the opposite side of the orange triangle. But exactly how much more (for opposite) or less (for adjacent)?

Here is where our green triangle is going to come in handy. One side of the green triangle is the length to add to the orange triangle find the total length of sin(a+b) and another side of the green triangle is the length needed to subtract from the orange triangle to find the total length of cos(a+b). The last side of the green triangle is its hypotenuse.

Here is the length of the hypotenuse of the green triangle. It ends up equaling the length of the opposite side of the yellow triangle, or sin(b).

And here is a better view.

To find the desired angle in the green triangle, start by pretending there is a rectangle that is made from the orange triangle (the grey arrows in this picture). Because the top and bottom of the rectangle are parallel, the diagonal line (the blue arrow) creates the SAME ANGLES as it crosses the top and bottom lines. Therefore the part of the yellow triangle showing below the green triangle and above the orange triangle forms an angle of a (or alpha). The corner of the green triangle directly above that completes the right angle of the yellow triangle, and therefore equals (90-alpha). Knowing two of the angles in the green triangle gives us the last angle: (sum of all angles in a triangle is 180)-(90 from right angle)-(90-alpha)=(180)-(90)-(90)+(alpha)=alpha.

Now that we know the angle and the length of the hypotenuse, we can find the remaining lengths using that same method as we used on the orange triangle. Shortcutting a little bit, let's go (length of opposite)=(length of hypotenuse)sin(some angle) which ends up equaling sin(b)sin(a); then the (length of adjacent)=(length of hypotenuse)cos(some angle) which ends up equaling sin(b)cos(a). And there's our finished green triangle!

Here are all of our triangles on top of each other.

Length of the adjacent side of the green triangle plus the opposite side of the orange triangle equals the opposite side of the white triangle. The formula looks like: sin(a+b)=[sin(b)cos(a)]+[cos(b)sin(a)].

Length of the adjacent side of the orange triangle minus the opposite side of the green triangle equals the adjacent side of the white triangle. The formula looks like: cos(a+b)=[cos(b)cos(a)]-[sin(b)sin(a)].

The End. <3