place to stand

Trigonometry 8:

By William E. Steinman:

July 23, 2007:

Now that we are armed with our new tools of trigonometry we can take a new look at the area of a triangle. This can get very involved so let’s begin with our basic statement of the area of a triangle.

Remember this?

The area K of any triangle is equal to 1/2 the product of the base and the altitude.

Okay, let’s take a case where we know two of the angles, A and B and one side of the triangle.

Then, the third angle C is easy enough to find.

C = 180⁰ – (A + B).

Now the area K of our triangle is equal to a side squared time the product of the sins of the angles which include that side divided by two times the sin of the angle opposite the side.

Huh!

Perhaps we should draw the triangle. Let’s make it easy and just copy the triangle from SN 033. Here it is.

All of the verbiage above comes out to this.

K = (a² sin B sin C)/2 sin A = (b² sin A sin C)/2 sin B = (c² sin A sin B)/2 sin C

Oh yeah!

How do we know that?

Let’s take the case K = (b² sin A sin C)/2 sin B

Look at the triangle above.

We know that the altitude h is equal to (b sin C)

We know that the area K = ½ ah

Substituting for h we get K = ½ a b sin C

Now, recall the law of sins.

a/sin A = b/sin B = c/sin C

So, a = b sin A/sin B

Substituting again we get K = ½ b sin C (b sin A/ sin B)

Finally K = (b² sin A sin C)/2 sin B

For your own amusement, you can prove the other two cases in the above statement.

Okay, if we are given two sides of a triangle and the angle opposite one of them, what can we do about the area?

Take the two sides as a and b and the opposite angle as A

We can use the law of sins to find the other opposite angle B.

That is sin B = b sin A/a

We can do a table look up to find A and then we can solve for sin B and look up B in the table.

Now we have two inclusive angles and a side.

This means our area K will be equal to one of the three cases we have already proven.

K = (a² sin B sin C)/2 sin A = (b² sin A sin C)/2 sin B = (c² sin A sin B)/2 sin C

Hot darn! Ain’t that swell?

Now, if we are given two sides and their included angle we can find the area using this formula, on case of which we have already proved above.

K = ½ ab sin C = ½ ac sin B = ½ bc sin A

Hot darn again!

We have one other case for solving for the area of a triangle but it’s rather tedious. We can get to that next time. By the time you prove all of this to your own satisfaction this will be quite enough for today.

You are doing the proofs, aren’t you?