8.2 The Law of Cosines

Greetings! It's Eleni.
Today, or yesterday, or whenever you choose to learn it, our class learned about the law of cosines.
Here's the link of a math teacher's lesson that Mr. Wilhelm gave us.
http://www.youtube.com/watch?v=24Ibcl3mBQU
This is a picture of the proof:

The a, b, and c are side lengths, while capital letters are angles.  Don't get caught up on which is a, b, or c; here's an easier way to think of it :
Side squared= the sum of the other two sides squared minus two times the same two sides and cos of the angle across from the initial side.
It may sound confusing now...but it actually helps and is better than having to memorize the formulas.

LAW OF COSINES 

When solving a triangle, the law of cosines can be used when you know SAS or SSS. Remember: the law of sines will also come in handy when doing these problems.
SAS
Example problem:
a=5.0, c=8.0, and B=77 degrees




1. draw a triangle and label

2. Since B is the angle between sides a and c, it is easiest to figure out b first.


law of cosines

2. Now that you know a side and it's angle (B and b) you can use the law of sines!

3. Lastly, knowing two out of the three angles of the triangle, we can easily find C.
180-77-35= 68 degrees =C 
SSS
Okay, so doing that last problem step-by-step was a lot of work soooo..... I'm just gonna give you the gist of it.
Example problem:
a=90 b=70 c=40

1. FIND THE LARGEST ANGLE FIRST
even if you have to find the smallest to answer the question, find the largest first.
* the largest angle is across from the largest side (angle A)

2. Now you can use the law of sines to find another angle

3. Subtract both angles from 180 to get the last angle

Another thing we learned in 8.2 was

Heron's Formula

This formula  finds the area of a triangle, any triangle.

S is one half the perimeter
a, b, and c are side lengths.
Another way to find the area of a triangle that's a lot more useful is

---where a and b are any two sides and y is the angle between them (SAS).
Also in the homework, they ask you to combine both of these concepts.
Example: Approximate the area of triangle ABC
A=35.7degrees
C=105.2 degrees
b=17.2
simply use the SAS version of the law of cosines
then use the law of sines like we did in the example above
after, use the a,b, and c side lengths to find the perimeter and therefore s
plug em in the equation.


sooo as we all know mr.w is returning in two days so I thought I'd map it out how far away from us he is.

(At least I think he's in San Diego)
According to google maps, that's only 2340 miles away
and by walking, it should only be roughly a 30 day journey
I hope wilhelm is a fast walker


That's all on 8.2 Bye!
---Eleni