8.6 Law of Cosines

• The Law of Cosines is more difficult to remember. It is also a formula that is much more difficult to derive/create. Remember these formulas are not must plucked out of the air, there is algebra behind each and every formula. Formulas are shortcuts that we recognize after doing a particular style of problem a numerous amount of times, the formula then allows us to complete such problems in a much quicker (sometimes more cryptic) method. One formula we know pretty well is the quadratic formula, its pretty cryptic, however it’s a shortcut that allows us to factor quadratics at a much more rapid pace..

• This is how I remember the formulas. The first letter and last letter are always the same.

• The middle numbers are then the left over letters. And their order does not matter. Must remember the operations of squaring and adding, and then subtracting 2 times their product.

• The Law of Cosines allows us to deal with non-right triangles that information is provided for in the form of SAS or SSS. Identify which of the 2 orientations is provided in the example and then solve the triangle.

Searching for b. So start with the formula, b2=a2+c2-2acCos(B)

Now that you know b, can you find the other angle measurements? Are there multiple methods to find m<A and m<C? Should they all give back the same values?

Finishing off the triangle

• Once you start finding out more things about your triangle then that usually opens the door for you to use different techniques/methods to find the remaining information. If I have to use Law of Cosines for something, I usually then solve the remaining parts using the Law of Sines simply because I think that the math for the Law of Sines is easier to work with. It is ultimately your choice.

Use Law of CosinesFind the measure of angle T.

Then find the measure of angle V.

How long is c?

• The N 75o E is a tricky concept sometimes to think about when you get into pre-calculus and physics we will discuss how “bearings” generally work. For questions like this I will provide you the (75o east of north) description