BQ# 1: Unit P Concept P: Law of Sines, Law of Cosines, Area formulas

BQ #1: Unit P Concept 1-5: Law of Sines, Law of Cosines, Area formulas

3. Law of Cosines - Why do we need it? How is it derived from what we already know?  

(http://mathworld.wolfram.com/LawofCosines.html)

Answer: The Law of Cosines is a very helpful law that helps us find answers to the triangle we're working on. The Law of Cosines can be used in any triangle. The formula for the Law of Cosines is    but the law itself can be arranged in two other ways. The Law of Cosines is useful for finding: The third side of a triangle when you know two sides and the angle between them. The angles of a triangle when you know all three sides. Also, you can rewrite the c² = a² + b² - 2ab cos(C) formula into "a²=" and "b²=" form.

4. Area formulas - How is the “area of an oblique” triangle derived?  How does it relate to the area formula that you are familiar with?

Answer: What if you wanted to find out the area of the triangle on your left, but did not know the value of h? Could we do it? So, the area of a triangle is A=1/2bh, where b is the base and h is the perpendicular height of the triangle. In the given triangle on the left, we know that sinC= h/a, and therefore h=asicC. Substituting in our regular area equation for h, we get A=1/2b(asinC). The area of an oblique (all sides different lengths!) triangle is one-half of the product of two sides and the sine of their included angle (the angle in between them). Depending on which sides and angles we know, the formula can be written in three ways:

Either: Area = 1 ab sin C 2

Or: Area = 1 bc sin A 2

Or: Area = 1 ca sin B 2

There are different ways to find an area of a triangle. You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been know for nearly 2000 years. :O 
References/Citations:
http://www.mathsisfun.com/algebra/trig-cosine-law.html
http://mathworld.wolfram.com/LawofCosines.html
http://www.mathsisfun.com/geometry/herons-formula.html
http://www.mathsisfun.com/algebra/trig-area-triangle-without-right-angle.html