BQ #1: Unit P Concepts 1-3 and 4-5: Law of Cosines (Derivation) and Area Formulas (Oblique Derivation)

Big Question Blog Post #1

Questions:

3. Law of Cosines - Why do we need it? How is it derived from what we already know?  The derivation must be shown either in a video or in multiple sequential pictures and it should include descriptions and information beyond what you can find in the SSS.

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?

Responses:

3. The Law of Cosines: First of all, before I explain why the Law of Cosines is needed, I must explain what it is; the Law of Cosines is a formula relates the lengths of the sides of a triangle to the cosine of one of its angles. The law of Cosines can be used to compute the remaining sides of a triangle when two sides and an angle are known(SAS) or to calculate the remaining angles when you know all three sides of a triangle(SSS). According to this law the following is true,

The Law of Cosines will be explained in the example(where α is the interior angle at A, β is the interior angle at B,  is the interior angle at C and c is the line AB), however we will only use variables for now:

4. Area Formulas:

Normally, if we wanted to find the area of a triangle we would use, A= 1/2bh. However, with an oblique triangle we are not given the value of "h" and so must compute for the area of a triangle without it and substitute our regular area equation for h. The way of finding the area of an oblique triangle without being given "h" will be shown in the next few steps.

 Using the labels in the triangle above, the altitude is h = a sin . Substituting this in the formula Area = ½ × b × h derived above, the area of the triangle can be expressed as:

(where α is the interior angle at A, β is the interior angle at B,  is the interior angle at C and c is the line AB).

Furthermore, since sin α = sin (π − α) = sin (β + ), and similarly for the other two angles:

Works Cited

1st Picture: Law of Cosines-http://www.mathwarehouse.com/trigonometry/law-of-cosines-formula-examples.php

2nd Picture: Cosines Triangle Example- http://en.wikipedia.org/wiki/Law_of_cosines

3rd Picture: Area Formula Triangle Example-
http://en.wikipedia.org/wiki/Triangle#Computing_the_area_of_a_triangle