BQ# 1: Unit P Concepts 1-5:Deriving Laws and Formulas

Big Question: Deriving oblique triangle laws and area formulas

PART 1 - Law of Sines:

Fig 1: http://hyperphysics.phy-astr.gsu.edu/hbase/imgmth/lsin.gif

The law of sines is needed when working with triangles that are not right triangles (those without a 90 degree angle measure). We need the law of sines to solve triangles (find all sides and angles), however, we can only use it after calculating the answers for corresponding sides and angles (i.e: knowing the values for angle B and side b). After having those values known, we can set up a proportion to solve for the rest. That proportion is derived from an oblique triangle and the result is what we know as the law of sines.

Deriving the Law of Sines: If we have an oblique triangle, we know that we can form a height from angle B that is perpendicular to the base to form two right triangles, like so:

If treated separately, those two triangles give different ratios for h. That is because in terms of sine, both triangles will use different values to form their ratios. In the end, the ratios are proportionate and h is the same value. With this relationship we can solve many triangles as long as we can calculate a relationship between a corresponding angle and side*.

*note: The law of sines will only work with AAS or ASA because these triangles are the only ones from which we can calculate a relationship corresponding with same angle and side. If it is not AAS or ASA the properties of the triangle would probably be found with law of cosines.  Examples of either are shown below:

Fig 2: http://www.mathwarehouse.com/trigonometry/law-of-sines/images/law-of-sines-and-cosines/law-of-sines-and-cosines-problem2.png

PART 2 - Area Formulas

The area of an oblique triangle is derived by substituting the value of h after forming two right triangles. We know that the traditional formula for the area of a triangle is:

Fig 3: http://www.calculateme.com/cArea/area-triangle-base-height.gif

If we do not know h, we can substitute it with a value of sine. The second cut triangle tells us h is equal to asinC.

The second cut triangle gives us a value for h:

This formula is the same as the formula for the area of a triangle, the value for h is just different, as it is written in terms of sine values. In the end, this formula works when the product of two sides and their included angle (i.e: SAS) is halved.

REFERENCES:
Fig 1: http://hyperphysics.phy-astr.gsu.edu/hbase/imgmth/lsin.gif
Fig 2: http://www.mathwarehouse.com/trigonometry/law-of-sines/images/law-of-sines-and-cosines/law-of-sines-and-cosines-problem2.png
Fig 3: http://www.calculateme.com/cArea/area-triangle-base-height.gif