BQ #1: Unit P Concepts 2 & 4: Law of Sines and Area of an Oblique Triangle

2. Law of Sines

Why is SSA ambiguous?

• SSA is has three possible solutions: one, none, or two. When solving a problem for this triangle, you need to solve it as if you're going to get two possible solutions. The only way you know there's only one or no solution is when you "hit a wall". A wall can either be when sine is less than -1 or more than 1. Remember that the trig function sine should be between -1 and 1 to be possible. Another wall can be if the angle of your triagle adds up to be greater than 180 degrees. We know that a triangle can only be 180 degrees so if two angles already add up to that amount, we will hit a wall. It is possible that when we find the two possible value of an angle (one acute in the first quadrant; one obtuse in the second quadrant), one can still work for a possible of one solution!

Two possible triangles-

When given the triangle, we match each angle to its corresponding side and we can indicate the "magical pair" and "bridge". The "magical pair" is the one with both given angles and side while the bridge is the one with either a given side or and angle. We use the law of sines using the pairs of B and C to find the value of angle B.

In order to make sure a wall was not hit, the Triangle Sum Theroem can be used. After finding the possible values of B, we add them to the given angle of C and make sure it does not go over 180 degrees. Afterwards, we can find the degree of angle A and A'. Afterwards we use the same maigcal pair and use the Law of Sines but with the unknown variable of side a. 

Our answer results with two possible triangles, one acute and the other obtuse.

One Possible Triangle-

As usual, we find the magical pairs and bridge. Then we use the Law of Sines to find the value of angle C. However, only one of the angles work in this problem because we have hit a wall. The value of C' is 159 and with that added to 84, it exceeds 180 degrees. So now we know that angle won't work for a triangle but we must continue to find the rest of the other triangle.

Using the Triangle Sum Theroem again, we find the value of angle B. Then we use the Law of Sines to find side b.

Our answer results in one possible triangle.

No Possible Triangle-

In this problem, when we find sin C, we result with 1.96 Remember that sine can only be -1< sine Ø < 1. Since sine is over 1, there is no solution and this results in no triangle.

4. Area Formulas

How is the “area of an oblique” triangle derived? 

• The area of an oblique triangle is derived from the area of a triangle formula.

Since we know the area of a triangle is A=1/2bh, we can make a substitution for h that will help us find the area of an oblique triangle. If we are looking for the angle C, we know that the sine formula will be h/a sine it's opposite over hypotenuse. After we multiple a to both sides, we get h alone and this is where "a sin C" can be subbed into h from the regular area of a triangle. From there on, we can find the area of an oblique triangle by plugging in the two sides and the sine of the included angle. 

It can also look like these options depending on what sides and angle is given:

(http://wps.prenhall.com/esm_blitzer_algtrig_2/13/3560/911520.cw/index.html)

How does it relate to the area formula that you are familiar with?

• This relates to the area of a triangle (A=1/2bh) because it is derived from it. There is both the base and height multipled together and divided by two, however, the oblique formula also has the sine of an angle multiplied.

Works Cited-

1. http://wps.prenhall.com/esm_blitzer_algtrig_2/13/3560/911520.cw/index.html