BQ: Law of Sin and Area Formula

Law of Sin

To being with, we will be looking at the law of sin and how it is derived to get the actual formula. The law of sin begins with  a triangle. When it is cut through, it give us a hypotenuse.

In trigonometry, we know that sin is opposite over hypotenuse. Therefore, we know that the sin of angle A is equal to h over c. You multiply both sides by c to get h alone 

Next we must find the law of sin for angle C. Using soh-cah-toa the sin of angle C is h over a. We must isolate the h by multiplying a to get aSinC.

Finally we can use the transitive property to derived to find the real sin formula. You take the answers to both angle and set it equal to each other. Divide by ca to both side to get SinA over side a and SinC equals to side c. There, you get the toe ratio of the law of sin 

You can only use the law of sin if you have AAS(angle, angle, side), ASA(angle, side, angle), and SSA(side, side, angle).

The case when you cannot use the law of sin is when you don't know the opposite side or angle. For instance like in this picture it is given the side but the angle is missing. In addition to, the angle is also given but the opposite side is missing. 

4.Area Formula

The area of an oblique triangle is derived with a combination of the law of sin and a regular area of a triangle. You would use the same formula of 1/2 times based times height with the substitution of h. To find h, you use the law of sin, h is equal to aSinC. There, you have the devriation of the oblique triangle.  Also for it to be an a oblique triangle, all sided must be different.

There is a several different form of the area depending in the angles you are looking for.

Reference:

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