Photo-credits: lovely, hand-drawn pictures by Mrs. Ventura
Using the picture from above, we will go step-by-step of the derivation of the equation. First things first, we must know that the law of sines is needed to solve missing parts of a non-right triangle. A regular right triangle can be easily solved because it has all the needed components, that being, most importantly, a right triangle. So, splitting a non-right triangle we get two right triangles. We solve for both angles A and C using sin. From there we get sinA=h/c and sinC=h/a. We take it one step further and we multiply by c and a ,for the two separate equations, in order to get h alone. Having h alone for both equations, we then combine them and move on over the a and c to their appropriate sides. From there we get the law of sines. Of course the law of sines is not just restricted to a or c, it also works for b as well. The law of sines fluctuates according to what specific information you are looking for.
Photo-credits: lovely, hand-drawn pictures by Mrs. Ventura
The area of an oblique triangle is derived from the area formula and a combination of a trig function. The area formula is A= 1/2bh. With a non-right triangle, trying to solve the area is much more difficult. Therefore, we split the triangle with a perpendicular line and get two triangles. From there, we find angle C. SinC=h/a, we multiply a to the other side in order to get h alone. Moreover, we plug in h into the original area formula. That, is the equation for solving the area of an oblique triangle.
It's similar to the area formula we know, except we manipulate it, in order to get what we need. The only difference is since we do not have h we use our sources and use it to the best of our abilities.
Below are other equations that are used to solve oblique triangles:
Photo-credits: lovely, hand-drawn pictures by Mrs. Ventura
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