The law of sines. In previous examples of trigonometry we were limited to right triangles. Now let's see how trig works in oblique (not right) triangles. You may recall from Plane Geometry that if you know any of the following information, you have a unique triangle. 1. ASA or angle side angle. 2. AAS or angle angle side. 3. ASS or angle side side (this is a special case as we shall see) 4. SSS or side side side. 5. SAS or side angle side. The Law of Sines can be used with the first three of these ASA, AAS, and ASS. While the law of cosines (which we'll cover later) covers SSS and SAS. The point to remember here is that given any three values in the above list, the triangle's other sides and angles can be found because the triangle's shape is fixed once you know any of these combinations. Notice that there is no AAA angle angle angle rule. That's because three angles determines if triangles are similar, but without at least one side, they can be any size. First let's derive the law of sines. Suppose we have a triangle ABC with sides abc. If we drop a perpendicular line from C to side c we divide the triangle into two right triangles. if the perpendicular is h, and we look at the two triangles separately, we can see that sin ( A ) = h / b and that sin( B ) = a / h. Solving for h in both equations gives us: h = b sin( A ) and h = a sin( B ). since both are equal to h set them equal to each other, giving
b sin( A ) = a sin( B ). Rearranging these by dividing both sides by ab gives us the relationship between angles and sides in a triangle known as the law of sines. We could drop a perpendicular from A to side a and show that sin ( C ) / c is equal to sin ( B ) / b as well which covers all three sides and all three angles. In order to use the law of sines it is necessary to be given at least one angle and the length of the side opposite to it. and one other angle or side. This is the basic expression for the law of sines. Now let's see what it can do for us. Suppose we have a triangle and are given the following information:
This is an Angle-Side-Angle or (ASA) problem. Given an angle, an adjacent side and the next angle, find the third angle (angle C) and the two unknown sides (a and b). First find angle C. Since a triangle has 180 degrees, subtract the two angles we are given from 180 and we get 80. S o angle C is 80. Now we are ready to apply the law of sines as we have one angle and its opposite side (C and c)
This was an ASA or angle side angle solution. You can easily see that given an AAS or angle angle side, you would just as easily solve for the other angle and sides using the law of sines. As I mentioned above, ASS or angle side side is a special case that also uses the law or sines and that is because it gives rise to three possible results. 1. no triangle is possible. 2. there is only one triangle and 3. there are two possible solutions (this is called the ambiguous situation). Let's examine all three of these.
Suppose we were given the angle at A, the length of b and the length of a. I like to think of this as a fishing pole scene. Angle A (where the fisherman is sitting at the water's edge) is the angle of the pole, b is the length of the pole and a is the length of the fishing line swinging off the end of the pole. Think of AB as being the surface of the water. You can see from the diagram that if the fishing line a is too short, it will not reach the water at all. This means its possible to invent an ASS probelm that has no solution. If a does not touch the water at some point (B) a triangle doesn't form. So how do we make sure that, given specific numbers angle A, side b and side a we actually have a triangle at all. Drop a height from C down to side AB. Then use the sin (A) = h / b to solve for h. This h is the shortest distance between C and line AB. If a is shorter than that line, then it cannot possibly reach AB and there is no triagle formed... ie no solution.
If a is exactly = to h, then the solution is a right triangle and it is then easy to calculate the remaining angle at C and the length of c which is AB. But suppose a is longer than h. Then we know we have at least one solution. Let me draw that solution. Now this is where we have to decide if we have two solutions of one solution. ie answer the question Is this an ambiguous solution? Look at the following diagrams and realize that it is possible to get either one solution or two solutions.
These are the two solution triangles. That's the theory. Now let's do an example of this ambiguous situation step by step. Given the ASS triangle angle = 55, side b = 37 and side a = 35.
At this point we have calculated angle B in the first triangle. Now
we can calculate angle C by subtracting 180 - (55 + 59.99 ) = 65.01 Now use the law of sines again to find side c in this first triangle. This concludes the first triangle. The second triangle starts by using the angle B1 to calculate angle B2.
This concludes the calculations for both triangles in the ambiguous case for the law of sines.