The Ambiguous Case for Sine Law - Nerdstudy

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Let's learn the Ambiguous Case for Sine Law!

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So, there are situations where the information we’re provided with for a triangle make it possible for there to actually be 2 DIFFERENT triangles created even after satisfying all the pieces of the given information. We call a situation like this, the “Ambiguous Case”. So in this lesson, we’ll explore more about this “Ambiguous Case” and see how we can identify these situations using the Sine Law.

Before we begin this lesson, we highly encourage you to watch our previous video on “Determining if a Triangle Exists” if you haven’t already done so, before moving on with this video in order to get some more context on what we’re about to teach here. Otherwise, let’s get right to it!

So if we have a triangle where we’re given the Acute angle A, the length of the side NEXT to the angle, and the length of the side OPPOSITE to the angle...it is important to compare the two values of these side lengths. If the opposite side length is longer than the side length of the side next to the angle, then we would definitely only have 1 possible triangle created AKA 1 solution. So we would know how to solve this angle HERE with the help of the Sine Law which is just Sin ‘a’ over ‘a’ equals Sine ‘b’ over ‘b’.

However, what happens if ‘a’ is shorter than ‘b’? Well, first of all we’ve already learned that the ‘a’, which is generally the side opposite to the angle, must be longer than the height of the triangle. If it’s not, then we wouldn’t even have a complete triangle all together. But assuming that it IS longer than the height, BUT shorter than the length of the side right beside the angle, we’d have ourselves what we’d call an ‘ambiguous case’ which is when it is possible to either have this length drawn like this to make a triangle, or drawn like THIS to become a completely different triangle. Notice how both triangles still maintained the values of every side and angle that we were required to respect. So, let’s learn how to solve a problem when we’re given a situation like this.

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So, the way to algebraically find out if we have an ambiguous case is this. We first assume the side length of 5 on this side. Now, this makes for an angle over here that is vv with this angle over here, since these two side lengths are the same. This would automatically mean that this angle over here would be 180-70.1.The answer to this angle becomes 109.9.

Now, here’s the key, when we add THESE two angles, if we get a value less than 180, then we’d know that there is room for THIS angle over here and that we have successfully found our second case in the ambiguous case. So since 109.9 plus 36 is equal to 145.9, which is less than 180, we know that this angle over here will be 180 minus 145.9, giving us 34.1 degrees for this angle! So there we have it.. Our 2 different triangles based on the same information provided.
So, let’s try one more example of an ambiguous case together.

Here is our triangle with the following information. So, to begin with let’s check to see if the side opposite to the angle, namely side ‘a’, is greater than the height of the triangle. What we do again is, identify the height and use Sine of ‘x’ equals Opposite over Hypotenuse to get the following. After simplifying and computing for ‘h’ we get the height as roughly 6.43, which is less than 7. So, would we be able to expect an ambiguous case here

Well, the answer is yes, because we can see that we have the length of side ‘a’ being greater than the height, as well as the length of side ‘b’ being greater than side ‘a’, along with our angle being acute, giving us the perfect situation where an ambiguous case can occur!

Now that we’ve confirmed this let’s solve for the two different angles that produce the 2 triangles. So, let’s use the Sine Law to get the following. Simplifying gives us this and computing this gives us roughly 66.67 degrees. So we know that THIS angle here is equal to 66.67 degrees. Assuming the side length of 7 on this side as well, we know this angle is also 66.67 degrees. Now, what is the process we would use next to find this angle over here?

Well, it would be to subtract 180 by 66.67 degrees to get 113.33 degrees here. Again, if these two angles added together are less than 180 degrees, then we can confirm for a fact that we have ourselves another triangle. And we can tell right away that these two do not add up to 180 degrees, so it seems to us that we have ourselves that second case, with angle B as either 113.33 degrees or 66.67 degrees!