When is the law of sines ambiguous

We can see right away that the problem that existed in Example 1 is not a problem with this triangle. The value of a is larger than the height from C 8 and a triangle will be formed. But, if we swing side a from point C to the left, can we form a second triangle? Looks good! But wait!

We know that the sine of an obtuse angle is the sine of its supplement. But since a is smaller than b , it can "swing" to the left of h and create a second triangle containing an obtuse angle. This type of triangle is called the Ambiguous Case! Ambiguous means that something is unclear or not exact or open to interpretation. So, if we encounter a triangle that has SSA congruency, we have an ambiguous triangle in the sense that we need to investigate more thoroughly.

Remember how the sine function is positive in both the first and second quadrants? Using this method, we find that the one triangle exists with an angle B equal to 48 degrees. Does another exist? To prove that another triangle exists, we must determine if another possible angle B exists that makes the triangle possible. Notice the isosceles triangle created by the Ambiguous Case above.

This means the second angle B is degrees. Since we know all angles in the triangle must add to degrees, a second triangle is not possible because angles A and B alone exceed degrees. If a second triangle was possible, angles A and B would sum to a number smaller than degrees. You should find that angle B is roughly 69 degrees. This time angle A and angle B sum to less than degrees, making an angle C possible it measures roughly 11 degrees.

Therefore, a second triangle is possible. High School. By Maryam Amr. A larger percentage of students also seemed to handle problems related to the ambiguous case better then their predecessors. I did not actually go back and compare grades; this observation is based solely on my memory. Click on the link below to see the demonstration GSP 4. At the top.