Aim: How do we handle the ambiguous case? Do Now: In ∆ABC, m  A = 30°, a = 6 and c = 10. Find  C to the nearest degree. We can use the Law of Sine to.

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Presentation transcript:

Aim: How do we handle the ambiguous case? Do Now: In ∆ABC, m  A = 30°, a = 6 and c = 10. Find  C to the nearest degree. We can use the Law of Sine to answer the problem. HW: p.574 # 6b,8b,10b,14b,16,18

Notice that sin C is positive 56° which can be in quadrant I or quadrant II. If angle C is in quadrant I then it is 56°. If 56° is in quadrant II then it is 180° – 56° = 124° Which angle measurement is correct for angle C? We call this situation as ambiguous case that means there are different choices for  C

We have to use the basic idea about the sum of the interior angles of a triangle to figure out. If  C = 56°, then  B = 180° – (56  + 30  ) = 94° If  C = 124°, then  B = 180° – (124° + 30  ) = 26° Both angle measurements satisfy the basic requirement of interior angles of a triangle. Therefore, we can say that there are two different triangles can be made with the given condition. How do we deal with this ambiguous case? We are given  A = 30 

Let’s see another example: In and c = 12 a) Find m  C b) How many triangles can be drawn?

or If then If Two interior angles of the ∆ABC are already over 180° Therefore, m  C =156° is invalid. The answer for a) is 24° only Therefore, there is only one triangle can be made with the given condition.

In ΔABC, b = 18, c = 10 and  B = 70 . How many triangles can be constructed? B A C   C undefined, therefore no triangle can be constructed

Application: 1.If side a = 16, side b = 20, m  B = 30°, how many distinct triangles can be constructed? 2. If m  A = 68°, side a = 10 and side b = 24, how many distinct triangles can be constructed? 1 triangle 0 triangle 3. If side a = 18, side b = 10 and m  C = 70°, how many distinct triangles can be constructed? 2 triangles