The Ambiguous Case

Two sides and the wrong angle can describe two triangles.

12 problems · about 22 minutes · G-SRT.D.11

NameDate

Figure — use these to answer the problems

Look at one hump: every height below the peak is reached twice, which is exactly the ambiguity.

y = 1 sin(0x) + 0

a, amplitude1

Point on the curve1

Figure 1

Straightforward practice. Get the method working first.

5 problems

An inverse sine gives 25°. What is the other candidate, in degrees?
Answer
2
Why does the Law of Cosines avoid the ambiguity?
- Cosine is negative for obtuse angles, so the sign settles it.
- It uses fewer steps.
An inverse sine gives 50°. The supplement, in degrees?
Answer
A = 30°, B could be 45° or 135°. How many triangles?
Answer
A = 100°, B could be 40° or 140°. How many triangles?
Answer

The same ideas with more to keep track of.

3 problems

The Law of Sines gives sin B = 1.4. How many triangles?
Answer
A = 40°, B = 53.4°. What is the third angle, in degrees to one decimal place?
Answer
8
Which configurations can ever be ambiguous?
- SSA
- SAS
- SSS
- ASA
- Tick every box that applies.

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Put the ambiguous case check in order.
Write 1 to 4 in the boxes to put these in order.
- Use the Law of Sines to find sin B.
- Take the inverse sine for the first candidate.
- Subtract from 180° for the second candidate.
- Keep the second only if the two known angles stay under 180°.
sin B = 0.5 and B must be obtuse. What is B, in degrees?
Answer
The Supplement
An inverse sine gives 35°. What is the supplement, in degrees?
Answerdegrees
The Count
A = 70° and the second candidate for B would be 130°. How many triangles are possible?
Answer