MCT4C
Ambiguous Case in the Sine Law
1. For each of the following, draw possible diagrams that match the given measurements. Then, calculate all possible lengths for side c.
a) , and
b) , and
c) , and
2. In , and
a) Draw all possible diagrams that match the given measurements.
b) Solve for all values of side c
c) If side a is 12 rather than 10, how many solutions are there? Explain.
d) Determine any solutions in part c)
3. A lighthouse at point L is 10 km from a yacht at point Y and 8 km from a sailboat at point B. From the yacht, the lighthouse and the sailboat are separated by an angle of 48º
a) Is it necessary to consider the ambiguous case? Explain.
b) Sketch all possible diagrams for this situation.
c) Determine all possible the distances from the yacht to the sailboat, to the nearest tenth of a kilometre.
4. Albert and Belle are part of a scientific team studying thunderclouds. The team is about to launch a weather balloon into an active part of the cloud. Albert’s rope is 7.8 m long and makes an angle of 36º with the ground. Belle’s rope is 5.9 m long.
a) Is it necessary to consider the ambiguous case? Explain.
b) Sketch all possible diagrams for this situation.
c) Determine all possible the distances between Albert and Belle to the nearest tenth of a meter.
5. A triangular plot of land is enclosed by a fence. Two sides of the fence are 9.8 m and 6.6 m long, respectively. The other side forms an angle of 40º with the 9.8 m side.
a) Draw a sketch of this situation
b) Calculate the height of the triangle to the nearest tenth. Compare it to the given sides.
c) How many lengths are possible for the third side? Explain.
d) Calculate all possible lengths to the nearest tenth of a meter.
6. Create a word problem that requires the use of the ambiguous case.
Answers: 1a) 24 or 4 b) 18.3 c) 26.5 or 10.3 2b) 15.6 or 2.8 c) one solution d) 15.3
3c) 9.7 km or 3.7 km 4c) 10.0 m or 2.6 m 5b) 6.3 m c) two d) 9.5 m or 5.6 m