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# Sine rule leading to two solutions for a missing angle watch

1. Hi,

The following question is from Edexcel C2, Exercise 2F, Question 1h.

In triangle ABC:
Angle A = 45 degrees
AB = 9.6 cm
AC = 4.8 cm
Find CB and the other two angles.

There are two solutions in the book:
B = 61.2, C = 73.7, BC = 7.07 cm, or
B = 28.7, C = 106, BC = 7.07 cm

I thought I understood how/when you got two solutions for a missing angle, but I can't get my head around this one. My thinking is that if the length of AB and AC and the angle between them is known, then surely there is only one possibility for the placement of the other side and therefore the size of the other angles. Or in other words, how can you draw two variations of a triangle when the lengths of two sides and the angle between them is known?

Any help would be appreciated.

Thanks
2. (Original post by -suze-)
Hi,

The following question is from Edexcel C2, Exercise 2F, Question 1h.

In triangle ABC:
Angle A = 45 degrees
AB = 9.6 cm
AC = 4.8 cm
Find CB and the other two angles.

There are two solutions in the book:
B = 61.2, C = 73.7, BC = 7.07 cm, or
B = 28.7, C = 106, BC = 7.07 cm

I thought I understood how/when you got two solutions for a missing angle, but I can't get my head around this one. My thinking is that if the length of AB and AC and the angle between them is known, then surely there is only one possibility for the placement of the other side and therefore the size of the other angles. Or in other words, how can you draw two variations of a triangle when the lengths of two sides and the angle between them is known?

Any help would be appreciated.

Thanks
3. You're entirely correct. Knowing the length of all three sides is also enough to uniquely determine the angles (and in fact there are various other combinations that work too).

The "B = 61.2, C = 73.7, BC = 7.07 cm" answer is wrong. (Try drawing two line segments, one twice the length of the other, at an angle of 45 degrees. If you add a third side, there's no way that you'll get angles anything like 61 or 74 degrees.)
4. the answer in the book for BC is only possible if the side AC is allowed to vary - i.e. if they TELL you from the start that BC = 7.07 and ask you for AC and angles aBc and aCb.

this is just a simple-ass application of the Cosine rule.
5. Thanks very much for your replies - glad I wasn't missing something here!

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Updated: January 21, 2014
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