Law of Sines - Ambiguous Case
Yes, there are times that the Law of Sines can produce incorrect answers.
Let's take for example, a triangle with sides a = 15, b = 35 and c = 25 and we have to find all 3 angles.
Let's find Angle A by using the Cosine Law
cos (A) = (b^2 + c^2 -a^2) / (2*b*c)
cos (A) = (35^2 + 25^2 -15^2) / (2*35*25)
cos (A) = (1,225 + 625 - 225) / (1,750)
cos (A) = (1,625) / (1,750)
cos (A) = 0.9285714286
Angle A = 21.787°
For the next step, most texts advise using the Law of Sines becuase the calculations are easier.|
(sine (A) / side a) = (sine (B) / side b)
(sine (21.787) / 15) = (sine (B) / 35)
sine (B) = (0.37116 * 35) / 15
sine (B) = .86604
Angle B = 60.00°
For the third angle, we could just subtract angle A and Angle B from 180°, but let's use the Law of Sines again.|
(sine (A) / side a) = (sine (C) / side c)
(sine (21.787) / 15) = (sine (C) / 25)
sine (C) = (0.37116 * 25) / 15
sine (C) = 0.6186
Angle C = 38.214°
Just to make sure that everything was calculated correctly, we should add up angles A, B and C and they should total 180°.
However, 21.787 plus 60.00 plus 38.214 sums to 120° so something has gone awry.
When we take the arc sine of a number, we actually get two angles that are between zero and 180°.
(For example, the arc sine of .5 equals 30° and 150°).
That means angle B could be (180 -60) or 120° and angle C could be (180 -38.214) or 141.786°.
It can be seen that it is angle B that is actually 120° because that makes all 3 angles sum to 180°.
So, is there any way that we can avoid the ambiguous case?
Actually there are several.
1) When you calculate the first angle using the Law of Cosines, always calculate the largest angle. (The largest angle will be the angle opposite the largest side.)
2) When using the Law of Sines, never use it to calculate the largest angle.
3) Always use the Law of Cosines for the calculations. Granted it's a little more difficult but you can be sure of the results.
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