Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe how the Law of cosines can be used to solve the ambiguous case of the oblique triangle \(A B C,\) where \(a=12\) feet, \(b=30\) feet, and \(A=20^{\circ} .\) Is the result the same as when the Law of sines is used to solve the triangle? Describe the advantages and the disadvantages of each method.

Step-by-Step Solution

Answer

Law of Cosines is more reliable for the ambiguous case because it provides a single accurate solution. Law of Sines, though simpler in application, can result in multiple or no solutions in the ambiguous case.

Step 1: Apply Law of Cosines to find unknown side

The Law of Cosines can be used here to find the length of the remaining side. We are given sides \(a\) and \(b\), as well as angle \(A\). Thus we can apply the formula \(c^2 = a^2 + b^2 - 2ab \cos(A)\), giving us: \(c^2 = 12^2 + 30^2 - 2*12*30 \cos(20)\). Solving this equation will give the length of side \(c\).

Step 2: Apply Law of Cosines to find remaining angles

Now we can apply the Law of Cosines again to find angles \(B\) and \(C\). The relevant formulas are: \(\cos(B) = (a^2 + c^2 - b^2) / (2ac)\) and \(\cos(C) = (b^2 + c^2 - a^2) / (2bc)\). Replacing the sides in these equations we've calculated before will yield the angles \(B\) and \(C\).

Step 3: Solving using the Law of Sines

The Law of Sines can be used in the following way: \(a/sin(A) = b/sin(B) = c/sin(C)\). However, in this ambiguous case, it might yield two possible values for angle \(B\), as \(\sin(x) = \sin(180 - x)\), and thus two different triangles. If \(b > a\sin(B)\) or \(a > b\sin(A)\), no solution exists.

Step 4: Comparing the methods

So, the Law of Sines might yield multiple or no solutions in the ambiguous case, which is a clear disadvantage. However, it's straightforward and easy to apply. Meanwhile, the Law of Cosines always yields one solution and always provides the correct response, but involves more calculations and steps than the Law of Sines.