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Chapter 5: Problem 72

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.

Short Answer

Expert verified
After calculating, you will get the unknown side c and angles B, C using the Law of Cosines and should verify that the same sides c, b and angles B, C can be found using the Law of Sines. The Law of Cosines is more versatile but slightly more complex in computation than the Law of Sines.

Step by step solution

01

Apply the Law of Cosines

To find the unknown side length c in triangle ABC, apply the Law of Cosines: \[ c^{2} = a^{2} + b^{2} - 2ab \cdot \cos(A) \]By substituting the known values \[ c^{2} = 12^{2} + 30^{2} - 2*12*30*\cos(20^{\circ}) \]Calculate the value of \( c^{2} \) and take the square root to find length c.
02

Calculate Remaining Angles

After getting side c, find the remaining angles B and C using the following formulas:\[ B = \cos^{-1}(\frac{a^{2}+c^{2}-b^{2}}{2ac}) \]\[ C = 180^{\circ} - A - B \]Substitute the known values and calculate the angles B and C.
03

Apply the Law of Sines

To compare with the Law of Cosines, calculate the sides using the Law of Sines:\[ b = \frac{{a \cdot \sin(B)}}{\sin(A)} \]\[ c = \frac{{a \cdot \sin(C)}}{\sin(A)} \]Substitute the known values and calculate the side lengths b and c.
04

Comparison of Methods

Compare the sides and angles values obtained from both methods. Moreover, discuss the advantages and disadvantages of both. For instance, the Law of Cosines is more versatile as it can be applied to all types of triangles and not just to oblique triangles unlike the Law of Sines. However, calculations with the Law of Cosines can be more complex due to the nature of the formula.

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