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Chapter 5: Problem 22

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$C=15^{\circ} 15^{\prime}, \quad a=7.45, \quad b=2.15$$

Short Answer

Expert verified
To get the solution, first convert the given angle to decimal form. Then use the Law of Cosines to find the length of side c. Once you have this, use the Law of Sines to find the magnitude of the other two angles. Add up the angles to confirm they sum to 180 degrees. The final solution will be the side lengths a, b, c and the angles A,B,C.

Step by step solution

01

Convert angle to decimal degree form

Before beginning with the calculations, it鈥檚 best to convert the given angle from degrees and minutes to only degrees. So, angle C = \(15^\circ 15' = 15^\circ + \frac{15}{60}^\circ = 15.25^\circ\).
02

Calculate side c using the Law of Cosines

The Law of Cosines can be stated as \(c = \sqrt{a^2 + b^2 - 2ab\cos(C)}\). Substituting the given values we get: \(c = \sqrt{(7.45)^2 + (2.15)^2 - 2*7.45*2.15*\cos(15.25)}\). Compute this to find the value of c.
03

Calculate angles A and B

Now that we have all three side lengths, we can use the Law of Sines to calculate the other two angles. The other two angles can be defined by the formulas \(A = \arcsin{(a\sin(C))/c)}\) and \(B = \arcsin{(b\sin(C))/c)}\). Remember to convert the results to degrees, if your calculator gives the results in radians. Adding all three angular degrees should sum to 180.
04

Check the results

We now have all the angles and sides of the triangle. Check the results: The sum of all angles should be equal to 180 degrees and applying the law of cosines using the obtained sides and angles should verify the initial given conditions.

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