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Chapter 6: Problem 23

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$a=63, b=22, c=50$$

Short Answer

Expert verified
After following these steps, the final solution will be the list of angles and side lengths, all to the appropriate level of precision.

Step by step solution

01

Apply the Law of Cosines to Find the Largest Angle

Since c = 50 is the largest side, thus an angle across it would be the largest. We can use the Law of Cosines to find out this angle. The Law of Cosines formula is \(c^2 = a^2 + b^2 - 2ab \cdot cos(C)\). We can rearrange this formula to solve for angle C: \(cos(C) = \frac{a^2 + b^2 - c^2}{2ab}\). After plugging the values we get \(cos(C) = \frac{63^2 + 22^2 - 50^2}{2 * 63 * 22}\)
02

Compute the Result and Convert to Degrees

Compute the above expression to get cos(C). Use the built-in cosine inverse function (often marked 'cos^-1' or 'arccos' on calculators) to find the degree measure of angle C. Make sure to round to the nearest degree.
03

Apply the Law of Cosines or Sines for Remaining Angles

Now, for the rest of the angles (A and B), we can use either the Law of Cosines similarly, or the Law of Sines could potentially be easier, due to its simpler formula: \( \frac{a}{sin(A)} = \frac{b}{sin(B)} = \frac{c}{sin(C)} \). Apply one of these formulas to find the remaining two angles. Keep in mind the sum of all angles in a triangle should be 180 degrees.
04

Find the Remaining Lengths to the Nearest Tenth

Once all three angles are known, compute corresponding side lengths, rounding your answers to the nearest tenth.

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