Problem 23 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

To solve the triangle with given sides \(a = 63\), \(b = 22\) and \(c = 50\), apply the Law of Cosines to find the first angle, then the Law of Sines to find the second angle. Find the third angle by subtracting the two found angles from 180.

Step 1: Calculation of first angle using Law of Cosines

Begin with calculating first angle A. The formula for the Law of Cosines in this case is \(A = cos^-1 \left( \frac{b^2 + c^2 - a^2}{2bc}\right)\). Substitute the provided values, \(b = 22\), \(c = 50\) and \(a = 63\) into the formula and solve for A.

Step 2: Calculation of second angle using Law of Sines

The Law of Sines can be used to calculate another angle. Take angle B for instance: The formula can be set as \(\frac{sin(B)}{b} = \frac{sin(A)}{a}\). Rearrange for \(sin(B)\) and put in the known values of \(b = 22\), \(a = 63\) and \(A\) (which was calculated in the previous step) and solve for \(B\). Don't forget to use the inverse sine to obtain the angle.

Step 3: Calculation of the third angle

As a triangle's inside angles always add up to 180 degrees, subtract the obtained angles A and B from 180 to get the third angle C: \(C = 180 - A - B\).