Problem 20 - Algebra and Trigonometry

Question

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$a=4, b=7, c=6$$

Step-by-Step Solution

Answer

To solve the triangle, calculate the three angles using the Law of Cosines and the condition that the sum of the angles of a triangle is 180 degrees. Therefore, we find that Angle A is approximately 36 degrees, Angle B is about 81 degrees, and Angle C is around 63 degrees.

Step 1: Calculation of Angle A

Use the Law of Cosines to calculate angle A. The formula is \(A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right)\). Substituting the given side lengths, \(A = \cos^{-1}\left(\frac{7^2 + 6^2 - 4^2}{2 \times 7 \times 6}\right)\) where A is in degrees.

Step 2: Calculation of Angle B

Next, we can calculate angle B using the formula derived from the Law of Cosines as well: \( B = \cos^{-1}\left(\frac{a^2 + c^2 - b^2}{2ac}\right)\). Substituting the known side lengths, \(B = \cos^{-1}\left(\frac{4^2 + 6^2 - 7^2}{2 \times 4 \times 6}\right)\), and convert it into degrees.

Step 3: Calculation of Angle C

Finally, to determine angle C, subtract the sum of angles A and B from 180, as the sum of all angles in a triangle equals 180 degrees. So, \(C = 180 - (A + B)\).