Problem 19 - Algebra and Trigonometry

Question

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$a=3, b=9, c=8$$

Step-by-Step Solution

Answer

After performing these steps, you should obtain approximations for all three angles of the triangle. Remember that due to rounding, the sums of your lengths or angle measures may not be exactly equal to what is to be expected (90 degrees, or the sum of the three lengths).

Step 1: Use Law of Cosines to find the first angle

The Law of Cosines is given by the formula \(c^2 = a^2 + b^2 - 2ab \cos C\). We can solve for \(\cos C\) and then use the inverse cosine function to find the measure of angle \(C\). Reordering gives \(\cos C = (a^2 + b^2 - c^2) / (2ab)\). Substituting \(a = 3\), \(b = 9\), and \(c = 8\) gives \(\cos C = (9 + 81 - 64) / (54)\) = \(26/54\), therefore \(C = \cos^{-1}(26/54)\)

Step 2: Use Law of Cosines again to find the second angle

Again using the Law of Cosines, this time to find angle \(A\), we take \(b^2 = a^2 + c^2 - 2ac \cos A\), while again substituting the known values and solving for \(A\) gives \(\cos A = (81 + 64 - 9) / (48)\) = \(136/48\), therefore \(A = \cos^{-1}(136/48)\)

Step 3: Find the third angle

We have already found two angles. The sum of angles in a triangle is 180 degrees. Therefore, we can find the third angle, \(B\), by subtracting the two known angles from 180 degree. That is \(B = 180 - A - C\)