Problem 55 - Algebra and Trigonometry

Question

Describe a strategy for solving an SAS triangle.

Step-by-Step Solution

Answer

A strategy for solving a SAS triangle is to first understand that SAS means Side-Angle-Side. With two sides known and the contained angle, apply the Law of Cosines to find the unknown side. Next, calculate one of the unknown angles using the Law of Sines. For the last angle, use the fact that the sum of all angles in a triangle is 180 degrees.

Step 1: Understanding SAS

SAS represents Side-Angle-Side. This means you are given two sides of a triangle and the angle that sits between them. Label your triangle with the given sides as 'a' and 'b', and the given angle as 'C'.

Step 2: Use the Law of Cosines

Use the Law of Cosines to find the length of the missing side. The Law of Cosines, in this case, is written as \(c^2 = a^2 + b^2 - 2ab\cos(C)\). Solve this equation for c, the length of the missing side.

Step 3: Use the Law of Sines

Now use the Law of Sines to find one of the unknown angles. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides of the triangle. In this case, we could for instance write it down as \(\sin(A) = sin(C)*a/c\). Solve this equation for 'A'.

Step 4: Find the Remaining Angle

In any triangle, all angles add up to 180 degrees. Therefore subtract the sum of angles 'A' and 'C' from 180 to get the measure of angle 'B'.