Problem 56 - Algebra and Trigonometry

Question

Describe a strategy for solving an SSS triangle.

Step-by-Step Solution

Answer

The strategy to solve an SSS triangle is to first use the law of cosines to find one angle, then use the law of sines or cosines to find the second angle, and finally use the fact that the sum of angles in a triangle is 180 to find the third angle.

Step 1: Find the first angle

To find the first angle (let's call it A), use the law of cosines. The law of cosines is expressed as: \( a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \) where \( a, b, c \) are the sides of the triangle. Since we know all the sides we can rearrange the formula to solve for \(\cos(A)\) and then use the inverse cosine to find angle A.

Step 2: Find the second angle

After finding angle A, now we turn our attention to finding the second angle (let's call it B). We could use the law of cosines again, but using the law of sines is generally easier. The law of sines is expressed as: \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)}\). We already know a, A, and b so we can solve for sin(B) and then use the inverse sine to find angle B.

Step 3: Find the third angle

Since we know that the sum of all angles in a triangle is 180 degrees, we can simply subtract the first two angles from 180 to get the third angle (let's call it C). So: \( C = 180 - A - B \)