Finding Angle Of Triangle Given 3 Sides

Howto Find the Angle of a Triangle When Given Three Sides

When working with triangles, one of the most common challenges is determining the measure of an angle when only the lengths of the three sides are known. Worth adding: fortunately, a powerful mathematical tool called the Law of Cosines provides a straightforward solution to this problem. This scenario arises frequently in fields like engineering, architecture, and physics, where precise angular measurements are critical. By applying this formula, you can calculate any angle of a triangle using only the side lengths, making it an essential skill for solving real-world problems.

Step-by-Step Guide to Finding an Angle of a Triangle Using Three Sides

To calculate the angle of a triangle when all three sides are known, follow these steps:

1. Identify the sides of the triangle
   Label the sides as $ a $, $ b $, and $ c $, where $ c $ is the side opposite the angle you want to find. As an example, if you’re solving for angle $ C $, side $ c $ will be the side directly across from it.

2. Label the angle you want to calculate
   Let’s say you’re solving for angle $ C $. This angle is opposite side $ c $, while sides $ a $ and $ b $ are adjacent to it.

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3. Apply the Law of Cosines formula
   The Law of Cosines states:
   $ c^2 = a^2 + b^2 - 2ab \cos(C) $
   Rearranging this formula to solve for $ \cos(C) $:
   $ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} $

4. Calculate the cosine value
   Plug the known side lengths into the formula. To give you an idea, if $ a = 5 $, $ b = 7 $, and $ c = 8 $:
   $ \cos(C) = \frac{5^2 + 7^2 - 8^2}{2 \cdot 5 \cdot 7} = \frac{25 + 49 - 64}{70} = \frac{10}{70} = \frac{1}{7} $

5. Find the angle using the inverse cosine function
   Use a calculator to compute $ \cos^{-1} $ (or arccosine) of the result:
   $ C = \cos^{-1}\left(\frac{1}{7}\right) \approx 81.79^\circ $

6. Verify your result
   Ensure the calculated angle makes sense based on the triangle’s side lengths. For example