Explain how you would solve a triangle if you knew the three sides lengths. Explain how you would find each of the three missing pieces.
If you know the three side lengths of a triangle, you can use the Law of Cosines to find each of the three missing angles. The Law of Cosines states that, for any triangle with sides of length a, b, and c, and angles A, B, and C opposite those respective sides:
a^2 = b^2 + c^2 - 2 * b * c * cos(A)
b^2 = a^2 + c^2 - 2 * a * c * cos(B)
c^2 = a^2 + b^2 - 2 * a * b * cos(C)
To find each of the three missing angles, follow these steps:
1. Choose one of the equations to start with, based on which angle you want to find first. For example, if you want to find angle A, start with the equation a^2 = b^2 + c^2 - 2 * b * c * cos(A).
2. Plug in the values of the side lengths a, b, and c.
3. Solve for cos(A) by rearranging the equation:
cos(A) = (b^2 + c^2 - a^2) / (2 * b * c)
4. Use the inverse cosine function (cos^(-1)) to find angle A:
A = cos^(-1)(cos(A))
5. Repeat steps 1-4 for angles B and C.
6. Now you have solved the triangle by finding all three angles A, B, and C.