Derivation of the Law of Sines and the Law of Cosines

1. Write out three formulas for the area of the triangle.
2. Set two of the are equations equal to each other and simplify.
3. Do the same thing for a different pair of equations.
4. Combine your equations above to arrive at the Law of Sines. Use your book to check your answer (see page 388).
5. Describe a way to remember the Law of Sines.

Suppose you know the values \(a,b,c\) and \(\gamma\) in the triangle above.

1. Draw a line segment perpendicular to \(b\) through the vertex at angle \(\beta\).
2. Label the line \(h\), and label the base of the right triangle on the left \(u\) and the base of the right triangle on the right \(v\).
3. Find a formula for \(h\) in known terms (\(a,b,c\) and \(\gamma\)).
4. Find a formula for \(v\) and \(u\) in known terms.
5. Apply the Pythagorean Theorem to the left triangle.
6. Simplify your answer to above so the formula is in known terms. This is the generalized Pythagorean Theorem, meaning it works even for triangles which are not right triangles. It is also called the Law of Cosines. Use your book to check your answer (see page 391).
7. What happens when \(\gamma\) is a right angle?
8. Derive two more formulas also called the Law of Cosines. For example suppose you know \(a,b,c\) and \(\alpha\), or \(a,b,c\) and \(\beta\).
9. Describe a way to remember the Law of Cosines (the three formulas found above).

1. Choose Example 1, 2 or 3 from your book (pages 389,390) and work it out. Don't look at the solution unless needed or to verify your solution.
2. Choose Example 4, 5, 6 or 7 from your book (pages 392-394) and work it out. Don't look at the solution unless needed or to verify your solution.