The document introduces the Sine Law, which states that for any triangle, the ratio of the sine of an angle to the side opposite it is equal to the ratio of the sine of any other angle to its opposite side. It demonstrates this law by considering a triangle ABC and its altitude to side BC, showing that the ratio of the sine of angle A to side a equals the ratio of the sine of angle B to side b. It then states the Sine Law formula that the ratio of the sine of any angle to its opposite side equals the same ratio for any other angle and side in the triangle.

1. The Sine Law (animated)

2. Sine Law Let’s begin with the triangle  ABC:

3. Sine Law Let’s begin with the triangle  ABC: A C B

4. Sine Law A C B c a b Let’s begin with the triangle  ABC:

5. Sine Law A C B c a b Let’s begin with the triangle  ABC: D altitude, h h

6. Sine Law A C B c a b Let’s begin with the triangle  ABC: D Note this ratio in triangle  ABD: h

7. Sine Law A C B c a b D Note this ratio in triangle  ABD: Note this ratio in triangle  ABC: h

8. Sine Law A C B c a b D Note this ratio in triangle  ABD: Note this ratio in triangle  ABD: h

9. Sine Law A C B c a b D h

10. Sine Law A C B c a b D Therefore, we get: h

11. Sine Law A C B c a b Therefore, we get: and, if we follow the same method for sides A and C: In conclusion: The Sine Law: