1. Right Triangle Trig Review Given the right triangle from the origin to the point (x, y) with the angle, we can find the following trig functions:

2. Replacing (x, y) with these new values, we get the point as: Moving to the circle centered at the origin

3. Moving to the circle centered at the origin with radius “r”, we find two points A and B.

4. We can use the distance formula to find the distance AB.

5. Next, construct the angle in a circle with the same radius r. Using the SAS property, the triangle AOB in the previous example is congruent to the triangle COD in this example. Therefore, the length of segment AB must equal the length of segment CD. It must also be true that

6. Finding points C and D and the length CD, we get:

7. By similar triangles, we know the length of AB = length of CD. We can square both sides to get rid of the square roots.

8. Simplifying by squaring each group, we get: Every term has an r 2. Divide each term by r 2. Using the pythagorean identity, we know

9. Simplifying, we get: Subtracting the 2’s from each side, we get: Each term has a -2, so divide out the -2.

10. However, recall that Replacing in the equation, we get:

11. To find a rule for, we replace v with –v. Simplifying with odd/even rules, we get:

12. To get the sum/difference rules for sin, we will use the co-function rule. Let’s use the cosine rule to find Using the cosine sum rule Using the co-function rules, we get:

13. Therefore: To get the sin(u+v) rule, Using the odd/even functions, we get: