How to Derive the Cosine Difference Formula

Step-by-Step Guide

1. 1

   Step 1: Draw your unit circle and label as needed.

   This gives us point x, which has the ordered pair (cosA, sinA).
   
   You know this from the definition of the trig functions. , This gives us point w, which has the ordered pair (cosB, sinB). , This gives you point u, which has the ordered pair (cos(A-B), sin(A-B)). , This creates a line segment that is opposite of angle A-B. , This also creates a chord opposite of the angle A-B.
   
   Knowing this information, we know that cord uy and cord wx are equivalent. ,
2. 2

   Step 2: Label the origin as point z with the ordered pair (0

3. 3

   Step 3: Label 0 degrees as point y with the ordered pair (1

4. 4

   Step 4: Draw a special angle from the unit circle and label it angle A.

5. 5

   Step 5: Construct another special angle from the unit circle and label it angle B.

6. 6

   Step 6: Construct an angle that is equal to angle A - angle B

7. 7

   Step 7: giving you the angle A-B.

8. 8

   Step 8: Draw a chord going from point u to point y.

9. 9

   Step 9: (Optional) Draw a chord going from point w to point x.

10. 10

    Step 10: Use the distance formula Sqrt((x2-x1)^2-(y2-y1)^2) to find the distance between the two points in either cord uy or cord wx.

Detailed Guide

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