1. Half-angle formulae
Trigonometry

2. Half-angle formula: sine
We start with the formula for the cosine of a double angle:
cos 2θ = 1− 2sin2 θ
Now, if we let θ=α/2, then 2θ = α and our formula becomes:
cos α = 1 − 2sin2 (α/2)

3. Solving for sin(α/2)
We now solve for sin(α/2) (that is, we get sin(α/2) on the left of the equation and everything else on the right): 2sin2(α/2) = 1 − cos α sin2(α/2) = (1 − cos α)/2 Taking the square root gives us the following sine of a half-angle identity:

4. Meaning of the ± sign
The sign of depends on the quadrant in which α/2 lies.
If α/2 is in the first or second quadrants, the formula uses the positive case:
If α/2 is in the third or fourth quadrants, the formula uses the negative case:

5. Half-angle formulae: cosine
Using a similar process, with the same substitution of θ=α/2(so 2θ = α) we substitute into the identity cos 2θ = 2cos2 θ − 1 and we obtain: cos α = 2 cos2(α/2) – 1 Reverse the equation: 2 cos2(α/2) – 1 = cos α

6. Half-angle cosine (continued)
Add 1 to both sides:
2 cos2(α/2) = cos α +1
Divide both sides by 2:
cos2(α/2) = (cos α +1)/2
Solving for cos(α/2), we obtain:

7. Meaning of the ± sign (cosine)
As before, the sign we need depends on the quadrant.
If α/2 is in the first or fourth quadrants, the formula uses the positive case:
If α/2 is in the second or third quadrants, the formula uses the negative case:

8. Half-angle formulae: tan
We can develop formulae for tan α/2 by using what we already know as trigonometry identities.

9. Half-angle tan (continued calculations)
We now try to get perfect squares inside the square root:

10. Sign of tangent?
Same considerations of quadrant/s have to be made as for sine and cosine

11. Equivalence of tan α/2
Another form:

12. Summary of tan of half-angles

13. Examples
Example 1: Find the value of sin 15°. Choose the positive values because 15° is in the 1st quadrant.

14. Example 2
Find the value of cos 165° using the cosine half-angle relationship. Answer: select α=330° so that α/2=165°. Minus sign because 165° is in the second quadrant (II).

15. Example 3
Show that Substitute for cos x/2:

16. Half-angle formulae and applications
Exercises

17. Exercise 1
Use the half-angle formula to evaluate sin 75°.

18. Exercise 2
Find the value of sin(α/2) if cos α = 12/13 where 0°< α <90°.

19. Exercise 3
Prove the identity

20. Exercise 4
Prove the identity