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Half-angle formulae

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

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Half-angle formulae

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  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: cosα = 2 cos2(α/2) – 1 Reverse the equation: 2 cos2(α/2) – 1 = cosα

  6. Half-angle cosine (continued) Add 1 to both sides: 2 cos2(α/2) = cosα +1 Divide both sides by 2: cos2(α/2) = (cosα +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

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