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Ken Ward's Mathematics Pages

Trigonometry

Trigonometry Contents

Page Contents

  1. Half Angle Formulae
    1. Sine
    2. Cosine
    3. Tangent
  2. Tangent Half Angle Formulae
    1. Sine
    2. Cosine
    3. Tangent
    4. Relationship Between Tangent of Half Angles

Half Angle Formulae

Sine

One form of the double-angle formula for the cosine is:
cos2aAsSine.gif [1.1]

If we rewrite α as α/2 
cos(2·α/2)=1-2·sin2(α/2)-1
As:
cos(2·α/2)=cos(α),
We have:
cos(α)=1−2·sin2(α/2) [1.2]
By rearranging :
2·sin2(α/2)=1-cos(α)

So:
sin2(α/2)=(1-cos(α))/2

And, by taking the square root, the sine half alpha is:
sinHalfAlpha.gif [1.3]
or
sinHalfAlpha2.gif [1.4]

Cosine

We again start with the double angle for cosine, but this time use this version:
doubleAngleCos2.gif [1.5]

So, by rearranging,
2·cos2α=cos2α+1

Rewriting α as α/2:
2·cos2(α)/2=cosα+1
Rearranging:
2cos2(α/2)=cosα+1
cos2(α/2)=(cosα+1)/2
And by taking the square root, we have two formulae (one for the + and one for the minus root):
cosHalfAnglePos.gif [1.6]

and
cosHalfAngleNeg.gif [1.7]

Tangent

To get the tangent of half an angle, knowing the angle, we divide the sine by the cosine formulae:
tan(α/2)=sin(α/2)/cos(α/2)
=√((1-cosα)/(1+cosα))

Multiply top and bottom by 1-cosα
=√((1-cosα)2/(1-cos2α))
As 1-cos2α=sin2α, we have:
= √((1-cosα)2/(sin2α))
Taking the square root:
= ±((1-cosα)/(sinα))
tanHalfAngle.gif [1.8]
or
tanHalfAngle2.gif [1.9]

Tangent Half Angle Formulae

Sine

Starting with the double angle formula for sine:
doubleAngleSin.gif

Writing α as α/2
sinα=2·sin(α/2)·cos(α/2)

Dividing by cos2(α/2)·sec2(α/2) (which is equal to 1):
sin(α)=2·tan(α/2)/sec2(α/2)
=2·tan(α/2)/1+tan2(α/2)
Because sec2(α/2)=1+tan2(α/2)

sinHalfTan.gif [2.1]

Cosine

Starting with the double angle formula for cosine:cos2aAsSine.gif [1.1, repeated]
Writing α as α/2
cos α=1−2·sin2(α/2)

Divide by cos2(α/2)·sec2(α/2) (which is equal to 1):
cos(α)=[sec2(α/2)−2·tan2(α/2)]/sec2(α/2)

We note
sec2(α/2)=1+tan2(α/2), so
cosaHalfTan.gif [2.2]

Tangent

Dividing sinα by cosα (Equation 2.1 by 2.2) gives us:
tanHalfTan.gif [2.3]

Alternatively, we note the tangent double angle formula is:
doubleAngleTan.gif

By setting α as α/2, we immediately get the half angle formula (Equation 2.3)

Relationship Between Tangent of Half Angles

The three values that occur in the half tangent formula are sides of a right angled triangle, so writing t=tan(α/2), and the hypotenuse, h=(1+t2), base, b=(1-t2), and perpendicular, p=2t, so
h2=b2+p2,

and substituting the half tangent, t, values, we get:
(1+t2)2=(1-t2)2+(2t)2





Trigonometry Contents

Ken Ward's Mathematics Pages


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