4. Half-Angle Formulas

by M. Bourne

We will develop formulas for the sine, cosine and tangent of a half angle.

Half Angle Formula - Sine

We start with the formula for the cosine of a double angle that we met in the last section.

cos 2θ = 1− 2sin2 θ

Formula Summary

We derive the following formulas on this page:

`sin (alpha/2)=+-sqrt((1-cos alpha)/2`

`cos (alpha/2)=+-sqrt((1+cos alpha)/2`

`tan (alpha/2)=(1-cos alpha)/(sin alpha`

Now, if we let

`theta=alpha/2`

then 2θ = α and our formula becomes:

`cos α = 1 − 2\ sin^2(α/2)`

We now solve for

`sin(alpha/2)`

(That is, we get `sin(alpha/2)` on the left of the equation and everything else on the right):

`2\ sin^2(α/2) = 1 − cos α`

`sin^2(α/2) = (1 − cos α)/2`

Solving gives us the following sine of a half-angle identity:

`sin (alpha/2)=+-sqrt((1-cos alpha)/2`

The sign (positive or negative) of `sin(alpha/2)` 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:

`sin (alpha/2)=sqrt(1-cos alpha)/2`

If `α/2` is in the third or fourth quadrants, the formula uses the negative case:

`sin (alpha/2)=-sqrt(1-cos alpha)/2`

Half Angle Formula - Cosine

Using a similar process, with the same substitution of `theta=alpha/2` (so 2θ = α) we subsitute into the identity

cos 2θ = 2cos2 θ − 1 (see cosine of a double angle)

We obtain

`cos alpha=2\ cos^2(alpha/2)-1`

Reverse the equation:

`2\ cos^2(alpha/2)-1=cos alpha`

Add 1 to both sides:

`2\ cos^2(alpha/2)=1+cos alpha`

Divide both sides by `2`

`cos^2(alpha/2)=(1+cos alpha)/2`

Solving for `cos(α/2)`, we obtain:

`cos (alpha/2)=+-sqrt((1+cos alpha)/2`

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:

`cos (alpha/2)=sqrt((1+cos alpha)/2`

If `α/2` is in the second or third quadrants, the formula uses the negative case:

`cos (alpha/2)=-sqrt((1+cos alpha)/2`

Half Angle Formula - Tangent

The tangent of a half angle is given by:

`tan (alpha/2)=(1-cos alpha)/(sin alpha)`

We can also write the tangent of a half angle as follows:

`tan (alpha/2)=(sin alpha)/(1+cos alpha)`

Summary of Tan of a Half Angle

`tan (alpha/2)=(1-cos alpha)/(sin alpha)=(sin alpha)/(1+cos alpha`

Using t

It is sometimes useful to define t as the tan of a half angle:

`t=tan (alpha/2)`

This gives us the results:

`sin a=(2t)/(1+t^2)`

`cos alpha=(1-t^2)/(1+t^2)`

`tan\ alpha=(2t)/(1-t^2)`

Tan of the Average of 2 Angles

With some algebraic manipulation, we can obtain:

`tan\ (alpha+beta)/2=(sin alpha+sin beta)/(cos alpha+cos beta)`

Example 1

Find the value of `sin 15^@` using the sine half-angle relationship given above.

Example 2

Find the value of `cos 165^@` using the cosine half-angle relationship given above.

Example 3

Show that `2\ cos^2(x/2)-cos x=1`

Exercises: Evaluating and Proving Half-Angle Identities

1. Use the half angle formula to evaluate `sin 75^@`.

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

3. Prove the identity: `2\ sin^2(x/2)+cos x=1`

4. Prove the identity: `2\ cos^2(theta/2)sec theta=sec theta+1`

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