# 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

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^2alpha/2-1

Reverse the equation:

2\ cos^2alpha/2-1=cos\ alpha

2cos^2alpha/2=1+cos\ alpha

Divide both sides by 2

cos^2alpha/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^2x/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^2x/2+cos\ x=1

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

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