# 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

Continued below

## 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

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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