Half angles:



Note: Most everything that is solved using Half angle formulas can also be solved using sum and difference identities. Not only that, but the answers are easier to understand. The following is a tutorial on how to use half angle identities.

Half Angle Formula-Sine

If we let
θ=α/2
then 2θ = α and our formula becomes:

cos α = 1 − 2sin2 (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


Solving gives us the following sine of a half-angle identity:
Half Angle Identity - Sine
Half Angle Identity - Sine

The sign we need depends on the quadrant. If α/2 is in the first or second quadrants, the formula uses the positive case, and if α/2 is in the third or fourth quadrants, the formula uses the negative case.

^Simply remember All Students Take Calculus (All/Sine/Tangent/Cosine)


Half Angle Formula-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
(see cosine of a double angle)

We obtain
cos
α = 2cos2 α/2 1

Reverse the equation:
2cos2 α/2 1 = cosα

Add 1 to both sides:
2cos2 α/2 = 1+cosα

Divide both sides by 2
cos
2 α/2 = (1+cosα)/2

Solving for cos(α/2), we obtain:
Half Angle Identity - Cosine
Half Angle Identity - Cosine



In the following verification, remember that 165° is in the second quadrant, and cosine functions in the second quadrant are negative. Also, 330° is in the fourth quadrant, and cosine functions in the fourth quadrant are positive. From Figure 2 , the reference triangle of 330° in the fourth quadrant is a 30°–60°–90° triangle. Therefore, cos 330° = cos 30°.




external image 11121.nfg003.jpg
using the half-angle identity for the cosine


external image 11049.nce030.jpg

To get Tan (X/2), divide the previous two identities like so...
Sin (X/2)/ Cos (X/2) = Tan (X/2)
± square root of [(1-cosx)/2] / ± square root of [(1+cosx)/2] = Tan (X/2)
± square root of [(1-cosx)/(1+cosx)] = Tan (X/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, and if α/2 is in the second or third quadrants, the formula uses the negative case.


Graphical Representation

external image halfAngle.png
Videos




This video is great for showing how to derive the half angle identities. It also has a couple of examples of how to use these identities. The answers for these examples are not quite complete though, because they have not been rationalized. When working through these types of problems, or any mathematical problem, make sure you rationalize.