Half Angle Properties

Introduction:

Let we will discuss about the half angle properties. The half angle is the angle that is half of original angle. Tha is, the product of two half angles at an edge is equal to the original(full) angle at that same edge. We will develop the properties formulas for half angle of both sine and cosine.


Half angle properties formula for sine:


We should start with formula for cosine of double angle. That is,
cos 2θ = 1 - 2sin^2 θ

Half angle properties formula - sine:

Let us consider,


Then 2θ = α and the formula becomes:


Solve for,


That is, we acquire sin(α/2) lying on the left of equation and everything else on right,



Solving of equation gives the following sine of half-angle identity:


The sign of sin α/2 represents on quadrant in which α/2 lies.
If α/2 is in the first or second quadrants then formula uses the positive case:


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


Example:

Find the value of sin 45o using the sine half-angle properties.

Solution:

The sine half angle formula is ,



Therefore,

sin 45o = `sqrt((1 - cos 90 )/(2))`

= `sqrt((1 - 0 )/(2))`

= `sqrt((1)/(2))`

= 0.707


Half angle properties formula for cosine:


Half angle properties formula - cosine:

Using same process, with the similar replacement of θ = α / 2. We want to substitute into the identity,


We get,


Reverse process of the equation should be,


Addition of 1 with both sides of an equation. We get,


Making division process on both sides of equation by 2


Solving for cos(α/2), we get,


As before, the sign we need depends on quadrant.
If α/2 is in first or fourth quadrants that formula uses in positive case:


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


Example:

Find the value of cos 115o using the cosine half-angle formula.

Solution:

We need to find cos 115o

That is, α = 230°, and so α/2 = 115°.

Therefore, cos 115o = `sqrt((1 + cos 230 )/(2))`

= `sqrt((1-0.643)/(2))`

= `sqrt((0.357)/(2))`

= 0.422

We have seen about the half angle properties.