Half Angle Formula

Half angle formula

Half angle formulas are studied in trigonometry. These formulas are usually derived from double angle formulas. These formulas are rudimentary parts of trigonometry. 

Half angle formulas are usually shown as, x/2, A/2 etc. Half angles are sub-multiple angles. These formulas are used to derive values of angles such as 22.5 degree (which is the half of angle 45 degree), 15 degree (half of angle 30 degree) etc. 

Formulas of half angles

In this part the sin, cos, tan values of half angles and their formulas are discussed below, 

The key formulas are, 

Sin (A/2) = √ (1 – cos A) /2 

Cos (A/2) = √ (1 + cos A) /2 

Tan (A/2) = √ (1 – cos A) / (1 + cos A) = 

Solved examples

1) Using the half angle formulas determine the value of cos 22.5 degree. 

Answer. Using the half angle formula of cos, we get, 

Cos (A/2) = √ (1 + cos A) /2 

Here A/2 = 22.5 degree

So, A = 45 degree

Cos (45/2) =√  (1 + cos 45) / 2 

= 0.8733

2) Using the half angle formulas determine the value of sin 15 degrees. 

Answer. Using the half angle formulas of sin, we get that, 

Sin (A/2) = √ (1 – cos A) / 2 

Here, A / 2 = 15

So, A = 30 degree

Sin (15) = sin (30/2) = √ (1 – cos 30) / 2 

= 0.6502

3) Prove that cos A / (1 + sin A) = tan (45 – A/2) 

Answer. So, the LHS is, 

Cos A / (1 + sin A) 

= (cos² A/2 – sin² A/2) / [1 + 2 sin (A/2) cos (A/2)]

We know that 1 = cos²(A/2) + sin²(A/2). So,

= [ (cos (A/2) + sin (A/2)) (cos (A/2) – sin (A/2))] / [cos²(A/2) + sin²(A/2) + 2 sin (A/2) cos (A/2)]

= [ (cos (A/2) + sin (A/2)) (cos (A/2) – sin (A/2))] / [cos (A/2) + sin (A/2)]²

= [cos (A/2) – sin (A/2)] / [cos (A/2) + sin (A/2)]

= [ cos (A/2) (1 – sin (A/2)/cos(A/2))] / [ cos (A/2) (1 + sin (A/2)/cos(A/2))]

= (1 – tan (A/2)) / (1 + tan (A/2))

We know that 1 = tan (180/4). So,

= (tan (π/4) – tan (A/2)) / (1 + tan (π/4) tan (A/2))

We have (tan A – tan B) / (1 + tan A tan B) = tan (A – B). So

= tan [ (π/4) – (A/2)]