Trigonometry Half angle formula
In trigonometry, we may come across half angle formula examples, where in we are required to find a particular trigonometric ratio of half of the given angle ratio. For example, we know that sin 45 degrees = 1/v2. Can we use that to find the value of sin (45/2) = sin (22.5 degrees)? That is what half angle formula is all about.
Sine half angle formula:
The sine half angle formula is like this:
Sin (b/2) = v[(1 – Cos b)/2]
Proof: This formula can be proved using the double angle formula for cosine. The double angle formula for cosine is like this:
Cos (2a) = 1 – 2 sin^2 (a)
Here, let 2a = b, then a = b/2. So now the formula would become,
Cos (b) = 1 – 2 sin^2 (b/2)
=> 2 sin^2 (b/2) = 1 – Cos b
=> Sin^2 (b/2) = (1 – Cos b)/2
=> Sin (b/2) = v[(1 – Cos b)/2].
So that is how we prove the formula of half angle of sine.
Cos half angle formula:
The half angle formula for cosine is as follows:
Cos (b/2) = v[(1 + Cos b)/2]
Proof: This formula can be proved using the double angle formula for cosine. The double angle formula that we will use here is:
Cos (2a) = 2 cos^2 (a) – 1
Here, let 2a = b, then a = b/2. So the above formula now would become,
Cos (b) = 2 cos^2 (b/2) – 1
=> Cos b + 1 = 2 cos^2 (b/2)
=> 2 cos^2 (b/2) = Cos b + 1
=> Cos^2 (b/2) = (Cos b + 1)/2
=> Cos (b/2) = v[(1 + Cos b)/2]
So we see how we can prove the half angle formula for cosine as well.
Half angle formula for tangent function:
This one can be stated like this:
Tan (b/2) = v[(1 – cos b)/(1 + cos b)]
Proof: As proved above we know that
Sin (b/2) = v[(1 – cos b)/2] and Cos (b/2) = v[(1 + Cos b)/2]
We also know that tan (b/2) = sin (b/2)/Cos (b/2)
Substituting values of sin and Cos into the above formula, we have,
tan (b/2) = {v[(1 – Cos b)/2}/{v[(1 + Cos b)/2]}
=> tan (b/2) = v{((1 – Cos b)/2) / ((1 + Cos b)/2)}
=> tan (b/2) = v[(1 – Cos b)/(1 + Cos b)]
Hence proved.
In trigonometry, we may come across half angle formula examples, where in we are required to find a particular trigonometric ratio of half of the given angle ratio. For example, we know that sin 45 degrees = 1/v2. Can we use that to find the value of sin (45/2) = sin (22.5 degrees)? That is what half angle formula is all about.
Sine half angle formula:
The sine half angle formula is like this:
Sin (b/2) = v[(1 – Cos b)/2]
Proof: This formula can be proved using the double angle formula for cosine. The double angle formula for cosine is like this:
Cos (2a) = 1 – 2 sin^2 (a)
Here, let 2a = b, then a = b/2. So now the formula would become,
Cos (b) = 1 – 2 sin^2 (b/2)
=> 2 sin^2 (b/2) = 1 – Cos b
=> Sin^2 (b/2) = (1 – Cos b)/2
=> Sin (b/2) = v[(1 – Cos b)/2].
So that is how we prove the formula of half angle of sine.
Cos half angle formula:
The half angle formula for cosine is as follows:
Cos (b/2) = v[(1 + Cos b)/2]
Proof: This formula can be proved using the double angle formula for cosine. The double angle formula that we will use here is:
Cos (2a) = 2 cos^2 (a) – 1
Here, let 2a = b, then a = b/2. So the above formula now would become,
Cos (b) = 2 cos^2 (b/2) – 1
=> Cos b + 1 = 2 cos^2 (b/2)
=> 2 cos^2 (b/2) = Cos b + 1
=> Cos^2 (b/2) = (Cos b + 1)/2
=> Cos (b/2) = v[(1 + Cos b)/2]
So we see how we can prove the half angle formula for cosine as well.
Half angle formula for tangent function:
This one can be stated like this:
Tan (b/2) = v[(1 – cos b)/(1 + cos b)]
Proof: As proved above we know that
Sin (b/2) = v[(1 – cos b)/2] and Cos (b/2) = v[(1 + Cos b)/2]
We also know that tan (b/2) = sin (b/2)/Cos (b/2)
Substituting values of sin and Cos into the above formula, we have,
tan (b/2) = {v[(1 – Cos b)/2}/{v[(1 + Cos b)/2]}
=> tan (b/2) = v{((1 – Cos b)/2) / ((1 + Cos b)/2)}
=> tan (b/2) = v[(1 – Cos b)/(1 + Cos b)]
Hence proved.
