tan A/2 (Half-Angle Formula)

tan A/2 (Half-Angle Formula)

How to solve tan A/2 (Half-Angle Formula) problems: formula, proof, example, and its solution.

Formula

tan A/2 = +-sqrt[(1 - cos A) / (1 + cos A)]

tan A/2 = ±√(1 - cos A) / (1 + cos A)

Proof

tan A/2 (Half-Angle Formula): Proof of the Formula

Quotient identities

sin A/2 (half-angle formula)

cos A/2 (half-angle formula)

Multiply 2
to both of the numerator and the denominator.

Then tan A/2 = ±√(1 - cos A) / (1 + cos A).

Example

If tan theta = 3/4 and pi <= theta <= 3pi/2, find the value of tan theta/2.

πθ ≦ 3π/2
So θ is in quadrant III.

tan θ = 3/4
And tangent: TOA.

So draw a right triangle in quadrant III
whose opposite side is -3
and whose adjacent side is -4.
[(-3)/(-4) = 3/4]

This right triangle is a (3, 4, 5) right triangle.

So (green side) = 5.

Pythagorean triples

The brown angle is the reference angle of ∠θ.

And cosine: CAH.

So cos θ = cos (brown)
= -4/5.

Next, find the sign of tan θ/2.

πθ ≦ 3π/2
So π/2 ≦ θ/2 ≦ 3π/4.

So θ/2 is in quadrant II.

Draw the axes of the coordinate plane
and write 'all, sin, tan, cos' like above.

This shows when the trigonometric function is (+):
for quadrant II, only sine is (+).

So tan θ/2 is (-).

cos θ = -4/5
And tan θ/2 is (-).

So tan θ/2 = -√(1 - [-4/5]) / (1 + [-4/5]).

Multiply 5
to both of the numerator and the denominator.