cos A/2 (Half-Angle Formula)

cos A/2 (Half-Angle Formula)

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

Formula

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

cos A/2 = ±√(1 + cos A) / 2

Proof

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

cos (2⋅[A/2]) = 2 cos2 [A/2] - 1

cos 2A (double-angle formula)

Move -1 to the left side.

Switch both sides.

Divide both sides by 2.

Square root both sides.

Then cos A/2 = ±√(1 + cos A) / 2.

Example

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

3π/2 ≦ θ ≦ 2π
So θ is in quadrant IV.

tan θ = -4/3
And tangent: TOA.

So draw a right triangle in quadrant IV
whose opposite side is -4
and whose adjacent side is 3.

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)
= 3/5.

Next, find the sign of cos θ/2.

3π/2 ≦ θ ≦ 2π
So 3π/4 ≦ θ/2 ≦ π.

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 cos θ/2 is (-).

cos θ = 3/5
And cos θ/2 is (-).

So cos θ/2 = -√(1 + 3/5) / 2.

Complex fraction

Rationalizing a denominator