cos A/2

How to find cos A/2 by using its formula (half-angle formula): formula, 1 example, and its solution.

Formula

Formula

cos A/2 = ±√(1 + cos A)/2

This is the half-angle formula of cosine.

To prove this formula,
put A/2 into cos 2A formula:
cos A = 2 cos2 A/2 - 1.

Example

Example

Solution

First find the sign of cos θ/2.

It says
3π/2 ≤ θ ≤ 2π.
Then
3π/4 ≤ θ/2 ≤ π.

Draw a simple coordinate plane.
Starting fron quadrant I,
write all, sin, tan, cos
in each quadrant.

This shows
when the trigonometric function is plus.

For example,
if an angle is in quadrant II,
then sin is plus.
(Cosine and tangent are minus.)

3π/4 ≤ θ/2 ≤ π

So draw a terminal side
in quadrant II.

Reference Angle

The terminal side is on [sin].
Then sine is plus.

So cos θ/2 is minus.

Next, find cos θ.

3π/2 ≤ θ ≤ 2π

So draw another coordinate plane.
And draw a terminal side in quadrant IV.

tan θ = -4/3

Tangent is TOA:
Tangent,
Opposite side (-4),
Adjacent side (3).

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

See the right triangle.

The sides are 3, -4, (hypotenuse).

So this is a [3, 4, 5] right triangle.

So the hypotenuse is 5.

Pythagorean Triple

Find cos θ.

Cosine is CAH:
Cosine,
Adjacent side (3),
Hypotenuse (5).

So cos θ = 3/5.

cos θ/2 is minus.
cos θ = 3/5

So cos θ/2 = -√(1 + 3/5)/2.

Multiply 5
to both of the numerator and the denominator.

(1 + 3/5)⋅5 = 5 + 3

2⋅5 = 10

5 + 3 = 8

8/10 = 4/5

-√4/5
= -√4/√5
= -√22/√5
= -2/√5.

Divide Radicals

To rationalize the denominator5,
multiply [√5/√5]

2⋅√5 = 2√5

5⋅√5 = 5

So cos θ/2 = -2√5/5.