How Can the Unit Circle Prove the Cosine Half-Angle Formula?

[Louis B]

Homework Statement

Prove that cos($$\frac{\theta}{2}$$) = $$\pm$$$$\sqrt{\frac{1+cos\theta}{2}}$$ using the unit circle.

Homework Equations

The Attempt at a Solution

I'm not sure if it's possible for you to provide a clear graph on here for the solution but a link would also be nice :)

View attachment half angle.bmp

 

[lurflurf]

consider the points A=(cos(t),sin(t)) B=(cos(t/2),sin(t/2)) C=(1,0) D=(0,0)
To find B show that the midpoint of the segment AC is on the segment BD

 

[Architeuthis Dux]

Dear student,

Thank you for your inquiry. I am happy to assist you with your request.

To prove the half angle formula for cosine, we will use the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. We will also use the Pythagorean identity, which states that sin^2θ + cos^2θ = 1.

First, let's draw a unit circle and label the points A, B, and C as shown in the diagram below:

_
/ \
/ \
A B C
|---|---|
1 1 1

Next, we will draw a line from point C to the x-axis, creating a right triangle. Label the angle at point A as θ/2 and the remaining angle at point C as θ/2. This is shown in the diagram below:

_
/ \
/ \
A B C
|---|---|
1 1 1
/|\
/ | \
/ | \
/ | \
/____|____\
1 1 1

Using the Pythagorean identity, we can find the value of sin(θ/2) and cos(θ/2) as follows:

sin^2(θ/2) + cos^2(θ/2) = 1
sin^2(θ/2) + cos^2(θ/2) = 1
sin^2(θ/2) + (1 - sin^2(θ/2)) = 1
2sin^2(θ/2) = 1
sin^2(θ/2) = 1/2
sin(θ/2) = ±√(1/2)

cos^2(θ/2) + sin^2(θ/2) = 1
cos^2(θ/2) + (1 - cos^2(θ/2)) = 1
2cos^2(θ/2) = 1
cos^2(θ/2) = 1/2
cos(θ/2) = ±√(1/2)

Now, we need to determine the signs of sin(θ/2) and cos(θ/2). Looking at