Step by step
The question
Question: find the exact value of cos 45°.
1 Draw the unit circle
The unit circle has its center at 0 and radius 1.
We turn 45° from the positive x-axis, against the clock.
The ray meets the circle at the point P.
Here is the key fact.
P is the point (cos 45°, sin 45°).
Across is cos.
Up is sin.
2 Use the special triangle
Draw a line from P straight down to the x-axis.
It makes a right triangle.
Its longest side goes from the center to P.
It is a radius: 1.
The angles are 45°, 45° and 90°: half of a square.
The two short sides are the same: x.
From the special triangles lesson: the sides are x, x and x root 2.
The longest side is x root 2.
The longest side is 1.
So x√2 = 1.
To remove times √2, we divide by √2.
We do it on both sides, so they stay equal.
√2 divided by √2 is 1.
So just x is left.
So x is 1 over √2.
Books write it with no root at the bottom.
We multiply top and bottom by √2: that is times 1.
√2 times √2 is 2.
So x = √2/2.
Both short sides are √2/2.
3 Read the point
Across, from the center to under P: √2/2.
Up, from the x-axis to P: √2/2.
So P = (√2/2, √2/2).
Careful!
The first number of the point is cos.
The second is sin.
Not the other way round.
Cos is the across number.
So cos 45° = √2/2.
4 Check
Let's check: P is on the circle, so across squared plus up squared must be 1.
Pythagoras, with the longest side 1.
It is 1.
So the point is right, and cos 45° = √2/2 is right!
Let's look back.
First: draw the unit circle.
Then: use the special triangle.
Then: read the point.
Last: check.