Step by step
The question
Question: find the exact value of sin π/6.
1 Change to degrees
The angle is in radians.
π is 180°, so we put 180° in place of π.
So π/6 = 30°.
2 Draw the unit circle
The unit circle has its center at 0 and radius 1.
We turn 30° 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 30°, sin 30°).
Across is cos.
Up is sin.
3 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 30°, 60° and 90°.
The short side x is across from the 30° angle.
From the special triangles lesson: the sides are x, x root 3 and 2x.
The longest side, 2x, is 1.
So 2x = 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 = 1/2: the short side.
The other side is x root 3: 1/2 times √3.
That is √3/2.
4 Read the point
Across, from the center to under P: √3/2.
Up, from the x-axis to P: 1/2.
So P = (√3/2, 1/2).
Careful!
Sin is the UP side, not the across side.
Up from the x-axis to P is 1/2.
Sin is the up number.
So sin π/6 = 1/2.
5 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 sin π/6 = 1/2 is right!
Let's look back.
First: change to degrees.
Then: draw the unit circle.
Then: use the special triangle.
Then: read the point.
Last: check.