Why does the Unit Circle work the way it does?
The unit circle is a circle with a radius of 1 that is centered on the origin.
The coordinates of the points work out where 𝑥 = cos 𝜃 𝑎𝑛𝑑 𝑦 = sin 𝜃 ,
but why is that? Let’s examine the 30-degree arc of the circle.
The hypotenuse is 1 since the radius of the circle is 1. The
value of the other sides can be calculated since we
already know the ratios of 30-60-90 and 45-45-90
triangles from Geometry. I put the triangles in the
bottom left for review.
( 3/2, 1/2)
1
sin 𝜃 = 𝑦/1
cos 𝜃 = 𝑥/1
tan 𝜃 = 𝑦/𝑥
𝑐𝑠𝑐 𝜃 = 1/𝑦
sec 𝜃 = 1/𝑥
cot 𝜃 = 𝑥/𝑦
60
1/2
30
(0, 0)
3/2
( 3/2, 0)
Using SOH-CAH-TOA, we can see that:
cos 30 =
𝑎𝑑𝑗
ℎ𝑦𝑝
=
3
2
1
=
3
2
1
𝑜𝑝𝑝 2 1
sin 30 =
= =
ℎ𝑦𝑝 1 2
Notice we only have angles on the unit circle from these two special right triangles.
150 degrees, for example, is just a 30-degree angle in Quadrant 2 of the graph. Notice
how the coordinates are the same except for the x-coordinate being negative because
its on the part of the graph where x is negative.