The Unit Circle

The Unit Circle

The unit circle is a staple of high school math. It is very easy to learn because of patterns involved. You should spend the time learning it and not just for when you are tested on it in school.

Cosine Values

Degrees	Radians	Cosine	Numerator
0°	0	1	√4
30°	π⁄6	√3⁄2	√3
45°	π⁄4	√2⁄2	√2
60°	π⁄3	1⁄2	√1
90°	π⁄2	0	√0
120°	2π⁄3	-1⁄2	-√1
135°	3π⁄4	-√2⁄2	-√2
150°	5π⁄6	-√3⁄2	-√3
180°	π	-1	-√4
210°	7π⁄6	-√3⁄2	-√3
225°	5π⁄4	-√2⁄2	-√2
240°	4π⁄3	-1⁄2	-√1
270°	3π⁄2	0	√0
300°	5π⁄3	1⁄2	√1
315°	7π⁄4	√2⁄2	√2
330°	11π⁄6	√3⁄2	√3
360°	2π	1	√4

If you pay close attention, you will see that there is a definite pattern. The numerator values for the cosines are clear square root countdowns of 4, 3, 2, 1, 0, -1, -2, -3, -4 (with the square root symbols omitted) before counting back up to 4. Every time there is radian denominator of 2, the cosine is 0. Every time there is a radian denominator of 3, the cosine is ±¹⁄₂. Every time there is a radian denominator of 4, the cosine is ±^√2⁄₂. Every time there is a radian denominator of 6, the cosine is ±^√3⁄₂.

Sine Values

Degrees	Radians	Sine	Numerator
0°	0	0	√0
30°	π⁄6	1⁄2	√1
45°	π⁄4	√2⁄2	√2
60°	π⁄3	√3⁄2	√3
90°	π⁄2	1	√4
120°	2π⁄3	√3⁄2	√3
135°	3π⁄4	√2⁄2	√2
150°	5π⁄6	1⁄2	√1
180°	π	0	√0
210°	7π⁄6	-1⁄2	-√1
225°	5π⁄4	-√2⁄2	-√2
240°	4π⁄3	-√3⁄2	-√3
270°	3π⁄2	-1	-√4
300°	5π⁄3	-√3⁄2	-√3
315°	7π⁄4	-√2⁄2	-√2
330°	11π⁄6	-1⁄2	-√1
360°	2π	0	√0

There is, of course, a pattern for the sine values as well. I will leave it to you to discover this pattern for yourself.