Unit Circle

Updated 2009-02-22 16:42

Unit Circle

When two angles are listed together below, it means they occur at the same x,y position on the unit circle. If you add or subtract multiples of 360 degrees or 2 pi to any angle, you end at the same x,y position on the unit circle.

AR x (180/pi)AR
0, 3600, 2 pi
30pi/6
45pi/4
60pi/3
90pi/2
1202 pi/3
1353 pi/4
1505 pi/6
180, -180pi, -pi
210, -1507 pi/6, -5 pi/6
225, -1355 pi/4, -3 pi/4
240, -1204 pi/3, -2 pi/3
270, -903 pi/2, -pi/2
300, -605 pi/3, -pi/3
315, -457 pi/4, -pi/4
330, -3011 pi/6, -pi/6

For the unit circle, the radius R is always sqrt(x2+y2)=1

while x=R cos(A), y=R sin(A), and tan(A)=y/x.

Quadrants I-IV tell in which region of 2-D space the angle A and its respective x,y value lie.

IAngle is 0 to 90 degrees
IAngle is 360 to 450 degrees
IAngle is 0 to pi/2 radians
IAngle is 2 pi to 5 pi/2 radians
Ix>0 and y>0
Icos>0, sin>0, tan>0
IIAngle is 90 to 180 degrees
IIAngle is 450 to 540 degrees
IIAngle is pi/2 to pi radians
IIAngle is 5 pi/2 to 3 pi radians
IIx<0 and y>0
IIcos<0, sin>0, tan<0
IIIAngle is 180 to 270 degrees
IIIAngle is -180 to -90 degrees
IIIAngle is pi to 3 pi/2 radians
IIIAngle is -pi to -pi/2 radians
IIIx<0 and y<0
IIIcos<0, sin<0, tan>0
IVAngle is 270 to 360 degrees
IVAngle is -90 to 0 degrees
IVAngle is 3 pi/2 to 2 pi radians
IVAngle is -pi/2 to 0 radians
IVx>0 and y<0
IVcos>0, sin<0, tan<0

Sines, Cosines, and Tangents

"SOHCAHTOA" is a useful Mnemonic for memorizing sines, cosines, and tangents:
Sine is Opposite/Hypotenuse.

In the following charts, sometimes several angles are listed as giving the same sine, cosine, or tangent. This holds because sine, cosine, and tangent are periodic functions that repeat every 360 degrees or 2 pi radians.

Some additional repetitions when the angle A is given in degrees follow:
Sin(A)=Sin(180-A)
Sin(A)=-Sin(-A)
Cos(A)=Cos(-A)
Cos(A)=-Cos(A+180)
Tan(A)=Tan(A+180)
Tan(A)=-Tan(-A)
If the angle A is given in radians, just replace each 180 above with pi.

Sines for Different Angles

Angle in DegreesSine
0, 180, -1800
30, 1501/2
45, 1352/2
60, 1203/2
901
-30, -150-1/2
-45, -135-√2/2
-60, -120-√3/2
-90-1

Cosines for Different Angles

Angle in DegreesCosine
01
30, -303/2
45, -452/2
60, -601/2
90, -900
120, -120-1/2
135, -135-√2/2
150, -150-√3/2
180, -180-1

Tangents for Different Angles

Angle in DegreesTangent
-180, 0, 180, 3600
-150, 30, 2103/3
-135, 45, 2251
-120, 60, 2403
-90, 90, 2701/0
-60, 120, 300-√3
-45, 135, 315-1
-30, 150, 330-√3/3

References:

http://en.wikipedia.org/wiki/Unit_circle
http://en.wikipedia.org/wiki/Cartesian_coordinate_system#Two-dimensional_coordinate_system