Unit Circle
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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.
| Angle in Degrees (memorize) | Angle in Radians (memorize) |
|---|---|
| AD | AD x (pi/180) |
| AR x (180/pi) | AR |
| 0, 360 | 0, 2 pi |
| 30 | pi/6 |
| 45 | pi/4 |
| 60 | pi/3 |
| 90 | pi/2 |
| 120 | 2 pi/3 |
| 135 | 3 pi/4 |
| 150 | 5 pi/6 |
| 180, -180 | pi, -pi |
| 210, -150 | 7 pi/6, -5 pi/6 |
| 225, -135 | 5 pi/4, -3 pi/4 |
| 240, -120 | 4 pi/3, -2 pi/3 |
| 270, -90 | 3 pi/2, -pi/2 |
| 300, -60 | 5 pi/3, -pi/3 |
| 315, -45 | 7 pi/4, -pi/4 |
| 330, -30 | 11 pi/6, -pi/6 |
Quadrants
For the unit circle, the radius R is always sqrt(x2+y2)=1while 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.
| Quadrant (memorize) | Property (memorize) |
|---|---|
| I | Angle is 0 to 90 degrees |
| I | Angle is 360 to 450 degrees |
| I | Angle is 0 to pi/2 radians |
| I | Angle is 2 pi to 5 pi/2 radians |
| I | x>0 and y>0 |
| I | cos>0, sin>0, tan>0 |
| II | Angle is 90 to 180 degrees |
| II | Angle is 450 to 540 degrees |
| II | Angle is pi/2 to pi radians |
| II | Angle is 5 pi/2 to 3 pi radians |
| II | x<0 and y>0 |
| II | cos<0, sin>0, tan<0 |
| III | Angle is 180 to 270 degrees |
| III | Angle is -180 to -90 degrees |
| III | Angle is pi to 3 pi/2 radians |
| III | Angle is -pi to -pi/2 radians |
| III | x<0 and y<0 |
| III | cos<0, sin<0, tan>0 |
| IV | Angle is 270 to 360 degrees |
| IV | Angle is -90 to 0 degrees |
| IV | Angle is 3 pi/2 to 2 pi radians |
| IV | Angle is -pi/2 to 0 radians |
| IV | x>0 and y<0 |
| IV | cos>0, sin<0, tan<0 |
Sines, Cosines, and Tangents
"SOHCAHTOA" is a useful Mnemonic for memorizing sines, cosines, and tangents:
Sine is Opposite/Hypotenuse.
Cosine is Adjacent/Hypotenuse.
Tangent is Opposite/Adjacent.
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 Degrees (memorize) | Sine (memorize) |
|---|---|
| 0, 180, -180 | 0 |
| 30, 150 | 1/2 |
| 45, 135 | √2/2 |
| 60, 120 | √3/2 |
| 90 | 1 |
| -30, -150 | -1/2 |
| -45, -135 | -√2/2 |
| -60, -120 | -√3/2 |
| -90 | -1 |
Cosines for Different Angles
| Angle in Degrees (memorize) | Cosine (memorize) |
|---|---|
| 0 | 1 |
| 30, -30 | √3/2 |
| 45, -45 | √2/2 |
| 60, -60 | 1/2 |
| 90, -90 | 0 |
| 120, -120 | -1/2 |
| 135, -135 | -√2/2 |
| 150, -150 | -√3/2 |
| 180, -180 | -1 |
Tangents for Different Angles
| Angle in Degrees (memorize) | Tangent (memorize) |
|---|---|
| -180, 0, 180, 360 | 0 |
| -150, 30, 210 | √3/3 |
| -135, 45, 225 | 1 |
| -120, 60, 240 | √3 |
| -90, 90, 270 | 1/0 |
| -60, 120, 300 | -√3 |
| -45, 135, 315 | -1 |
| -30, 150, 330 | -√3/3 |
