In trigonometry, the unit circle has a radius of 1 and is centered at the origin. It’s useful for learning ratios like sine and cosine. It also illustrates the relationship between angles in degrees and radians.
| Degrees | Radians | Radians (π) | Cos (X Coord) | Sin (Y Coord) | Tan | Cot | Cos2+Sin2 | Quadrant | CAST |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 | 0 | # | * | ||
| 10 | 0.17453292519943295 | 0.984807753012208 | 0.17364817766693033 | 0.17632698070846498 | 1.5423510453569202 | 1 | I | A | |
| 20 | 0.3490658503988659 | 0.9396926207859084 | 0.3420201433256687 | 0.36397023426620234 | 0.4469951089489167 | 1 | I | A | |
| 30 | 0.5235987755982988 | π/6 | 0.8660254037844387 | 0.49999999999999994 | 0.5773502691896257 | -0.15611995216165922 | 1 | I | A |
| 40 | 0.6981317007977318 | 0.766044443118978 | 0.6427876096865393 | 0.8390996311772799 | -0.8950829176379128 | 1 | I | A | |
| 45 | 0.7853981633974483 | π/4 | 0.7071067811865476 | 0.7071067811865475 | 1 | 0.6173696237835551 | 1 | I | A |
| 50 | 0.8726646259971648 | 0.6427876096865394 | 0.766044443118978 | 1.19175359259421 | -3.677814450850569 | 1 | I | A | |
| 60 | 1.0471975511965976 | π/3 | 0.5000000000000001 | 0.8660254037844386 | 1.7320508075688767 | 3.124605622242308 | 1 | I | A |
| 70 | 1.2217304763960306 | 0.3420201433256688 | 0.9396926207859083 | 2.7474774194546216 | 0.8183574478651037 | 1 | I | A | |
| 80 | 1.3962634015954636 | 0.17364817766693041 | 0.984807753012208 | 5.671281819617707 | 0.11106600664299159 | 1 | I | A | |
| 90 | 1.5707963267948966 | π/2 | 0 | 1 | ∞ | -0.5012027833801532 | * | ||
| 100 | 1.7453292519943295 | -0.1736481776669303 | 0.984807753012208 | -5.671281819617711 | -1.702956919426469 | 1 | II | S | |
| 110 | 1.9198621771937625 | -0.3420201433256687 | 0.9396926207859084 | -2.7474774194546225 | 22.580477867856576 | 1 | II | S | |
| 120 | 2.0943951023931953 | 2π/3 | -0.4999999999999998 | 0.8660254037844387 | -1.7320508075688783 | 1.4022826164313726 | 1 | II | S |
| 130 | 2.2689280275926285 | -0.6427876096865394 | 0.766044443118978 | -1.19175359259421 | 0.3948919264315488 | 1 | II | S | |
| 135 | 2.356194490192345 | 3π/4 | -0.7071067811865475 | 0.7071067811865476 | -1.0000000000000002 | -11.27195479594305 | 1 | II | S |
| 140 | 2.443460952792061 | -0.7660444431189779 | 0.6427876096865395 | -0.8390996311772804 | -0.20180123513591974 | 1 | II | S | |
| 150 | 2.6179938779914944 | 5π/6 | -0.8660254037844387 | 0.49999999999999994 | -0.5773502691896257 | -0.9781422040185391 | 1 | II | S |
| 160 | 2.792526803190927 | -0.9396926207859083 | 0.3420201433256689 | -0.36397023426620256 | -4.446294469482043 | 1 | II | S | |
| 170 | 2.9670597283903604 | -0.984807753012208 | 0.17364817766693028 | -0.1763269807084649 | 2.7058884335316002 | 1 | II | S | |
| 180 | 3.141592653589793 | π | -1 | 0 | 0 | 0.7469988144140444 | * | ||
| 190 | 3.3161255787892263 | -0.984807753012208 | -0.17364817766693047 | 0.1763269807084651 | 0.06645310311247116 | 1 | III | T | |
| 200 | 3.490658503988659 | -0.9396926207859084 | -0.34202014332566866 | 0.3639702342662023 | -0.5578715021347701 | 1 | III | T | |
| 210 | 3.6651914291880923 | 7π/6 | -0.8660254037844386 | -0.5000000000000001 | 0.577350269189626 | -1.8897636901659693 | 1 | III | T |
| 220 | 3.839724354387525 | -0.766044443118978 | -0.6427876096865393 | 0.8390996311772799 | 11.268095912734225 | 1 | III | T | |
| 225 | 3.9269908169872414 | 5π/4 | -0.7071067811865477 | -0.7071067811865475 | 0.9999999999999997 | -0.39492677189913306 | 1 | III | T |
| 230 | 4.014257279586958 | -0.6427876096865395 | -0.7660444431189779 | 1.1917535925942093 | 1.2785939135279258 | 1 | III | T | |
| 240 | 4.1887902047863905 | 4π/3 | -0.5000000000000004 | -0.8660254037844384 | 1.7320508075688754 | 0.34457980332273175 | 1 | III | T |
| 250 | 4.363323129985824 | -0.34202014332566855 | -0.9396926207859084 | 2.7474774194546243 | -0.24830638624841692 | 1 | III | T | |
| 260 | 4.537856055185257 | -0.17364817766693033 | -0.984807753012208 | 5.67128181961771 | -1.0687232505188873 | 1 | III | T | |
| 270 | 4.71238898038469 | 3π/2 | 0 | -1 | ∞ | -5.591619519586494 | * | ||
| 280 | 4.886921905584122 | 0.17364817766692997 | -0.9848077530122081 | -5.671281819617723 | 2.376784911280432 | 1 | IV | C | |
| 290 | 5.061454830783556 | 0.342020143325669 | -0.9396926207859083 | -2.74747741945462 | 0.6802103121702459 | 1 | IV | C | |
| 300 | 5.235987755982989 | 5π/3 | 0.5000000000000001 | -0.8660254037844386 | -1.732050807568877 | 0.02210201570902426 | 1 | IV | C |
| 310 | 5.410520681182422 | 0.6427876096865393 | -0.7660444431189781 | -1.1917535925942102 | -0.6174112582889291 | 1 | IV | C | |
| 315 | 5.497787143782138 | 7π/4 | 0.7071067811865474 | -0.7071067811865477 | -1.0000000000000004 | 0.8950286238945064 | 1 | IV | C |
| 320 | 5.585053606381854 | 0.7660444431189778 | -0.6427876096865396 | -0.8390996311772806 | -2.1106940440151623 | 1 | IV | C | |
| 330 | 5.759586531581287 | 11π/6 | 0.8660254037844384 | -0.5000000000000004 | -0.5773502691896265 | 7.487434832947242 | 1 | IV | C |
| 340 | 5.934119456780721 | 0.9396926207859084 | -0.3420201433256686 | -0.3639702342662022 | 1.168162023308022 | 1 | IV | C | |
| 350 | 6.1086523819801535 | 0.984807753012208 | -0.1736481776669304 | -0.176326980708465 | 0.2957801336815787 | 1 | IV | C | |
| 360 | 6.283185307179586 | 2π | 1 | 0 | 0 | -0.295845697968555 | * |
Table Legend
* These parts of the unit circle are not triangles; they are lines/vectors. Therefore, their length, by default being part of a unit circle, is one.
# Division by zero is undefined (See Why Dividing by Zero is Undefined?)
Quadrant IV – (C)osine ratio is positive
Quadrant I – (A)ll ratios are positive
Quadrant II – (S)ine ratio is positive
Quadrant III – (T)angent ratio is positive
Suggested Reading
Abramson, Jay. 2015. “Unit Circle”. openstax . https://opentextbc.ca/algebratrigonometryopenstax/chapter/unit-circle/.
“Exsecant – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Exsecant.
“The CAST Method”. 2021. mathonweb.com. https://mathonweb.com/help_ebook/html/cast.htm.
“Trig Cheat Sheet”. 2021. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf.
“Trig. Equations Examples Using CAST Diagrams (Solutions, Examples, Worksheets, Videos, Activities)”. 2021. onlinemathlearning.com. https://www.onlinemathlearning.com/trig-equations-cast.html.
“Unit Circle”. 2021. mathsisfun.com. https://www.mathsisfun.com/geometry/unit-circle.html.
“Unit Circle: Sine And Cosine Functions | Precalculus II”. 2021. courses.lumenlearning.com. https://courses.lumenlearning.com/precalctwo/chapter/unit-circle-sine-and-cosine-functions/.
“UNIT CIRCLE TRIGONOMETRY”. 2021. online.math.uh.edu. https://online.math.uh.edu/MiddleSchool/Modules/Module_4_Geometry_Spatial/Content/UnitCircleTrigonometry-TEXT.pdf.
“Versine – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Versine.
Videos
Unit circle trigonometry comes up a lot in geometry, precalculus, and even calculus problems.
Extending SOH CAH TOA so that we can define trig functions for a broader class of angles.
In this lesson, we will learn what a unit circle is and why it is crucial to master in trigonometry, pre-calculus, and calculus. The unit circle is a circle of radius 1 (a unit radius) that is superimposed on to an xy plane. The points on the circle itself represent the sine and cosine values of the angles around the unit circle. In this lesson, we will spend our time finding the sine and cosine of the angles in quadrant one, and we will fill out a table of trigonometric functions at each of these angles. This is critical to master as understanding the first quadrant will help us find trig functions at any other angle around the unit circle.
Mathematical Mysteries 