The Unit Circle

Sources and simplifications

Every reference cited on this site, and every place a figure idealizes the real system.

References

  1. Abramson, J., et al. (2021). Precalculus 2e (§5.1 Angles, §5.2 Unit circle, §5.3 The other trigonometric functions, §6.1–6.2 Graphs, §6.3 Inverse trigonometric functions, §7.2–7.3 Identities, §8.1–8.2 Non-right triangles, §8.3–8.4 Polar coordinates). OpenStax, Rice University. textbook (Unit circle)
  2. Weber, K. (2005). Students' understanding of trigonometric functions. Mathematics Education Research Journal 17(3), 91–112. doi:10.1007/BF03217423. primary (Unit circle)
  3. Maknun, C. L., Rosjanuardi, R., Jupri, A. (2020). Didactical design on drawing and analysing trigonometric functions graph through a unit circle approach. International Electronic Journal of Mathematics Education 15(3), em0614. doi:10.29333/iejme/9275. primary (Unit circle)
  4. Roy, R., Olver, F. W. J. (chapter authors); Olver, F. W. J., et al. (eds.) (2026). NIST Digital Library of Mathematical Functions, Chapter 4: Elementary Functions (§4.14 definitions, §4.21 identities). National Institute of Standards and Technology, Release 1.2.8. reference (Unit circle)
  5. O'Connor, J. J., Robertson, E. F. (1996). The trigonometric functions. MacTutor History of Mathematics, University of St Andrews. scholarship (Unit circle)
  6. Taub, L. (1999). Hipparchus and trigonometry (Starry Messenger). Department of History and Philosophy of Science, University of Cambridge. museum (Unit circle)
  7. Klintberg, B. (2005). Hipparchus's 3600′-based chord table and its place in the history of ancient Greek and Indian trigonometry. Indian Journal of History of Science 40(2), 169ff. (INSA repository record). primary (Unit circle)
  8. O'Connor, J. J., Robertson, E. F. (1999). Claudius Ptolemy (biography). MacTutor History of Mathematics, University of St Andrews. scholarship (Unit circle)
  9. O'Connor, J. J., Robertson, E. F. (2000). Aryabhata I (biography). MacTutor History of Mathematics, University of St Andrews. scholarship (Unit circle)
  10. Joyce, D. E. (2015). Aryabhata's trig table (Math 105, History of Mathematics; quotes the Āryabhaṭīya in Clark's 1930 translation). Clark University. scholarship (Unit circle)
  11. Heydari-Malayeri, M. (2026). An Etymological Dictionary of Astronomy and Astrophysics: sine (Sanskrit jiyā, 'chord, bow-string'). Observatoire de Paris. reference (Unit circle)
  12. Euler, L. (1748). Introductio in analysin infinitorum, vol. 1, ch. VIII, §126–128 (English translation by Ian Bruce). Lausanne: Bousquet; translation at 17centurymaths.com. primary (Unit circle)
  13. Akkoç, H. (2008). Pre-service mathematics teachers' concept images of radian. International Journal of Mathematical Education in Science and Technology 39(7), 857–878. doi:10.1080/00207390802054458. primary (Radians)
  14. Strang, G., Herman, E. (2016). Calculus Volume 1, §2.3 The Limit Laws (sin θ < θ < tan θ and lim sin θ/θ = 1). OpenStax, Rice University. textbook (Radians)
  15. Bureau International des Poids et Mesures (2024). The International System of Units (SI Brochure), 9th edition, version 3.01 (radian: m/m = 1). BIPM. standard (Radians)
  16. Quincey, P. (2016). The range of options for handling plane angle and solid angle within a system of units. Metrologia 53(2), 840–845. doi:10.1088/0026-1394/53/2/840. primary (Radians)
  17. Miller, J. (2023). Earliest Known Uses of Some of the Words of Mathematics: R (entry RADIAN). MacTutor History of Mathematics, University of St Andrews. scholarship (Radians)
  18. O'Connor, J. J., Robertson, E. F. (2005). Roger Cotes (biography). MacTutor History of Mathematics, University of St Andrews. scholarship (Radians)
  19. Strang, G., Herman, E. (2016). Calculus Volume 1, §3.5 Derivatives of Trigonometric Functions (d/dx sin x = cos x, d/dx cos x = −sin x, x in radians). OpenStax, Rice University. textbook (Graphs)
  20. Ling, S. J., Sanny, J., Moebs, W. (2016). University Physics Volume 1, §15.1 Simple Harmonic Motion and §15.3 Comparing Simple Harmonic Motion and Circular Motion. OpenStax, Rice University. textbook (Graphs)
  21. Ling, S. J., Sanny, J., Moebs, W. (2016). University Physics Volume 2, §15.1 AC Sources (v = V₀ sin ωt; 60 or 50 Hz; amplitude 170 or 311 V). OpenStax, Rice University. textbook (Graphs)
  22. NOAA Center for Operational Oceanographic Products and Services (2024). What are Harmonic Constituents? (tide predictions as the sum of 37 cosine curves). National Oceanic and Atmospheric Administration. institutional (Graphs)
  23. O'Connor, J. J., Robertson, E. F. (1997). Jean Baptiste Joseph Fourier (biography: the 1807 memoir on heat, Théorie analytique de la chaleur, 1822). MacTutor History of Mathematics, University of St Andrews. scholarship (Graphs)
  24. Weisstein, E. W. (2024). Wilbraham–Gibbs Constant (the limiting overshoot of Fourier partial sums at a jump, Si(π) ≈ 1.851937). MathWorld, Wolfram Research. reference (Graphs)
  25. Carleton College Library (2021). Oliver Byrne, The First Six Books of the Elements of Euclid, in which coloured diagrams and symbols are used instead of letters (London: Pickering, 1847). Carleton College, Gould Library news. museum (Identities)
  26. Euclid; Joyce, D. E. (ed.) (1996). Euclid's Elements, Book II, Propositions 12 and 13 (with commentary relating them to the law of cosines). Clark University. primary (Triangles)
  27. Euclid; Joyce, D. E. (ed.) (1996). Euclid's Elements, Book III, Propositions 21 (angles in the same segment are equal) and 31 (the angle in a semicircle is right). Clark University. primary (Triangles)
  28. O'Connor, J. J., Robertson, E. F. (1999). Al-Tusi Nasir (biography; the Treatise on the Quadrilateral and the sine formula for plane triangles). MacTutor History of Mathematics, University of St Andrews. scholarship (Triangles)
  29. Complete Dictionary of Scientific Biography (2008). Snel (Snellius or Snel van Royen), Willebrord (triangulation of 1615, Eratosthenes Batavus 1617). Charles Scribner's Sons (via Encyclopedia.com; MacTutor mirror). scholarship (Triangles)
  30. Todhunter, I. (1886). Spherical Trigonometry, for the use of colleges and schools, 5th ed. (Art. 20: sides measured by the angles they subtend at the centre; Art. 32: the three angles together exceed two right angles). London: Macmillan; Project Gutenberg eBook 19770. textbook (Triangles)
  31. Kahan, W. (2014). Miscalculating Area and Angles of a Needle-like Triangle (lecture notes, §1–5 and Table 1). University of California, Berkeley. primary (Triangles)
  32. Strang, G., Herman, E. (2016). Calculus Volume 2, §7.4 Area and Arc Length in Polar Coordinates (A = ½∫ r² dθ) and §7.5 Conic Sections (r = ep/(1 ± e cos θ)). OpenStax, Rice University. textbook (Polar)
  33. DPA Microphones (2024). 1st order pattern (Mic University dictionary: omni 1 + 0 cos θ, cardioid 0.5 + 0.5 cos θ, supercardioid 0.37 + 0.63 cos θ, hypercardioid 0.25 + 0.75 cos θ, figure-of-eight cos θ). DPA Microphones. institutional (Polar)
  34. Williams, D. R. (2024). Earth Fact Sheet (orbital eccentricity 0.0167; perihelion 147.095 and aphelion 152.100 million km). NASA Goddard Space Flight Center, NSSDCA. institutional (Polar)
  35. Coolidge, J. L.; MacTutor (1952). The origin of polar coordinates (notes on J. L. Coolidge's 1950 ICM lecture, published in Amer. Math. Monthly 59 (1952) 78–85). MacTutor History of Mathematics, University of St Andrews. scholarship (Polar)
  36. Roediger, H. L., Karpicke, J. D. (2006). Test-enhanced learning: taking memory tests improves long-term retention. Psychological Science 17(3), 249–255. doi:10.1111/j.1467-9280.2006.01693.x. primary (Unit circle game)
  37. Yale Center for British Art (1847). Byrne, The First Six Books of the Elements of Euclid (London: Pickering; Chiswick: C. Whittingham, 1847), catalogue record with notes ("Each proposition is set in Caslon italic"). Yale Center for British Art, Paul Mellon Collection. museum (About)

Simplifications