The Unit Circle

Reference

Trig identities cheat sheet

All the standard identities on one page, each with the one-line reason it is true. Prints on a single sheet.

Each identity is a fact about one point, (cos θ, sin θ), on a circle of radius 1.

Definitions

  • tan θ = sin θ / cos θ   cot θ = cos θ / sin θSimilar triangles: tan θ is the height where the radius meets the line x = 1 (Chapter 1)
  • sec θ = 1 / cos θ   csc θ = 1 / sin θWhere the tangent line at the point meets the axes (all six functions)

Pythagorean

  • sin²θ + cos²θ = 1The point is on the unit circle: Pythagoras with hypotenuse 1 (the squares)
  • 1 + tan²θ = sec²θDivide the first by cos²θ, or use the triangle on the tangent line
  • 1 + cot²θ = csc²θDivide the first by sin²θ

Symmetry and period

  • sin(−θ) = −sin θ   cos(−θ) = cos θ   tan(−θ) = −tan θ−θ is the mirror image of θ in the x-axis
  • sin(θ + 2π) = sin θ   cos(θ + 2π) = cos θA full turn brings the point back
  • tan(θ + π) = tan θOpposite points lie on the same line through the center
  • sin(π − θ) = sin θ   cos(π − θ) = −cos θMirror image in the y-axis

Cofunctions

  • sin(π/2 − θ) = cos θ   cos(π/2 − θ) = sin θReflecting in the line y = x swaps the coordinates
  • tan(π/2 − θ) = cot θThe ratio of the swapped coordinates

Sum and difference

  • sin(α ± β) = sin α cos β ± cos α sin βThe rectangle around two stacked right triangles (the picture proof)
  • cos(α ± β) = cos α cos β ∓ sin α sin βThe top side of the same rectangle
  • tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β)Divide the sine formula by the cosine formula, then top and bottom by cos α cos β

Double angle

  • sin 2θ = 2 sin θ cos θSum formula with β = α
  • cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θSum formula, then trade squares with the Pythagorean identity
  • tan 2θ = 2 tan θ / (1 − tan²θ)Tangent sum with β = α

Half angle and power reduction

  • sin²(θ/2) = (1 − cos θ) / 2   cos²(θ/2) = (1 + cos θ) / 2Solve the double-angle forms of cos for the squares
  • sin²θ = (1 − cos 2θ) / 2   cos²θ = (1 + cos 2θ) / 2The same, written for θ; used to integrate sin² and cos²
  • tan(θ/2) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θThe angle at the edge of the circle is half the angle at the center

Triple angle

  • sin 3θ = 3 sin θ − 4 sin³θSum formula for 2θ + θ, then the double-angle and Pythagorean identities
  • cos 3θ = 4 cos³θ − 3 cos θThe same route for cosine

Product to sum

  • sin α sin β = ½[cos(α − β) − cos(α + β)]Subtract the two cosine sum formulas
  • cos α cos β = ½[cos(α − β) + cos(α + β)]Add them
  • sin α cos β = ½[sin(α + β) + sin(α − β)]Add the two sine sum formulas

Sum to product

  • sin α + sin β = 2 sin((α + β)/2) cos((α − β)/2)Product to sum, read backwards with α and β renamed
  • sin α − sin β = 2 cos((α + β)/2) sin((α − β)/2)
  • cos α + cos β = 2 cos((α + β)/2) cos((α − β)/2)
  • cos α − cos β = −2 sin((α + β)/2) sin((α − β)/2)Two nearly equal tones add to a fast wave inside a slow one: the beats a piano tuner listens for

Any triangle

  • a/sin A = b/sin B = c/sin C = 2REach side over its opposite sine is the diameter of the circle through the corners (Chapter 5)
  • c² = a² + b² − 2ab cos CPythagoras with a correction for the angle; Euclid II.12–13
  • Area = ½ab sin CHalf the base times the height, and the height is b sin C

The small line under each says why it is true and links to the figure that proves it. Every formula here is standard (NIST’s Digital Library of Mathematical Functions, §4.21, lists them all[1]), and the tester at the foot of the page checks the main ones numerically at 4,096 angles (all but the tangent sum and half-angle forms, for a reason given there).

Test an identity yourself

A list of formulas invites a nagging doubt: is this one really right, or did I copy it wrong? The tester below evaluates both sides at 4,096 angles from −4π to 4π. A true identity agrees everywhere to within the rounding error of the computer’s arithmetic; a false one, such as the three tempting mistakes at the end of the list, fails at once, and the figure shows you where. The identities chapter has the pictures that prove the important ones.

Figure 1 Choose any identity from the list, including the three classic false ones at the end. Top: both sides plotted over the range tested. Bottom: the gap between them on a logarithmic scale, which sits at the floor of rounding error when the identity is true.#

The two formulas on the sheet that the tester leaves out, the tangent sum and the half-angle tangent, are left out for an honest numerical reason rather than a mathematical one: close to the angles where tangent blows up, both sides are huge and the computer cannot evaluate them to the same last digit, so the strict test would flag rounding error as a failure. They are standard results, given in the same references[1][2].

Sources

  1. 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
  2. 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