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Reference · Trigonometry · Formula Sheet

The Trig Cheat Sheet, Live

Every box from the classic Dawkins trig cheat sheet — definitions, the unit circle, identities, angle-sum formulas, inverse functions, and the Laws of Sines & Cosines — rebuilt so you can drag an angle and watch both sides of a formula compute in real time.

Definition of the Trig Functions

Two equivalent pictures of the same six ratios: a right triangle (for 0°<θ<90°) and the unit circle (for any angle at all). Drag either slider.

Right-triangle definition

Unit-circle definition

Domain, Range & Period

Pick a function and stretch it with ω. The dashed vertical lines mark where it's undefined (outside its domain); the shaded band shows its range; the amber ticks mark one computed period T.

Function: ω =

The Full Unit Circle

All the standard angles at once. Pick one to see its exact coordinates and exact sin/cos/tan values — for any point (x,y) on the unit circle, cos(θ)=x and sin(θ)=y.

Angle:

Formulas & Identities

Set θ and every identity below computes both sides live — a match confirms the identity holds for that angle (undefined cases are shown as "—" rather than a false mismatch).

θ = °
IdentityLHSRHSMatch

Degrees ↔ Radians

degrees  |  radians (×π) π →

Double, Half, Sum & Product Formulas

Set α and β and watch double-angle, sum/difference, product-to-sum and sum-to-product formulas all check out numerically at once.

α = °    β = °
FormulaLHSRHSMatch

Half angle (± sign depends on quadrant of θ/2)

sin(θ/2) = ± √[(1−cosθ)/2]
cos(θ/2) = ± √[(1+cosθ)/2]
tan(θ/2) = ± √[(1−cosθ)/(1+cosθ)]

Cofunction formulas

sin(π/2−θ) = cos θ
tan(π/2−θ) = cot θ
sec(π/2−θ) = csc θ

Remaining sum-to-product

sinα−sinβ = 2cos(½(α+β))sin(½(α−β))
cosα−cosβ = −2sin(½(α+β))sin(½(α−β))

Inverse Trig Functions & Solving Triangles

Inverse function calculator

Enter x. Each inverse function only accepts x in its domain — out-of-range boxes are grayed out.

x =
FunctionDomainRange
y = sin⁻¹(x)−1 ≤ x ≤ 1−π/2 ≤ y ≤ π/2
y = cos⁻¹(x)−1 ≤ x ≤ 10 ≤ y ≤ π
y = tan⁻¹(x)−∞ < x < ∞−π/2 < y < π/2

Law of Sines, Cosines & Tangents — solve a triangle (SAS)

Give two sides and the angle between them (γ, opposite side c); the rest is computed and checked with the Law of Sines and Mollweide's formula.

a = b = γ = °