Trig Functions & Identities — Interactive Unit Circle

Trig Functions, Right Triangles, and Identities

Rotate a ray by an angle θ in standard position (start along +x, counter‑clockwise positive). On a circle of radius r, the endpoint has coordinates (x,y) = (r·cosθ, r·sinθ). The ratios sin, cos, tan — and their reciprocals csc, sec, cot — are defined below and visualized on the diagram.

Angle ° or rad Radius r

Blue = adjacent (x), pink = opposite (y), green = hypotenuse (r). The vertical tangent at x=r visualizes tan and sec. Values update as you change θ or r.

Controls & Quick angles

θ = 45.0°

Triangle (lengths)

adj = x0.7071

opp = y0.7071

hyp = r1.0000

Trig ratios

cos θ = x/r0.7071

sin θ = y/r0.7071

tan θ = y/x1.0000

sec θ = 1/cos θ1.4142

csc θ = 1/sin θ1.4142

cot θ = 1/tan θ1.0000

Definitions (right triangle)

cosθ = adj/hyp,   sinθ = opp/hyp,   tanθ = opp/adj

secθ = hyp/adj,   cscθ = hyp/opp,   cotθ = adj/opp

Using coordinates (x, y, r)

cosθ = x/r,   sinθ = y/r,   tanθ = y/x

secθ = r/x,   cscθ = r/y,   cotθ = x/y

adjacent (x)

opposite (y)

hypotenuse (r)

tan & vertical segment at x=r

sec length along the ray

Common angles table (exact values)

Angles in degrees and radians, with principal trig function values. "undef" indicates undefined (division by 0).

Key Identities

Symmetry

sin(−θ)=−sinθ,  cos(−θ)=cosθ,  tan(−θ)=−tanθ

sin(π/2−θ)=cosθ,  cos(π/2−θ)=sinθ,  tan(π/2−θ)=cotθ

Pythagorean

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

Sum & difference

sin(a±b) = sin a cos b ± cos a sin b

cos(a±b) = cos a cos b ∓ sin a sin b

tan(a±b) = (tan a ± tan b)/(1 ∓ tan a tan b)

Double‑angle

sin 2θ = 2 sinθ cosθ

cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1

tan 2θ = 2 tanθ / (1 − tan²θ)

Law of Sines & Law of Cosines

For any triangle with sides a,b,c opposite angles A,B,C:

Law of Sines

sin A / a = sin B / b = sin C / c

Law of Cosines

a² = b² + c² − 2bc cos A

b² = a² + c² − 2ac cos B

c² = a² + b² − 2ab cos C

Glossary — Core Trig Terms

Clear, short definitions that match the diagram and formulas above.

Hypotenuse (r)

The longest side of a right triangle, opposite the right angle. In the unit‑circle model it’s the radius. With endpoint (x, y), its length is r = √(x² + y²).

cosine (cosθ)

Ratio of adjacent to hypotenuse: cosθ = adj/hyp = x/r. On the unit circle (r=1), it equals the x‑coordinate of the point.

sine (sinθ)

Ratio of opposite to hypotenuse: sinθ = opp/hyp = y/r. On the unit circle, it’s the y‑coordinate.

tangent (tanθ)

Opposite over adjacent and slope of the radius ray: tanθ = opp/adj = y/x = sinθ/cosθ. On the unit circle, it equals the signed length of the vertical segment where the ray meets the line x = 1.

secant (secθ)

Reciprocal of cosine: secθ = 1/cosθ = hyp/adj = r/x. Geometrically, it’s the length along the ray from the origin to where it hits x = r, divided by r.

cosecant (cscθ)

Reciprocal of sine: cscθ = 1/sinθ = hyp/opp = r/y.

cotangent (cotθ)

Reciprocal of tangent: cotθ = 1/tanθ = adj/opp = x/y.