Trigonify

Topics · 3 identities

Reciprocal

Secant, cosecant, and cotangent defined as reciprocals of the primary ratios.

Secant identity: sec θ = 1 / cos θ

sec⁡θ=1cos⁡θ\sec\theta = \dfrac{1}{\cos\theta}

Secant is defined as the reciprocal of cosine, so sec θ = 1/cos θ. It is undefined wherever cos θ = 0 - at π/2, 3π/2 and every odd multiple of π/2 - which is exactly where the secant graph has its vertical asymptotes.

Open identity

Cosecant identity: csc θ = 1 / sin θ

csc⁡θ=1sin⁡θ\csc\theta = \dfrac{1}{\sin\theta}

Cosecant is the reciprocal of sine, so csc θ = 1/sin θ. It is undefined wherever sin θ = 0 - at 0, π and every multiple of π. Note the mismatch students trip on: cosecant goes with sine, not cosine.

Open identity

Cotangent reciprocal identity: cot θ = 1 / tan θ

cot⁡θ=1tan⁡θ\cot\theta = \dfrac{1}{\tan\theta}

Cotangent is the reciprocal of tangent, so cot θ = 1/tan θ. The reciprocal form and the quotient form cot θ = cos θ / sin θ agree everywhere both are defined, but the quotient form is the safer one to use at angles where tan θ is zero or undefined.

Open identity