Graph-Based Reciprocal Reasoning

Cosecant blows up where sine hits zero.

Secant blows up where cosine hits zero.

Cotangent is undefined where sine hits zero.

Cosine pairs with its reciprocal secant.

Sine pairs with its reciprocal cosecant.

Where cosine is at a minimum, secant is at a maximum.

Where sine is at a maximum, cosecant is at a maximum.

Cotangent’s asymptotes line up with multiples of π.

Secant is undefined where cosine is zero.

Cosecant is undefined where sine is zero.

Cotangent is undefined where sine is zero.

The “U-shaped branches” flanking cosine’s zeros are secant.

The alternating “cup/cap” shapes around sine’s zeros are cosecant.

The reciprocal of a tangent graph is cotangent.

The reciprocal of a sine graph is cosecant.

The reciprocal of a cosine graph is secant.

Among the reciprocal graphs, the one with asymptotes at every multiple of π is cosecant.

The reciprocal graph with asymptotes at every multiple of π/2 is secant.

Where sine crosses zero (0, π, 2π), cosecant has vertical asymptotes.

Where cosine is zero (π/2, 3π/2), secant has vertical asymptotes.