Graph-Based Reciprocal Reasoning

Cotangent is undefined where sine is zero.

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

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

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

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

The reciprocal of a cosine graph is secant.

The reciprocal of a sine graph is cosecant.

The reciprocal of a tangent graph is cotangent.

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

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

Cosecant blows up where sine hits zero.

Cosecant is undefined where sine is zero.

Secant is undefined where cosine is zero.

Cotangent’s asymptotes line up with multiples of π.

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

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

Sine pairs with its reciprocal cosecant.

Cosine pairs with its reciprocal secant.

Cotangent is undefined where sine hits zero.

Secant blows up where cosine hits zero.