Reciprocal Trigonometric Functions

This lesson covers the three reciprocal trigonometric functions — sec, cosec, and cot — their definitions, graphs, domains, ranges, and the key identities associated with them. These functions extend your trigonometric toolkit and are essential for proving identities, solving equations, and performing integration at A-Level.

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Definitions

The reciprocal trigonometric functions are defined as follows:

sec θ = 1/cos θ       (secant)
cosec θ = 1/sin θ     (cosecant)
cot θ = 1/tan θ = cos θ/sin θ   (cotangent)

Note: these are reciprocals, not inverses. sec θ ≠ cos⁻¹ θ. The inverse function arccos is a completely different concept.

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Domains

Each reciprocal function is undefined where its corresponding function is zero:

Function	Undefined when	Domain exclusions
sec θ	cos θ = 0	θ ≠ π/2 + nπ (i.e., θ ≠ 90° + 180°n)
cosec θ	sin θ = 0	θ ≠ nπ (i.e., θ ≠ 180°n)
cot θ	sin θ = 0 (or tan θ = 0)	θ ≠ nπ (i.e., θ ≠ 180°n)

where n is any integer.

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Ranges