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Article 11 — Appendix A.16

csc or cosec — trigonometric cosecant function

Category. Mathematics.

Abstract. Trigonometric cosecant: definition, graph, properties, identities and table of values for some angles.

References. This article is a part of scientific calculator Li-L, scientific calculator Li-X, scientific calculator Li-Lc and scientific calculator Li-Xc products.

1. Definition

Cosecant of the angle is ratio of the hypotenuse to the opposite leg.

2. Graph

Cosecant is 2π periodic function defined everywhere on real axis, except its singular points πn, n=0, ±1, ±2, ... — so, function domain is (πn, π(n + 1)), n∈N. Its stalactite-stalagmite graph is depicted below — fig. 1.

Fig. 1. Graph of the cosecant function y = csc x Fig. 1. Graph of the cosecant function y = cscx.

Function codomain is entire real axis with gap in the middle: (−∞, −1]∪[1, +∞).

3. Identities

Base:

csc−2φ + sec−2φ = 1

and its consequences:

cscφ = ±1 /√(1 − cos2φ)
cscφ = ±√(1 + tan2φ) / tanφ
cscφ = ±√(1 + cot2φ)
cscφ = ±secφ /√(sec2φ − 1)

By definition:

cscφ ≡ 1 /sinφ

Properties — symmetry, periodicity, etc.:

csc−φ = −cscφ
cscφ = csc(φ + 2πn), where n = 0, ±1, ±2, ...
cscφ = csc(π − φ)
cscφ = −csc(π + φ)
cscφ = sec(π/2 − φ)

Sum of angles:

csc(φ + ψ + χ) = secφ secψ secχ / (tanφ + tanψ + tanχ − tanφ tanψ tanχ)

Some angles:

Angle φValue cscφ
π/12√6 + √2
π/10√5 + 1
π/8√(4 + 2√2)
π/62
π/5√(50 + 10√5) /5
π/4√2
3π/10√5 − 1
π/32√3 /3
3π/8√(2 − √2)
2π/5√(50 − 10√5) /5
5π/12√6 − √2
π/21
Table 1. Cosecant for some angles.

4. Support

Trigonometric cosecant function csc or cosec is supported in:

Trigonometric cosecant function of the complex argument csc or cosec is supported in:

5. How to use

To calculate cosecant of the number:

csc(-1);

To calculate cosecant of the current result:

csc(Rslt);

To calculate cosecant of the angle φ in memory:

csc(Mem[φ]);