The Helpful Mathematics
+38 044 577 28 76

Article 11 — Appendix A.3

arccsc or arccosec — trigonometric arc cosecant function

Category. Mathematics.

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

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

Arc cosecant is inverse of the cosecant function.

2. Graph

Arc cosecant is antisymmetric function defined everywhere on real axis, ecxept the range (-1, 1) — so, function domain is (−∞, −1]∪[1, +∞). Its graph is depicted below — fig. 1

Fig. 1. Graph of the arc cosecant function y = arccsc x Fig. 1. Graph of the arc cosecant function y = arccscx.

Function codomain is limited to the range [−π/2, 0)∪(0, π/2].

3. Identities

Complementary angle:

arcsecx + arccscx = π/2

and as consequence:

arccsc sec φ = π/2 − φ

Negative argument:

arccsc(−x) = −arccscx

Reciprocal argument:

arcsc(1/x) = arcsinx

Sum and difference:

arccscx + arccscy = arccsc{xy / [x√(1 − 1 /x2) + y√(1 − 1 /y2)]}
arccscx − arccscy = arccsc{xy / [y√(1 − 1 /y2) − x√(1 − 1 /x2)]}

Some argument values:

Argument xValue arccscx
1π/2
√6 − √25π/12
√(50 − 10√5) /52π/5
√(2 − √2)3π/8
2√3 /3π/3
√5 − 13π/10
√2π/4
√(50 + 10√5) /5π/5
2π/6
√(4 + 2√2)π/8
√5 + 1π/10
√6 + √2π/12
Table 1. Arc cosecant for some argument values.

4. Support

Trigonometric arc cosecant function arccsc or arccosec is supported in:

Trigonometric arc cosecant function of the complex argument arccsc or arccosec is supported in:

5. How to use

To calculate arc cosecant of the number:

arccsc(-1);

To calculate arc cosecant of the current result:

arccsc(Rslt);

To calculate arc cosecant of the number x in memory:

arccsc(Mem[x]);