Download PDF

cyclometric functions∗
pahio†
2013-03-21 17:52:36
The trigonometric functions are periodic, and thus get all their values infinitely many times. Therefore their inverse functions, the cyclometric functions, are multivalued, but the values within suitable chosen intervals are unique;
they form single-valued functions, called the branches of the multivalued functions.
The principal branches of the most used cyclometric functions are as follows:
• arcsin x is the angle y satisfying sin y = x and − π2 < y 5
−1 5 x 5 1)
π
2
(defined for
• arccos x is the angle y satisfying cos y = x and 0 5 y < π (defined for
−1 5 x 5 1)
• arctan x is the angle y satisfying tan y = x and − π2 < y <
the whole R)
π
2
(defined in
• arccot x is the angle y satisfying cot y = x and 0 < y < π (defined in
the whole R)
Those functions are denoted also sin−1 x, cos−1 x, tan−1 x and cot−1 x. We
here use these notations temporarily for giving the corresponding multivalued
functions (n = 0, ±1, ±2, ...):
sin−1 x = nπ + (−1)n arcsin x
cos−1 x = 2nπ ± arccos x
tan−1 x = nπ + arctan x
cot−1 x = nπ + arccot x
∗ hCyclometricFunctionsi created: h2013-03-21i by: hpahioi version: h36169i Privacy
setting: h1i hDefinitioni h26A09i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1
Some formulae
π
2
π
arctan x + arccot x =
2
Z x
dt
√
arcsin x =
dt
1
− t2
0
Z x
dt
dt
arctan x =
2
0 1+t
arcsin x + arccos x =
arcsin x = x +
1 x3
1·3 x5
1·3·5 x7
·
+
·
+
·
+ ...
2 3
2·4 5
2·4·6 7
(|x| 5 1)
x5
x7
x3
+
−
+ − . . . (|x| 5 1)
3
5
7
d
1
arccos x = − √
(|x| < 1)
dx
1 − x2
arctan x = x −
1
d
arccot x = −
(∀x ∈ R)
dx
1 + x2
The classic abbreviations of the cyclometric functions are usually explained
as follows. The values of the trigonometric functions are got from the unit
circle by means of its arc (in Latin arcus) with starting point (1, 0). The arc
represents the angle (which may be thought as a central angle of the circle), and
its end point (ξ, η) is achieved when the starting point has circulated along the
circumference anticlockwise for positive angle (and clockwise for negative angle).
Then the cosine of the arc (i.e. angle) is the abscissa ξ of the end point, the
sine of the arc is the ordinate η of it. Correspondingly, one can get the tangent
and cotangent of the arc by using certain points on the tangent lines x = 1 and
y = 1 of the unit circle.
The word cosine is in Latin cosinus, its genitive form is cosini. So e.g.
“arccos” of a given real number x means the ‘arc of the cosine value x’, in Latin
arcus cosini x.
2