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The trigomertric principal inverses are listed in the following table.
Name 
Notation 
Definition 
Domain of x
for real result 
Range of usual
principal value
( radians) 
Range of usual
principal value
( degrees ) 
arcsine 
y = arcsin x 
x = sin y 
−1 ≤ x ≤ 1 
− π /2 ≤ y ≤ π /2 
−90° ≤ y ≤ 90° 
arccosine 
y = arccos x 
x = cos y 
−1 ≤ x ≤ 1 
0 ≤ y ≤ π 
0° ≤ y ≤ 180° 
arctangent 
y = arctan x 
x = tan y 
all real numbers 
− π /2 < y < π /2 
−90° < y < 90° 
arccotangent 
y = arccot x 
x = cot y 
all real numbers 
0 < y < π 
0° < y < 180° 
arcsecant 
y = arcsec x 
x = sec y 
x ≤ −1 or 1 ≤ x 
0 ≤ y < π /2 or π /2 < y ≤ π 
0° ≤ y < 90° or 90° < y ≤ 180° 
arccosecant 
y = arccsc x 
x = csc y 
x ≤ −1 or 1 ≤ x 
− π /2 ≤ y < 0 or 0 < y ≤ π /2 
90° ≤ y < 0° or 0° < y ≤ 90° 
(Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π /2 or π ≤ y < 3 π /2 ), because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example using this range, tan(arcsec( x ))=√ x 2 1 , whereas with the range ( 0 ≤ y < π /2 or π /2 < y ≤ π ), we would have to write tan(arcsec( x ))=±√ x 2 1 , since tangent is nonnegative on 0 ≤ y < π /2 but nonpositive on π /2 < y ≤ π . For a similar reason, the same authors define the range of arccosecant to be (  π < y ≤  π /2 or 0 < y ≤ π /2 ).)
If x is allowed to be a complex number , then the range of y applies only to its real part.
Relationships between trigonometric functions and inverse trigonometric functions
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a rightangled triangle, with one side of length 1, and another side of length x (any real number between 0 and 1), then applying the Pythagorean theorem and definitions of the trigonometric ratios. Purely algebraic derivations are longer.
