Trigonometry Solutions And Relationships Chart Table

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 right-angled 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.

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