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Trigonometry Solutions And Relationships Chart Table

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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 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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