### Trigonometry Solutions And Relationships Chart Table

The trigomertric principal inverses are listed in the following table.

Name |
Notation |
Definition |
Domain of xfor 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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