## Inverse Trigonometric Functions

# Basic Concepts of Inverse Trigonometric Functions

- Sin
^{−1}(Sin x) = x ∀ x ∈ \left[-\frac{\pi}{2},\frac{\pi}{2}\right]

- Sin (Sin
^{−1}x) = x ∀ x ∈ [−1, 1] - Cos
^{−1}(Cos x) = x ∀ x ∈ [0, π]

Cos(Cos^{−1}x) = x ∀ x ∈ [−1, 1] - Tan
^{−1}(Tan x) = x ∀ x ∈ \left(-\frac{\pi}{2},\frac{\pi}{2}\right)

Tan (Tan^{−1}x) = x ∀ x ∈ R - Cot
^{−1}(Cot x) = x ∀ x ∈ (0, π)

Cot(Cot^{−1}x) = x ∀ x ∈ R - Sec
^{−1}(Sec x) = x ∀ x ∈ [0, π] − \left\{\frac{\pi}{2}\right\}

Sec(Sec^{−1}x) = x ∀ x ∈ R − (−1, 1) - Cosec
^{−1}(Cosec x) = x ∀ x ∈ \left[-\frac{\pi}{2},\frac{\pi}{2}\right] - {0}

Cosec(Cosec^{−1 }x) = x ∀ x ∈ R − (−1, 1)

### Part1: View the Topic in this video From 00:40 To 57:48

### Part2: View the Topic in this video From 00:40 To 16:30

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- Domain and Range of Inverse Trigonometric functions:

FUNCTION |
DOMAIN |
RANGE |

Sin^{−1} x |
[−1, 1] | \left[-\frac{\pi}{2},\frac{\pi}{2}\right] |

Cos^{−1} x |
[−1, 1] | [0, π] |

Tan^{−1} x |
R | \left(-\frac{\pi}{2},\frac{\pi}{2}\right) |

Cot^{−1} x |
R | (0, π) |

Sec^{−1} x |
R − (−1, 1) | [0, π] − \left\{\frac{\pi}{2}\right\} |

Cosec^{−1} x |
R − (−1, 1) | \left[-\frac{\pi}{2},\frac{\pi}{2}\right] - {0} |