Inverse Trigonometric Functions

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Trigonometric functions are not one-one (injective) over their natural domains. For a function to have an inverse, it must be both one-one and onto (bijective). Since sin x, cos x, etc. repeat their v

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Domain: [-1, 1] Range: [-π/2, π/2] This range is called the principal value branch of sin⁻¹x. The function sin⁻¹: [-1, 1] → [-π/2, π/2]

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Domain: [-1, 1] Range: [0, π] This range is the principal value branch of cos⁻¹x. The function cos⁻¹: [-1, 1] → [0, π]

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Domain: R (all real numbers) Range: (-π/2, π/2) This range is the principal value branch of tan⁻¹x. The function tan⁻¹: R → (-π/2, π/2)

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Domain: R (all real numbers) Range: (0, π) This range is the principal value branch of cot⁻¹x. The function cot⁻¹: R → (0, π)

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Domain: R - (-1, 1) or (-∞, -1] ∪ [1, ∞) Range: [0, π] - {π/2} This range is the principal value branch of sec⁻¹x. The function sec⁻¹: R - (-1, 1) → [0, π] - {π/2}

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Domain: R - (-1, 1) or (-∞, -1] ∪ [1, ∞) Range: [-π/2, π/2] - {0} This range is the principal value branch of cosec⁻¹x. The function cosec⁻¹: R - (-1, 1) → [-π/2, π/2] - {0}

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The principal value of an inverse trigonometric function is the value that lies in the range of the principal value branch of that function. When we write sin⁻¹x, cos⁻¹x, etc., without specifying a br