{"id":7964,"date":"2026-02-15T20:12:56","date_gmt":"2026-02-15T14:42:56","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=7964"},"modified":"2026-02-15T20:13:07","modified_gmt":"2026-02-15T14:43:07","slug":"domain-and-range-of-inverse-trigonometric-functions","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/domain-and-range-of-inverse-trigonometric-functions\/","title":{"rendered":"Domain and Range Of Inverse Trigonometric Functions"},"content":{"rendered":"\n

Inverse trigonometric functions are the inverse functions of the basic trigonometric functions: sine, cosine, and tangent. To define these inverse functions, we need to restrict the domain and range of the original functions so that they become one-to-one and onto.<\/p>\n\n\n\n

Here are the domain and range for each of the primary inverse trigonometric functions:<\/h2>\n\n\n\n
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  1. Inverse Sine function ($sin^{-1}$ or arcsin):<\/li>\n<\/ol>\n\n\n\n

    Domain:<\/strong> $-1 \\leq x \\leq 1$, as the sine function takes values in this range. <\/p>\n\n\n\n

    Range<\/strong>: $-\\pi\/2 \\leq y \\leq \\pi\/2$, which corresponds to the interval over which sine is a strictly increasing function.<\/p>\n\n\n\n

      \n
    1. Inverse Cosine function ($cos^{-1}$ or arccos):<\/li>\n<\/ol>\n\n\n\n

      Domain<\/strong>: $-1 \\leq x \\leq 1$, as the cosine function takes values in this range.
      Range<\/strong>: $0 \\leq y \\leq \\pi$, which corresponds to the interval over which cosine is a strictly decreasing function.<\/p>\n\n\n\n

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      1. Inverse Tangent function ($tan^{-1}$ or arctan):<\/li>\n<\/ol>\n\n\n\n

        Domain<\/strong>: $-\\infty < x < \\infty$, as the tangent function takes values over the entire real line.
        Range<\/strong>: $-\\pi\/2 < y < \\pi\/2$, which corresponds to the interval over which tangent is a strictly increasing function.<\/p>\n\n\n\n

        There are also three other inverse trigonometric functions corresponding to the reciprocal functions of sine, cosine, and tangent: inverse cosecant, inverse secant, and inverse cotangent. Their domain and range are as follows:<\/p>\n\n\n\n

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        1. Inverse Cosecant function ($csc^{-1}$ or arccsc):<\/li>\n<\/ol>\n\n\n\n

          Domain<\/strong>: $x \\geq -1$ or $x \\geq 1$, as the cosecant function takes values in this range.
          Range<\/strong>: $-\\pi \/2 \\leq y < 0$ or $0 < y \\leq \\pi\/2$, which corresponds to the interval over which cosecant is a strictly decreasing function.<\/p>\n\n\n\n

            \n
          1. Inverse Secant function ($sec^{-1}$ or arcsec):<\/li>\n<\/ol>\n\n\n\n

            Domain<\/strong>: $x \\leq -1$ or $x \\geq 1$, as the secant function takes values in this range.
            Range<\/strong>: $0 \\leq y < \\pi\/2$ or $\\pi\/2 < y \\leq \\pi$, which corresponds to the interval over which secant is a strictly decreasing function.<\/p>\n\n\n\n

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            1. Inverse Cotangent function ($cot^{-1}$ or arccot):<\/li>\n<\/ol>\n\n\n\n

              Domain<\/strong>: $-\\infty < x < \\infty$, as the cotangent function takes values over the entire real line.
              Range<\/strong>: $0 < y < \\pi$, which corresponds to the interval over which cotangent is a strictly decreasing function.<\/p>\n\n\n\n

              \"Domain<\/a><\/figure>\n\n\n\n

              How to remember it<\/h2>\n\n\n\n