{"id":3606,"date":"2021-01-30T12:28:24","date_gmt":"2021-01-30T12:28:24","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3606"},"modified":"2023-06-22T17:28:26","modified_gmt":"2023-06-22T11:28:26","slug":"%e0%a6%ac%e0%a6%bf%e0%a6%aa%e0%a6%b0%e0%a7%80%e0%a6%a4-%e0%a6%a4%e0%a7%8d%e0%a6%b0%e0%a6%bf%e0%a6%95%e0%a7%8b%e0%a6%a3%e0%a6%ae%e0%a6%bf%e0%a6%a4%e0%a6%bf-%e0%a6%ab%e0%a6%be%e0%a6%82%e0%a6%b6%e0%a6%a8","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/%e0%a6%ac%e0%a6%bf%e0%a6%aa%e0%a6%b0%e0%a7%80%e0%a6%a4-%e0%a6%a4%e0%a7%8d%e0%a6%b0%e0%a6%bf%e0%a6%95%e0%a7%8b%e0%a6%a3%e0%a6%ae%e0%a6%bf%e0%a6%a4%e0%a6%bf-%e0%a6%ab%e0%a6%be%e0%a6%82%e0%a6%b6%e0%a6%a8\/","title":{"rendered":"\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b2\u09c7\u0996 (Graphs of inverse trigonometric functions)"},"content":{"rendered":"

\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b2\u09c7\u0996 <\/b>(Graphs of inverse trigonometric functions):<\/b>\u00a0<\/b><\/h2>\n

\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b2\u09c7\u0996 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09a4\u09c7 <\/span> y=\\sin ^{-1} x <\/span> \u0993\u00a0 y=\\cos ^{-1} x <\/span><\/span>\u00a0\u098f\u09b0 \u099c\u09a8\u09cd\u09af<\/span> x \\in[-1,1] <\/span><\/span>, y=\\tan ^{-1} x <\/span><\/span>\u00a0<\/span>\u0993 y=\\cot ^{-1} <\/span><\/span>\u00a0<\/span>\u098f\u09b0 \u099c\u09a8\u09cd\u09af x \\in \\mathbb{R} <\/span><\/span> \u098f\u09ac\u0982 y=\\sec ^{-1} x <\/span><\/span>\u0993 y=\\operatorname{cosec}^{-1} x <\/span>\u00a0\u098f\u09b0 \u099c\u09a8\u09cd\u09af x\u2264-1 <\/span> \u09ac\u09be x\u22651<\/span> \u098f\u09b0 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af y <\/span> \u098f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b0\u09c2\u09aa\u09c0 \u09ae\u09c2\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09af\u09bc\u0964 x <\/span> \u0993 y <\/span> \u09b8\u09ae\u09a8\u09cd\u09ac\u09af\u09bc\u09c7 \u0997\u09a0\u09bf\u09a4 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf \u099b\u0995 \u0995\u09be\u0997\u099c\u09c7 \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09c7 \u09b8\u09cd\u09a5\u09be\u09aa\u09bf\u09a4 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf \u09ae\u09c1\u0995\u09cd\u09a4 \u09b9\u09b8\u09cd\u09a4\u09c7 \u09ac\u0995\u09cd\u09b0\u09be\u0995\u09be\u09b0\u09c7 \u09b8\u0982\u09af\u09c1\u0995\u09cd\u09a4 \u0995\u09b0\u09b2\u09c7 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u0985\u0999\u09cd\u0995\u09bf\u09a4 \u09b9\u09af\u09bc\u0964<\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3:<\/b> y=\\sin ^{-1} x <\/span><\/span>\u00a0\u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0, \u09af\u0996\u09a8 x \\in[-1,1] <\/span><\/span><\/strong><\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8<\/b>: \u09a8\u09bf\u099a\u09c7\u09b0 \u09a4\u09be\u09b2\u09bf\u0995\u09be\u09af\u09bc <\/span>x\u2208[-1,\u00a01] sine <\/span><\/span> \u098f\u09b0 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af sine <\/span><\/span>\u00a0\u098f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b0\u09c2\u09aa\u09c0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09bf\u0964<\/span><\/p>\n

\"\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4<\/p>\n

\u098f\u0995\u099f\u09bf \u099b\u0995 \u0995\u09be\u0997\u099c\u09c7 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995\u09c7\u09b0 \u0985\u0995\u09cd\u09b7 \u09b0\u09c7\u0996\u09be<\/span> XOX <\/span>‘<\/span> \u0993 <\/span> YOY <\/span>‘<\/span> \u0986\u0981\u0995\u09bf\u0964<\/span><\/p>\n

\u09b8\u09cd\u0995\u09c7\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a7\u09be\u09b0\u09a3<\/b>:<\/span> x <\/span>-\u0985\u0995\u09cd\u09b7 \u09ac\u09b0\u09be\u09ac\u09b0 \u099b\u09cb\u099f \u09ac\u09b0\u09cd\u0997\u09c7\u09b0<\/span>\u00a0<\/span>10<\/span> \u09ac\u09be\u09b9\u09c1<\/span>\u00a0<\/span>=1<\/span> \u098f\u09ac\u0982<\/span> y <\/span>–<\/span>\u0985\u0995\u09cd\u09b7 \u09ac\u09b0\u09be\u09ac\u09b0 \u099b\u09cb\u099f \u09ac\u09b0\u09cd\u0997\u09c7\u09b0 \u098f\u0995 \u09ac\u09be\u09b9 <\/span> =15\u00b0 <\/span><\/span><\/h4>\n

\u098f\u0996\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a7\u09be\u09b0\u09bf\u09a4 \u09b8\u09cd\u0995\u09c7\u09b2 \u0985\u09a8\u09c1\u09af\u09be\u09af\u09bc\u09c0 \u09a4\u09be\u09b2\u09bf\u0995\u09be\u09ad\u09c1\u0995\u09cd\u09a4 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09cb \u099b\u0995 \u0995\u09be\u0997\u099c\u09c7 \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09bf\u0964 \u09b8\u09cd\u09a5\u09be\u09aa\u09bf\u09a4 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09cb \u09ae\u09c1\u0995\u09cd\u09a4 \u09b9\u09b8\u09cd\u09a4\u09c7 \u09ac\u0995\u09cd\u09b0\u09be\u0995\u09be\u09b0\u09c7 \u09af\u09cb\u0997 \u0995\u09b0\u09c7 y=\\sin ^{-1} x <\/span><\/span>\u00a0\u098f\u09b0 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09be \u09b9\u09b2\u0964<\/span>\"\"<\/p>\n

\u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0\u09c7\u09b0 \u09ac\u09c8\u09b6\u09bf\u09b7\u09cd\u099f\u09cd\u09af (Chart features):<\/strong> \u09b2\u09c7\u0996\u099f\u09bf \u0985\u09ac\u09bf\u099a\u09cd\u099b\u09bf\u09a8\u09cd\u09a8 \u098f\u09ac\u0982 \u09a4\u09be \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09af\u09bc\u09c7 \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

\u09b8\u09ae\u09b8\u09cd\u09af\u09be (Problem): y=\\tan ^{-1} x <\/span><\/strong>\u00a0\u098f\u09b0 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/strong><\/h3>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8: \u09a4\u09a4\u09cd\u09a4\u09cd\u09ac:<\/strong>\u00a0<\/span> tanx <\/span> \u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf \\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)<\/span><\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \u098f\u0995-\u098f\u0995 \u098f\u09ac\u0982 \u098f\u09b0 \u09b0\u09c7\u099e\u09cd\u099c \\mathbb{R} <\/span><\/span>\u0964 \u0995\u09be\u099c\u09c7\u0987 <\/span> \\tan ^{-1} x <\/span> <\/span>\u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09a1\u09cb\u09ae\u09c7\u09a8<\/span> = \\mathbb{R} <\/span> <\/span>\u098f\u09ac\u0982 \u09b0\u09c7\u099e\u09cd\u099c \\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right) <\/span><\/span><\/p>\n

\u0989\u09aa\u0995\u09b0\u09a3: <\/strong>\u099b\u0995 \u0995\u09be\u0997\u099c, \u09aa\u09c7\u09a8\u09cd\u09b8\u09bf\u09b2, \u09b0\u09be\u09ac\u09be\u09b0, \u09b8\u09be\u09af\u09bc\u09c7\u09a8\u09cd\u099f\u09bf\u09ab\u09bf\u0995 \u0995\u09cd\u09af\u09be\u09b2\u0995\u09c1\u09b2\u09c7\u099f\u09b0\u0964<\/span><\/p>\n

\u0995\u09be\u09b0\u09cd\u09af\u09aa\u09a6\u09cd\u09a7\u09a4\u09bf (Procedure):<\/strong><\/p>\n

    \n
  1. \u0997\u09cd\u09b0\u09be\u09ab \u0995\u09be\u0997\u099c\u09c7 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac<\/span> XOX' <\/span><\/span> \u098f\u09ac\u0982 <\/span> YOY' <\/span><\/span> \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc \u0986\u0981\u0995\u09bf\u0964<\/span><\/li>\n
  2. x <\/span> \u098f\u09b0 \u0995\u09af\u09bc\u09c7\u0995\u099f\u09bf \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af <\/span><\/span> y=\\tan ^{-1} x <\/span><\/span><\/strong>\u09b9\u09a4\u09c7 y <\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09bf\u0964<\/li>\n
  3. \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4<\/span> (x,\u00a0y) <\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf \u099b\u0995 \u0995\u09be\u0997\u099c\u09c7 \u09b8\u09c1\u09ac\u09bf\u09a7\u09be\u099c\u09a8\u0995 \u09b8\u09cd\u0995\u09c7\u09b2\u09c7 \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09bf\u0964\u00a0<\/span><\/li>\n
  4. \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09c1 \u09b6\u09bf\u09b7\u09af\u09c1\u0995\u09cd\u09a4 \u09aa\u09c7\u09a8\u09cd\u09b8\u09bf\u09b2 \u09a6\u09bf\u09af\u09bc\u09c7 \u09ae\u09b8\u09c3\u09a8 \u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf \u09af\u09cb\u0997 \u0995\u09b0\u09c7 \u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf\u09b0 \u09b2\u09c7\u0996 \u0986\u0981\u0995\u09bf\u0964<\/span><\/li>\n<\/ol>\n

    \u09ab\u09b2 \u09b8\u0982\u0995\u09b2\u09a8:<\/strong><\/p>\n

    \"\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4<\/p>\n

    \u09b8\u09a4\u09b0\u09cd\u0995\u09a4\u09be<\/b>:<\/span><\/p>\n

      \n
    1. \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc\u09c7 \u09b8\u09a4\u09b0\u09cd\u0995\u09a4\u09be \u0985\u09ac\u09b2\u09ae\u09cd\u09ac\u09a8 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/li>\n
    2. \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3 \u0995\u09b0\u09c7 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b2\u09c7\u0996\u09c7\u09b0 \u09b8\u09a4\u09cd\u09af\u09a4\u09be \u09af\u09be\u099a\u09be\u0987 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/li>\n
    3. \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf \u09af\u09cb\u0997\u09c7\u09b0 \u09b8\u09ae\u09af\u09bc \u09b8\u09be\u09ac\u09a7\u09be\u09a8\u09a4\u09be \u0985\u09ac\u09b2\u09ae\u09cd\u09ac\u09a8 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/li>\n<\/ol>\n

      \"\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4<\/p>\n

      \u09ac\u09c8\u09b6\u09bf\u09b7\u09cd\u099f\u09cd\u09af:<\/b><\/p>\n

        \n
      • \u00a0 x <\/span> \u098f\u09b0 \u09b8\u0995\u09b2 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4\u0964<\/span><\/li>\n
      • \\left[0, \\frac{\\pi}{2}\\right) <\/span><\/span>\u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ae\u09be\u09a8 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u0985\u09b8\u09c0\u09ae\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09a7\u09be\u09ac\u09bf\u09a4 \u09b9\u09af\u09bc \u098f\u09ac\u0982 \\left[0, \\frac{\\pi}{2}\\right) <\/span>\u00a0\u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ae\u09be\u09a8 \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u0985\u09b8\u09c0\u09ae\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09a7\u09be\u09ac\u09bf\u09a4 \u09b9\u09af\u09bc\u0964<\/li>\n
      • x \\rightarrow \\infty <\/span> \u09b9\u09b2\u09c7 \\tan ^{-1} x \\rightarrow\\frac{\\pi}{2} <\/span><\/span> \u098f\u09ac\u0982 x \\rightarrow -\\infty <\/span><\/span>\u00a0\u09b9\u09b2\u09c7 \\tan ^{-1} x \\rightarrow-\\frac{\\pi}{2} <\/span><\/span> \u09b9\u0993\u09af\u09bc\u09be\u09af\u09bc x=\\frac{\\pi}{2} <\/span><\/span> \u098f\u09ac\u0982 x=\\frac{\\pi}{2} <\/span>\u00a0\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0985\u09b8\u09c0\u09ae\u09a4\u099f \u09b0\u09c7\u0996\u09be\u0964<\/li>\n
      • \\tan \\left(\\tan ^{-1} x\\right)= <\/span> \u09af\u0996\u09a8 -\\infty<x<\\infty <\/span><\/span>\u00a0<\/span><\/li>\n
      • \\tan \\left(\\tan ^{-1} x\\right)= <\/span> \u09af\u0996\u09a8 <\/span> \\frac{\\pi}{2}<x<\\frac{\\pi}{2} <\/span><\/span><\/li>\n
      • \\operatorname{Arctan}(x)=t <\/span> \u09ac\u09be, \\tan ^{2} x=t <\/span><\/span> \u09af\u09a6\u09bf \u098f\u09ac\u0982 \u0995\u09c7\u09ac\u09b2 \u09af\u09a6\u09bf \\frac{\\pi}{2}<t<\\frac{\\pi}{2} <\/span><\/span>\u00a0<\/span> \u098f\u09ac\u0982 -\\tan (t)=x <\/span><\/span><\/li>\n
      • \\operatorname{Arctan}(x) <\/span> \u09ac\u09be, \\tan ^{-1} x <\/span><\/span>\u00a0\u098f\u0995\u099f\u09bf \u0985\u09af\u09c1\u0997\u09cd\u09ae \u09ab\u09be\u0982\u09b6\u09a8\u0964<\/span><\/li>\n<\/ul>\n

        Note: \\operatorname{Arctan}(x) <\/span><\/span>\u00a0= \\tan ^{-1} x <\/span><\/span>\u00a0<\/span><\/p>\n

        \u09b8\u09ae\u09b8\u09cd\u09af\u09be (Problem): \u098f\u0995\u0987 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0\u09c7 f(x)=\\cos x <\/span> \u098f\u09ac\u0982 \u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ab\u09be\u0982\u09b6\u09a8 f^{-1}(x) <\/span>= \\cos ^{-\\mathbf{1}} x <\/span> <\/strong>\u098f\u09b0 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/strong><\/h3>\n

        \u09b8\u09ae\u09be\u09a7\u09be\u09a8<\/b>: <\/span>\u09a4\u09a4\u09cd\u09a4\u09cd\u09ac<\/b>:<\/span>\u00a0<\/span> y=\u00a0f(x)=\\cos x <\/span> \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09a1\u09cb\u09ae\u09c7\u09a8 = \\mathbb{R} <\/span><\/span>\u00a0\u098f\u09ac\u0982 \u09b0\u09c7\u099e\u09cd\u099c<\/span>\u00a0<\/span>=[-1,\u00a01] <\/span><\/span> \u0995\u09bf\u09a8\u09cd\u09a4\u09c1<\/span> [0, \\pi] <\/span><\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 <\/span> f(x) <\/span><\/span> \u098f\u0995-\u098f\u0995 \u098f\u09ac\u0982 \u09b8\u09be\u09b0\u09cd\u09ac\u09bf\u0995\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 f^{-1}(x) <\/span><\/span>\u00a0\u098f\u09b0 \u09a1\u09cb\u09ae\u09c7\u09a8<\/span>\u00a0<\/span>= [-1,\u00a01]\u00a0<\/span><\/span> \u098f\u09ac\u0982 \u09b0\u09c7\u099e\u09cd\u099c <\/span>= [0, \\pi] <\/span><\/span>\u0964 f(x) <\/span> \u0993 f^{-1}(x) <\/span> \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b2\u09c7\u0996 y=x <\/span> \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b8\u09ae\u0964 y=x <\/span> \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 f(x) <\/span> \u09b0\u09c7\u0996\u09be\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u09ab\u09b2\u09bf\u09a4 \u099a\u09bf\u09a4\u09cd\u09b0 \u0986\u0981\u0995\u09b2\u09c7\u00a0 f^{-1}(x) <\/span>\u098f\u09b0 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09ac\u09c7\u0964<\/p>\n

        \u0989\u09aa\u0995\u09b0\u09a3<\/b>: <\/span>\u099b\u0995 \u0995\u09be\u0997\u099c, \u09aa\u09c7\u09a8\u09cd\u09b8\u09bf\u09b2, \u09b8\u09cd\u0995\u09c7\u09b2, \u09b0\u09be\u09ac\u09be\u09b0, \u09b8\u09be\u09af\u09bc\u09c7\u09a8\u09cd\u099f\u09bf\u09ab\u09bf\u0995 \u0995\u09cd\u09af\u09be\u09b2\u0995\u09c1\u09b2\u09c7\u099f\u09b0\u0964<\/span><\/p>\n

        \u0995\u09be\u09b0\u09cd\u09af\u09aa\u09a6\u09cd\u09a7\u09a4\u09bf<\/b>:<\/span><\/p>\n

          \n
        1. \u0997\u09cd\u09b0\u09be\u09ab \u0995\u09be\u0997\u099c\u09c7 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac<\/span> XOX' <\/span><\/span> \u098f\u09ac\u0982 <\/span> YOY' <\/span><\/span> \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc \u0986\u0981\u0995\u09bf\u0964<\/span><\/li>\n
        2. y=cosx <\/span> \u098f\u09b0 \u09ab\u09be\u0982\u09b6\u09a8 \u09a5\u09c7\u0995\u09c7 [0, \\pi] <\/span><\/span>\u00a0\u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7<\/span> x <\/span><\/span> \u098f\u09b0 \u0995\u09af\u09bc\u09c7\u0995\u099f\u09bf \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af <\/span>\u00a0 y <\/span>\u098f\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09bf\u0964\u00a0<\/span><\/li>\n
        3. \u099b\u0995 \u0995\u09be\u0997\u099c\u09c7 x <\/span><\/span>-\u0985\u0995\u09cd\u09b7 \u0993 y <\/span>-\u0985\u0995\u09cd\u09b7 \u09a8\u09bf\u09b0\u09cd\u09a7\u09be\u09b0\u09a3 \u0995\u09b0\u09c7 \u09b8\u09c1\u09ac\u09bf\u09a7\u09be\u099c\u09a8\u0995 \u09b8\u09cd\u0995\u09c7\u09b2\u09c7 ( x <\/span>-\u0985\u0995\u09cd\u09b7 \u09ac\u09b0\u09be\u09ac\u09b0 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u00a05 \u09ac\u09be\u09b9\u09c1 =6 \u0993 \u00a0 y <\/span> \u0985\u0995\u09cd\u09b7 \u09ac\u09b0\u09be\u09ac\u09b0 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u00a010 \u09ac\u09be\u09b9\u09c1 =1 \u098f\u0995\u0995) \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09bf\u0964<\/li>\n
        4. \u09b8\u09b0\u09c1 \u0995\u09b0\u09c7 \u0995\u09be\u0981\u099f\u09be \u09aa\u09c7\u09a8\u09cd\u09b8\u09bf\u09b2 \u09a6\u09bf\u09af\u09bc\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf \u09b8\u0982\u09af\u09cb\u0997 \u0995\u09b0\u09c7<\/span> f(x) <\/span><\/span>= cosx <\/span> \u098f\u09b0 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf\u0964<\/span><\/li>\n
        5. y <\/span>= x <\/span><\/span> \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 f(x) <\/span><\/span>\u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0989\u09aa\u09b0\u09b8\u09cd\u09a5 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09bf\u0964<\/li>\n
        6. \u09aa\u09cd\u09b0\u09a4\u09bf\u09ab\u09b2\u09bf\u09a4 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf \u09aa\u09c7\u09a8\u09cd\u09b8\u09bf\u09b2 \u09a6\u09bf\u09af\u09bc\u09c7 \u09af\u09cb\u0997 \u0995\u09b0\u09c7 f^{-1}(x) <\/span>\u00a0\u098f\u09b0 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf\u0964<\/li>\n<\/ol>\n

          \u09ab\u09b2 \u09b8\u0982\u0995\u09b2\u09a8<\/b>:<\/span>\"\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4<\/p>\n

          \"\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4<\/p>\n

          \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0\u09c7\u09b0 \u09ac\u09c8\u09b6\u09bf\u09b7\u09cd\u099f\u09cd\u09af<\/b>:<\/span><\/p>\n

            \n
          1. \u09b2\u09c7\u0996 \u09a5\u09c7\u0995\u09c7 \u098f\u099f\u09be \u09b8\u09cd\u09aa\u09b7\u09cd\u099f \u09af\u09c7,<\/span>\u00a0<\/span> [0, \\pi] <\/span><\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 f(x) <\/span><\/span>= cosx <\/span><\/span>\u00a0\u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf \u0985\u09a8\u09a8\u09cd\u09af\u0964<\/span><\/li>\n
          2. f^{-1}(x) <\/span> \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09a1\u09cb\u09ae\u09c7\u09a8 <\/span>[-1,\u00a01]<\/span> \u098f\u09ac\u0982 \u09b0\u09c7\u099e\u09cd\u099c <\/span>= [0, \\pi] <\/span><\/span><\/li>\n
          3. cos<\/span>-1<\/span>x<\/span>=t<\/span> \u09af\u09a6\u09bf \u098f\u09ac\u0982 \u0995\u09c7\u09ac\u09b2 \u09af\u09a6\u09bf<\/span> 0 \\leq t \\leq \\pi <\/span><\/span>\u00a0\u098f\u09ac\u0982 \\cos (t) <\/span><\/span>\u00a0= (x) <\/span><\/span><\/li>\n
          4. \\cos \\left(\\cos ^{-1}(x)\\right) <\/span> = x <\/span>\u09af\u09a6\u09bf \u098f\u09ac\u0982 \u0995\u09c7\u09ac\u09b2 \u09af\u09a6\u09bf<\/span> -1\u2264x\u22641 <\/span><\/span><\/li>\n
          5. \\cos ^{-1}(\\cos (x)) <\/span>\u09af\u09a6\u09bf \u098f\u09ac\u0982 \u0995\u09c7\u09ac\u09b2 \u09af\u09a6\u09bf 0 \\leq x \\leq \\pi <\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

            \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b2\u09c7\u0996 (Graphs of inverse trigonometric functions):\u00a0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b2\u09c7\u0996 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09a4\u09c7 \u0993\u00a0 \u00a0\u098f\u09b0 \u099c\u09a8\u09cd\u09af , \u00a0\u0993 \u00a0\u098f\u09b0 \u099c\u09a8\u09cd\u09af \u098f\u09ac\u0982 \u0993 \u00a0\u098f\u09b0 \u099c\u09a8\u09cd\u09af \u09ac\u09be \u098f\u09b0 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u098f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b0\u09c2\u09aa\u09c0 \u09ae\u09c2\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09af\u09bc\u0964 \u0993 \u09b8\u09ae\u09a8\u09cd\u09ac\u09af\u09bc\u09c7 \u0997\u09a0\u09bf\u09a4<\/p>\n

            Read More<\/a><\/p>\n","protected":false},"author":56,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[3026,4272,3038,50],"tags":[2567,2497,2564,2566,2565],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3606"}],"collection":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/users\/56"}],"replies":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/comments?post=3606"}],"version-history":[{"count":15,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3606\/revisions"}],"predecessor-version":[{"id":7867,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3606\/revisions\/7867"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=3606"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=3606"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=3606"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}