{"id":3595,"date":"2021-01-30T12:30:31","date_gmt":"2021-01-30T12:30:31","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3595"},"modified":"2022-03-23T19:29:12","modified_gmt":"2022-03-23T19:29:12","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%95","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%95\/","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 \u0993 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (Inverse Trigonometric Functions and Trigonometric Equations)"},"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(<\/b>Inverse trigonometric functions):<\/b><\/h2>\n

\\sin \\theta=x:(-1 \\leq x \\leq 1 \\text { \u098f\u09ac\u0982 } x \\in \\mathbb{R}) <\/span> \u09b9\u09b2\u09c7 \u0986\u09ae\u09b0\u09be \u09ac\u09c1\u099d\u09bf <\/span> \u098f\u0995\u099f\u09bf \u0995\u09cb\u09a3 \u09af\u09be\u09b0 sine <\/span> <\/span>\u098f\u09b0 \u09ae\u09be\u09a8 x <\/span> <\/span>\u098f\u09b0 \u09b8\u09ae\u09be\u09a8\u0964 \u098f \u0995\u09a5\u09be\u099f\u09bf\u0995\u09c7 \u0989\u09b2\u09cd\u099f\u09cb\u09ad\u09be\u09ac\u09c7 \\theta=\\sin ^{-1} x <\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be\u0993 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \\sin ^{-1} x <\/span> <\/span>\u09aa\u09cd\u09b0\u09a4\u09c0\u0995\u099f\u09bf \u098f\u09ae\u09a8 \u098f\u0995\u099f\u09bf \u0995\u09cb\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7 \u09af\u09be\u09b0<\/span> sine <\/span><\/span>\u00a0\u0985\u09a8\u09c1\u09aa\u09be\u09a4<\/span> x <\/span><\/span> \u098f\u09b0 \u09b8\u09ae\u09be\u09a8\u0964 \u09a4\u09be\u0987 \u09a6\u09c7\u0996\u09be \u09af\u09be\u099a\u09cd\u099b\u09c7 \u09af\u09c7, \\sin ^{-1} x <\/span><\/span>\u098f\u0995\u099f\u09bf \u0995\u09cb\u09a3\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \\sin \\theta\u00a0 <\/span><\/span> \u098f\u0995\u099f\u09bf \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982, \\sin \\theta = x <\/span>\u00a0<\/span>\u098f\u09ac\u0982 \\theta=\\sin ^{-1} x <\/span><\/span> \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09a6\u09cd\u09ac\u09af\u09bc \u09b8\u09ae\u09a4\u09c1\u09b2\u09cd\u09af\u0964 \u098f\u09a6\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09a5\u09c7\u0995\u09c7 \u0985\u09aa\u09b0\u099f\u09bf \u09b8\u09b9\u099c\u09c7\u0987 \u09aa\u09cd\u09b0\u09a4\u09bf\u09aa\u09be\u09a6\u09a8 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964 \\sin ^{-1} x <\/span> <\/span>\u0995\u09c7 \u09b8\u09be\u0987\u09a8 \u0987\u09a8\u09ad\u09be\u09b0\u09cd\u09b8 x <\/span><\/span>\u00a0( sine <\/span> inverse x <\/span>)<\/span> \u09ac\u09be \u0987\u09a8\u09ad\u09be\u09b0\u09cd\u09b8 \u09b8\u09be\u0987\u09a8 \u0985\u09ab x <\/span><\/span>\u00a0(inverse sine <\/span> of x <\/span>)<\/span> \u09aa\u09a1\u09bc\u09be \u09b9\u09af\u09bc\u0964 \\sin ^{-1} x <\/span> <\/span>\u098f\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af \u09ae\u09be\u09a8\u0995\u09c7 \u0985\u09a8\u09c7\u0995 \u09b8\u09ae\u09af\u09bc <\/span> arc\u00a0sinx <\/span>\u098f\u09ac\u0982 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09ae\u09be\u09a8\u0995\u09c7<\/span> Arc sinx <\/span><\/span> \u09b2\u09c7\u0996\u09be \u09b9\u09af\u09bc\u09c7 \u09a5\u09be\u0995\u09c7\u0964 \\cos ^{-1} x <\/span><\/span>, \\tan ^{-1} x <\/span><\/span>, \\cosec ^{-1} x <\/span><\/span>\u00a0\u09aa\u09cd\u09b0\u09ad\u09c3\u09a4\u09bf \u0995\u09cb\u09a3\u0995\u09c7 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8 \u09ac\u09be \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c0\u09af\u09bc \u09ab\u09be\u0982\u09b6\u09a8 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span>\u00a0<\/span><\/p>\n

\u09ac\u09bf.\u09a6\u09cd\u09b0.<\/b> (i) \\sin ^{-1} x <\/span> <\/span>\u098f\u09ac\u0982<\/span> (\\sin x)^{-1} <\/span><\/span> \u098f\u0995 \u09a8\u09af\u09bc\u0964 \u09aa\u09cd\u09b0\u09a5\u09ae\u099f\u09bf \u098f\u0995\u099f\u09bf \u0995\u09cb\u09a3 \u098f\u09ac\u0982 \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc\u099f\u09bf \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09b6\u09c1\u09a6\u09cd\u09a7 \u09b0\u09be\u09b6\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \\sin ^{-1} x <\/span>\u00a0\u098f\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7\u00a0(\\sin x)^{-1} <\/span> \u09ac\u09be \\frac{1}{\\sin x}<\/span>\u00a0\u00a0\u09b2\u09c7\u0996\u09be \u09af\u09be\u09ac\u09c7 \u09a8\u09be\u0964<\/p>\n

(ii) \\frac{1}{\\sin ^{2} x} <\/span><\/span>\u00a0\u0995\u09c7<\/span>\u00a0<\/span> (\\sin x)^{-2} <\/span> \u09b2\u09c7\u0996\u09be \u09af\u09be\u09ac\u09c7 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \\frac{1}{\\sin ^{2} x} <\/span>\u00a0<\/span> \u0995\u09c7<\/span>\u00a0<\/span> \\sin ^{-2} x <\/span> <\/span>\u09b2\u09c7\u0996\u09be \u09af\u09be\u09ac\u09c7 \u09a8\u09be\u0964 \u0985\u09a8\u09cd\u09af\u09be\u09a8\u09cd\u09af \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u098f\u0995\u0987 \u09a8\u09bf\u09df\u09ae \u09aa\u09cd\u09b0\u09af\u09cb\u099c\u09cd\u09af\u0964<\/span><\/p>\n

(iii)\\sin ^{-1} x <\/span> <\/span>, x <\/span><\/span>\u00a0\u098f\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4 \u09a8\u09af\u09bc\u0964 \u09af\u09c7\u09ae\u09a8<\/span> x=2 <\/span><\/span> \u09b9\u09b2\u09c7 \\sin ^{-1} x <\/span><\/span>\u00a0\u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4 \u09b9\u09af\u09bc \u09a8\u09be\u0964 \u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09c7 \\cos ^{-1} x <\/span>, \\sec ^{-1} x <\/span>, \\cosec ^{-1} x <\/span>\u00a0<\/span>\u098f\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4 \u09b9\u09af\u09bc \u09a8\u09be\u0964<\/span><\/p>\n

\u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 <\/b>(Principal Value)<\/b>: \u098f\u0995\u099f\u09bf \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09b9\u09b2 \u09b8\u09c7\u0987 \u09ae\u09be\u09a8 (\u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995\/\u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995) \u09af\u09be\u09b0 \u09b8\u09be\u0982\u0996\u09cd\u09af\u09bf\u0995 \u09ae\u09be\u09a8 \u09b8\u09ac \u09b8\u09be\u0982\u0996\u09cd\u09af\u09bf\u0995 \u09ae\u09a7\u09cd\u09af\u09c7 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae\u0964<\/span><\/p>\n

\u09ae\u09c1\u0996\u09cd\u09af \u09ae\u09be\u09a8\u0997\u09c1\u09b2\u09cb\u09b0 \u09b8\u09c0\u09ae\u09be \u09b9\u099a\u09cd\u099b\u09c7 <\/b>(Range of Principal values):<\/b><\/h3>\n
    \n
  1. \u0995\u09cb\u09a8 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be x <\/span>, -1 \\leq x \\leq <\/span> 1 <\/span>\u09b6\u09b0\u09cd\u09a4 \u09b8\u09bf\u09a6\u09cd\u09a7 \u0995\u09b0\u09b2\u09c7 \\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]<\/span><\/span>\u00a0\u09ac\u09a6\u09cd\u09a7 \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \\sin ^{-1} x <\/span> <\/span>\u00a0<\/span>\u098f\u09b0 \u09af\u09c7 \u09ae\u09be\u09a8 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u09a4\u09be\u0995\u09c7 \\sin ^{-1} x <\/span> <\/span>\u00a0\u098f\u09b0<\/span>\u00a0<\/span>\u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09ac\u09b2\u09c7\u0964 \u09af\u09c7\u09ae\u09a8: \u00a0 \\sin ^{-1}\\left(\\frac{1}{2}\\right) <\/span><\/span>\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \\frac{\\pi}{6} <\/span><\/span>\u098f\u09ac\u0982\u00a0 \u00a0 \\sin ^{-1}\\left(-\\frac{1}{2}\\right) <\/span><\/span>\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \\frac{-\\pi}{6} <\/span><\/span><\/li>\n
  2. \u0995\u09cb\u09a8 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be -1 \\leq x \\leq <\/span><\/span>\u09b6\u09b0\u09cd\u09a4 \u09b8\u09bf\u09a6\u09cd\u09a7 \u0995\u09b0\u09b2\u09c7 [0, \\pi] <\/span><\/span>,<\/span> \u09ac\u09a6\u09cd\u09a7 \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \\cos ^{-1} x <\/span><\/span>\u00a0<\/span>\u098f\u09b0 \u09af\u09c7 \u09ae\u09be\u09a8 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u09a4\u09be\u0995\u09c7 \\cos ^{-1} x <\/span><\/span>\u00a0\u098f\u09b0<\/span>\u00a0<\/span>\u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09ac\u09b2\u09c7\u0964 \u09af\u09c7\u09ae\u09a8: \\cos ^{-1}\\left(\\frac{1}{2}\\right) <\/span><\/span>\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 <\/span>3<\/span> \u098f\u09ac\u0982 \\sin ^{-1}\\left(-\\frac{1}{2}\\right) <\/span><\/span>\u00a0\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 <\/span>2<\/span>3<\/span><\/li>\n
  3. \u0995\u09cb\u09a8 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be x <\/span><\/span>, -\\infty<x<\\infty <\/span><\/span> \u09b6\u09b0\u09cd\u09a4 \u09b8\u09bf\u09a6\u09cd\u09a7 \u0995\u09b0\u09b2\u09c7 \\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)<\/span><\/span> \u09ac\u09a6\u09cd\u09a7 \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \\tan ^{-1} x <\/span><\/span>\u098f\u09b0 \u09af\u09c7 \u09ae\u09be\u09a8 <\/span>\u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u09a4\u09be\u0995\u09c7 \\tan ^{-1} x <\/span><\/span>\u098f\u09b0<\/span>\u00a0<\/span>\u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09ac\u09b2\u09c7\u0964 \\tan ^{-1} 1 <\/span><\/span>\u00a0<\/span>\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \\frac{\\pi}{4} <\/span><\/span>\u098f\u09ac\u0982 \\tan ^{-1} (-1 )<\/span><\/span> \u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 -\\frac{\\pi}{4} <\/span><\/span><\/li>\n
  4. \u0995\u09cb\u09a8 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be x \\leq-1<\/span><\/span> or x \\leq1<\/span><\/span> \u09b6\u09b0\u09cd\u09a4 \u09b8\u09bf\u09a6\u09cd\u09a7 \u0995\u09b0\u09b2\u09c7 <\/span>\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)<\/span><\/span>\u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \\cosec ^{-1} x <\/span><\/span>\u00a0<\/span>\u098f\u09b0 \u09af\u09c7 \u09ae\u09be\u09a8 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u09a4\u09be\u0995\u09c7 \\cosec ^{-1} x <\/span><\/span>\u00a0\u098f\u09b0<\/span>\u00a0<\/span>\u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09ac\u09b2\u09c7\u0964 \\cosec ^{1} (2) <\/span><\/span>\u00a0<\/span>\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \\frac{\\pi}{3} <\/span><\/span> \u098f\u09ac\u0982 \\cosec ^{-1} (-2) <\/span><\/span>\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 -\\frac{\\pi}{3} <\/span><\/span><\/li>\n
  5. \u0995\u09cb\u09a8 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be <\/span> x \\leq-1<\/span><\/span>\u00a0or x \\leq1<\/span><\/span>\u09b6\u09b0\u09cd\u09a4 \u09b8\u09bf\u09a6\u09cd\u09a7 \u0995\u09b0\u09b2\u09c7[0, \\pi] <\/span><\/span>,<\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \\sec ^{-1} x <\/span><\/span>\u00a0<\/span>\u098f\u09b0 \u09af\u09c7 \u09ae\u09be\u09a8 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u09a4\u09be\u0995\u09c7 \\sec ^{-1} x <\/span><\/span>\u00a0\u098f\u09b0<\/span>\u00a0<\/span>\u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09ac\u09b2\u09c7\u0964 \u09af\u09c7\u09ae\u09a8: \\sec ^{-1} 2 <\/span><\/span>\u00a0<\/span>\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \\frac{\\pi}{3} <\/span><\/span> \u098f\u09ac\u0982 \\sec ^{-1} (-2) <\/span><\/span>\u00a0<\/span>\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \\frac{2\\pi}{3} <\/span><\/span><\/li>\n
  6. \u0995\u09cb\u09a8 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be <\/span> x \\leq-1<\/span><\/span>\u00a0or x \\leq1<\/span><\/span>\u09b6\u09b0\u09cd\u09a4 \u09b8\u09bf\u09a6\u09cd\u09a7 \u0995\u09b0\u09b2\u09c7\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)<\/span><\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \\cot ^{-1} x <\/span><\/span>\u00a0<\/span>\u098f\u09b0 \u09af\u09c7 \u09ae\u09be\u09a8 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u09a4\u09be\u0995\u09c7 \\cot ^{-1} x <\/span><\/span>\u098f\u09b0<\/span>\u00a0<\/span>\u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09ac\u09b2\u09c7\u0964 \u09af\u09c7\u09ae\u09a8: [<\/span>katex]\\cot ^{-1} 1 [\/katex]<\/span>\u00a0<\/span>\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 -\\frac{\\pi}{3} <\/span><\/span>\u098f\u09ac\u0982\\cot ^{-1} (-2) <\/span><\/span>\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 -\\frac{\\pi}{4} <\/span><\/span><\/li>\n<\/ol>\n

    Tips<\/b>:<\/span><\/p>\n

      \n
    1. \\ sin^{-1} x<\/span><\/span>\u00a0 \u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af \u09ae\u09be\u09a8 \u03b1<\/span><\/span><\/span><\/span><\/span> \u09b9\u09af\u09bc \u09a4\u09ac\u09c7\u00a0 \\sin ^{-1} x <\/span><\/span>\u00a0<\/span>\u098f\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09ae\u09be\u09a8 \u09b9\u09ac\u09c7 n \\pi \\pm(-1)^{n} \\alpha<\/span> \u09ae\u09a7\u09cd\u09af\u09c7 \u09af\u09c7 \u0995\u09cb\u09a3\u099f\u09bf\u09b0 \u09ae\u09be\u09a8 \u09b8\u09be\u0982\u0996\u09cd\u09af\u09bf\u0995\u09ad\u09be\u09ac\u09c7 (numerical) \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09b8\u09c7\u0987 \u0995\u09cb\u09a3\u099f\u09bf\u0995\u09c7 (\u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09ac\u09be \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae) \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c0\u09af\u09bc \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a7\u09be\u09a8 \u09ac\u09be \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 (Principle value) \u09a7\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u09af\u09a6\u09bf \u0995\u09cb\u09a8 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09c3\u09a4\u09c0\u09af\u09bc \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09ae\u09be\u09a8\u099f\u09bf \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09ac\u09be \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u0989\u09ad\u09af\u09bc\u0987 \u09b9\u09af\u09bc \u09af\u09be\u09a6\u09c7\u09b0 \u09b8\u09be\u0982\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09b8\u09ae\u09be\u09a8 \u09b8\u09c7\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09ae\u09be\u09a8\u099f\u09bf\u0995\u09c7 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09a7\u09b0\u09be \u09b9\u09af\u09bc\u0964<\/li>\n<\/ol>\n

      \u09af\u09c7\u09ae\u09a8: \\cos \\frac{\\pi}{3}=\\frac{1}{2}<\/span><\/span>\u00a0\u098f\u09ac\u0982<\/span> \\cos \\frac{-\\pi}{3}=\\frac{1}{2}<\/span><\/span>\u00a0\u09b8\u09c1\u09a4\u09b0\u09be\u0982<\/span> \\cos ^{-1} \\frac{1}{2}<\/span><\/span> \u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09b9\u09ac\u09c7 \\frac{\\pi}{3} <\/span><\/span><\/p>\n

      \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u09aa\u09cd\u09b0\u0995\u09be\u09b0 \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 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 (<\/b>Different types of inverse trigonometric functions<\/b>)<\/b><\/h3>\n

      \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09b8\u09be\u0987\u09a8 \u09ab\u09be\u0982\u09b6\u09a8 <\/b>(Inverse sine function)<\/b><\/p>\n

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

      \u099a\u09bf\u09a4\u09cd\u09b0 \u09b9\u09a4\u09c7 \u09a6\u09c7\u0996\u09be \u09af\u09be\u09af\u09bc f(x)=\\sin x <\/span> <\/span>\u09ab\u09be\u0982\u09b6\u09a8 \\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]<\/span><\/span>\u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \u098f\u0995-\u098f\u0995\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce<\/span> \\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]<\/span><\/span>\u00a0\u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7<\/span>\u00a0<\/span>x<\/span> \u098f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u099f\u09bf \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af<\/span> f(x)=\\sin x <\/span><\/span>\u098f\u09b0 \u09aa\u09c3\u09a5\u0995 \u09aa\u09c3\u09a5\u0995 \u09ae\u09be\u09a8 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc, \u09af\u09be <\/span>[-1,\u00a01]<\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u098f\u09ac\u0982<\/span>\u00a0<\/span>x<\/span> \u098f\u09b0 \u098f\u0987 \u09ae\u09be\u09a8\u0995\u09c7 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09ac\u09be \u09ae\u09c1\u0996\u09cd\u09af \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

      \u09af\u09a6\u09bf\u00a0<\/span> f:\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right] \\rightarrow[-1,1]<\/span><\/span>\u00a0\u09af\u09c7\u0996\u09be\u09a8\u09c7 f(x)=\\sin x <\/span> \u0995\u09cb\u09a8\u09cb \u09ab\u09be\u0982\u09b6\u09a8 \u09b9\u09af\u09bc \u09a4\u09be\u09b9\u09b2\u09c7f(x) <\/span>\u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ab\u09be\u0982\u09b6\u09a8 f^{-1}:[-1,1] \\rightarrow\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right] <\/span>\u09af\u09c7\u0996\u09be\u09a8\u09c7 f^{-1}(x)=\\sin ^{-1} x\\left(\\neq \\frac{1}{\\sin x}\\right) <\/span>, \u098f\u0995\u09c7 \u09b8\u09be\u0987\u09a8 \u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ab\u09be\u0982\u09b6\u09a8 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 y=\\sin ^{-1} x <\/span>\u21d4x=\\sin y <\/span><\/p>\n

      \u09af\u09c7\u09ae\u09a8: y=\\sin ^{-1} \\frac{1}{2} \\Rightarrow \\sin y=\\frac{1}{2} \\Rightarrow y=\\frac{\\pi}{6}<\/span><\/span><\/p>\n

      \u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \\sin ^{-1} x <\/span><\/span>\u2208\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]<\/span><\/span>,<\/span> \u0985\u09a4\u098f\u09ac \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09b8\u09be\u0987\u09a8 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8<\/span> \\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]<\/span><\/span>\u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u098f\u09ac\u0982 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ae\u09be\u09a8\u0997\u09c1\u09b2\u09bf \u09aa\u09cd\u09b0\u09a5\u09ae \u09ac\u09be \u09e9\u09af\u09bc \u099a\u09a4\u09c1\u09b8\u09cd\u0995\u09cb\u09a3\u09c7 \u09a5\u09be\u0995\u09ac\u09c7\u0964 [\u099a\u09bf\u09a4\u09cd\u09b0 \u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af]<\/span><\/p>\n

      \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u0995\u09cb\u09b8\u09be\u0987\u09a8 \u09ab\u09be\u0982\u09b6\u09a8 <\/b>(Inverse cosine function)<\/b><\/p>\n

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

      \u099a\u09bf\u09a4\u09cd\u09b0 \u09b9\u09a4\u09c7,[0, \\pi]<\/span><\/span>,<\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \\cos x <\/span><\/span>\u098f\u09b0 \u09ae\u09be\u09a8 \u098f\u0995-\u098f\u0995\u0964 \u098f \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7<\/span> \\cos x <\/span><\/span>\u00a0\u098f\u09b0 \u09ae\u09be\u09a8 <\/span>[-1,\u00a01]<\/span> \u098f\u09ac\u0982 \u0985\u09a8\u09a8\u09cd\u09af \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u0986\u099b\u09c7\u0964 \u09af\u09a6\u09bf g:[0, \\pi] \\rightarrow[-1,1] <\/span><\/span>\u00a0\u09af\u09c7\u0996\u09be\u09a8\u09c7<\/span> g(x)=\\cos x<\/span><\/span>\u00a0\u09b9\u09af\u09bc \u09a4\u09be\u09b9\u09b2\u09c7<\/span>\u00a0<\/span>g(x)<\/span> \u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ab\u09be\u0982\u09b6\u09a8<\/span> g^{-1}:[-1,1] \\rightarrow[0, \\pi] <\/span><\/span>,<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 g^{-1}(x)=\\cos ^{-1} x <\/span><\/span><\/p>\n

      \u0986\u09ac\u09be\u09b0, y=\\cos ^{-1} x <\/span><\/span>\u21d4x=\\text { cosy } <\/span><\/span><\/p>\n

      \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \\text { cosy } <\/span><\/span> \u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 [0, \\pi]<\/span><\/span>,<\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u098f\u09ac\u0982 \u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf\u09b0 \u09b2\u09c7\u0996 \u09b8\u09b0\u09cd\u09ac\u09a6\u09be\u0987 \u09e7\u09ae \u09ac\u09be \u09ea\u09b0\u09cd\u09a5 \u099a\u09a4\u09c1\u09b7\u09cd\u0995\u09cb\u09a3\u09c7 \u09a5\u09be\u0995\u09ac\u09c7\u0964<\/span><\/p>\n

      \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u099f\u09cd\u09af\u09be\u09a8\u099c\u09c7\u09a8\u09cd\u099f \u09ab\u09be\u0982\u09b6\u09a8 <\/b>(Inverse tangent function):<\/b><\/p>\n

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

      f(x)=\\tan x <\/span> \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ae\u09be\u09a8 \\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right) <\/span><\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \u0985\u09a8\u09a8\u09cd\u09af\u0964 \u098f\u0987 \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \\tan x <\/span><\/span>\u00a0\u098f\u09b0<\/span>\u00a0<\/span>\u09ae\u09be\u09a8 (-\u221e, \u221e) <\/span><\/span>\u0964 \u098f\u0987 \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0995\u09c7<\/span> \\tan x <\/span><\/span>\u00a0\u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

      \u09af\u09a6\u09bfh: \\rightarrow\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right) \\rightarrow \\mathbb{R} <\/span><\/span>\u00a0\u09af\u09c7\u0996\u09be\u09a8\u09c7<\/span> h(x)=\\tan x <\/span><\/span>\u0995\u09cb\u09a8\u09cb \u09ab\u09be\u0982\u09b6\u09a8 \u09b9\u09af\u09bc \u09a4\u09be\u09b9\u09b2\u09c7<\/span> h(x) <\/span><\/span> \u098f\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ab\u09be\u0982\u09b6\u09a8 h^(-1):R\u2192(-\u03c0\/2, \u03c0\/2) <\/span><\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 h^{-1}(x)=\\tan ^{-1} x <\/span><\/span>\u00a0<\/span>\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09be \u09b9\u09af\u09bc \u098f\u09ac\u0982 y=\u3016tan\u3017^(-1) x\u21d4x=tany <\/span><\/span>\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \\tan ^{-1} x <\/span><\/span> \u098f\u09b0 \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right) <\/span><\/span>\u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u098f\u09ac\u0982 \u09e7\u09ae \u0993 \u09e9\u09af\u09bc \u099a\u09a4\u09c1\u09b7\u09cd\u0995\u09cb\u09a3\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964<\/span><\/p>\n\n\n\n\n\n\n\n\n\n
      \n

      \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8<\/b><\/p>\n<\/td>\n

      		\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8<\/b><\/td>\n\u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09af\u0996\u09a8,<\/b><\/p>\n

      \u00a0<\/b>x\u22650<\/span><\/td>\n

      		\u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u09af\u0996\u09a8,\u00a0<\/b><\/p>\n

      x<0<\/span><\/td>\n<\/tr>\n

      \n

      x=\\sin( y)<\/span><\/p>\n<\/td>\n

      		y=\\sin ^{-1} x=\\arcsin (x) <\/span> \u00a0<\/span><\/td>\n0 \\leq \\sin ^{-1} x \\leq \\frac{\\pi}{2} <\/span> <\/span><\/td>\n-\\frac{\\pi}{2} \\leq \\sin ^{-1} x<0 <\/span><\/span>\u00a0<\/span><\/td>\n<\/tr>\n
      \n

      x=\\cos( y) <\/span><\/p>\n<\/td>\n

      		y=\\cos ^{-1} x=\\arccos (x)<\/span><\/td>\n0 \\leq \\cos ^{-1} x \\leq \\frac{\\pi}{2}<\/span> <\/span><\/td>\n-\\frac{\\pi}{2} \\leq \\cos ^{-1} x<0 <\/span><\/span><\/td>\n<\/tr>\n
      \n

      x=\\tan( y)<\/span> <\/span><\/i><\/p>\n<\/td>\n

      		y=\\tan ^{-1} x=\\arctan (x) <\/span><\/td>\n0 \\leq \\tan ^{-1} x \\leq \\frac{\\pi}{2} <\/span> <\/span><\/td>\n\n

      -\\frac{\\pi}{2} \\leq \\tan ^{-1} x<0 <\/span> <\/span><\/p>\n<\/td>\n<\/tr>\n

      x=\\cot( y) <\/span><\/td>\n y=\\cot ^{-1} x=\\operatorname{arccot}(x)<\/span><\/td>\n0 \\leq \\cot ^{-1} x \\leq \\frac{\\pi}{2} <\/span><\/span><\/td>\n\n

      -\\frac{\\pi}{2} \\leq \\cot ^{-1} x<0 <\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

      x=\\sec( y)<\/span><\/td>\ny=\\sec ^{-1} x=\\operatorname{arcsec}(x) <\/span><\/td>\n0 \\leq \\sec ^{-1} x \\leq \\frac{\\pi}{2} <\/span> <\/span><\/td>\n\n

      -\\frac{\\pi}{2} \\leq \\sec ^{-1} x<0 <\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

      x=\\cosec( y) <\/span> <\/span><\/i><\/td>\ny=\\operatorname{cosec}^{-1} x=\\operatorname{arccosec}(x) <\/span><\/td>\n0 \\leq \\cosec\u00a0 ^{-1} x \\leq \\frac{\\pi}{2} <\/span><\/span><\/td>\n\n

      -\\frac{\\pi}{2} \\leq \\cosec ^{-1} x<0 <\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      \u09ae\u09c1\u0996\u09cd\u09af \u09b8\u09c0\u09ae\u09be\u09af\u09bc \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u0997\u09c1\u09b2\u09bf\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 (<\/b>Relation of inverse trigonometric functions in the principal range):<\/b><\/h3>\n

      (i) \\sin ^{-1} x=\\operatorname{cosec}^{-1} \\frac{1}{x} <\/span><\/span><\/b>;-1 \\leq x \\leq 1<\/span><\/span><\/p>\n

      (ii) \\operatorname{cosec}^{-1} x=\\sin ^{-1} \\frac{1}{x} <\/span><\/span><\/b>;x \\leq-1 <\/span><\/span>\u00a0\u0985\u09a5\u09ac\u09be <\/b>x\u22651<\/span><\/p>\n

      (iii) \\cos ^{-\\mathbf{1}} x=\\sec ^{-\\mathbf{1}} \\frac{1}{x}<\/span><\/span><\/b>;-1 \\leq x \\leq 1 <\/span><\/span>\u00a0(iv) \\sec ^{-\\mathbf{1}} x=\\cos ^{-1} \\frac{1}{x} <\/span><\/span><\/b>;x\u2264-1<\/span> \u0985\u09a5\u09ac\u09be <\/b>x\u22651<\/span><\/p>\n

      (v) \\tan ^{-1} x=\\cot ^{-1} \\frac{1}{x} <\/span><\/span><\/b>\u00a0(vi) \\cot ^{-1} x=\\tan ^{-1} \\frac{1}{x} <\/span><\/span><\/b><\/p>\n

      \u09aa\u09cd\u09b0\u09ae\u09be\u09a3:<\/b> (i) \u09a7\u09b0\u09bf, \\sin \\theta <\/span><\/span>=x <\/span><\/span> \u2234\\theta <\/span>=\\sin ^{-1} x <\/span><\/span><\/p>\n

      \u0986\u09ac\u09be\u09b0, \\sin \\theta <\/span><\/span>=x <\/span>\u21d2\\frac{1}{\\operatorname{cosec} \\theta} <\/span><\/span>=x <\/span>\u21d2\\operatorname{cosec} \\theta <\/span>=\\frac{1}{x} <\/span><\/span>\u21d2 \\theta<\/span>= \\operatorname{cosec}^{-1} \\frac{1}{x}<\/span><\/span><\/p>\n

      \u2234\\sin ^{-1} x <\/span><\/span>= \\operatorname{cosec}^{-1} \\frac{1}{x}<\/span><\/span><\/p>\n

      (ii) \u09a7\u09b0\u09bf, \\operatorname{cosec} \\theta <\/span><\/span>=x <\/span> <\/span>\u2234 \\theta <\/span>= \\operatorname{cosec}^{-1} \\frac{1}{x}<\/span><\/span><\/p>\n

      \u0986\u09ac\u09be\u09b0, \\operatorname{cosec} \\theta <\/span><\/span>=x <\/span> \u21d2\\frac{1}{\\sin \\theta} <\/span><\/span>=x <\/span>\u21d2\\sin \\theta <\/span>=\\frac{1}{x} <\/span><\/span>\u21d2 \\theta <\/span>=\\sin ^{-1} \\frac{1}{x} <\/span><\/span>\u00a0\u2234\\operatorname{cosec} \\theta <\/span><\/span>= \\sin ^{-1} \\frac{1}{x}<\/span><\/span><\/p>\n

      \u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7 \u0985\u09a8\u09cd\u09af\u09be\u09a8\u09cd\u09af \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u0997\u09c1\u09b2\u09bf \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964<\/span>\u00a0<\/span><\/p>\n

      (i) <\/b>\\sin \\left(\\sin ^{-1} x\\right) <\/span>=x <\/span><\/span>\u00a0(ii) <\/b>\\cos \\left(\\cos ^{-1} x\\right) <\/span>=x <\/span><\/span><\/p>\n

      (iii) <\/b>\\tan \\left(\\tan ^{-1} x\\right) <\/span>=x <\/span><\/span>\u00a0(iv) <\/b>\\cot \\left(\\cot ^{-1} x\\right) <\/span>=x <\/span><\/span><\/p>\n

      (v)<\/b> \\sec \\left(\\sec ^{-1} x\\right) <\/span>=x <\/span><\/span>\u00a0(vi) <\/b>\\operatorname{cosec}\\left(\\operatorname{cosec}^{-1} x\\right) <\/span>=x <\/span><\/span><\/p>\n

      \u09aa\u09cd\u09b0\u09ae\u09be\u09a3:<\/b> (ii) \u09a7\u09b0\u09bf, \\cos \\theta <\/span><\/span>=x <\/span><\/span>\u2234 \\theta <\/span>=\\cos ^{-1} x <\/span><\/span><\/p>\n

      \u098f\u0996\u09a8 <\/span> \u03b8<\/span><\/span><\/span><\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \\cos \\theta <\/span><\/span>=x <\/span><\/span> \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 \u09ac\u09b8\u09bf\u09af\u09bc\u09c7 \u09aa\u09be\u0987, \\cos \\left(\\cos ^{-1} x\\right) <\/span><\/span>=x <\/span><\/span><\/p>\n

      \u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7 \u0985\u09a8\u09cd\u09af\u09be\u09a8\u09cd\u09af \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u0997\u09c1\u09b2\u09bf \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964<\/span>\u00a0<\/span><\/p>\n

      (i) <\/b> \\sin ^{-1}(\\sin x)<\/span>=x <\/span>\u00a0(ii) \\cos ^{-1}(\\cos x) <\/span>=x <\/span><\/strong><\/p>\n

      (iii) \\tan ^{-1}(\\tan x) <\/span>=x <\/span>\u00a0(iv) \\cot ^{-1}(\\cot x) <\/span>=x <\/span><\/strong><\/p>\n

      (v) \\sec c^{-1}(\\sec x) <\/span>=x <\/span>\u00a0(iv) \\operatorname{cosec}^{-1}(\\operatorname{cosec} x)<\/span>=x <\/span><\/strong><\/p>\n

      \u09aa\u09cd\u09b0\u09ae\u09be\u09a3:<\/b> (iii) \u09a7\u09b0\u09bf, <\/span>tanx=\u03b8<\/span> \u2234x <\/span>= \\tan ^{-1} \\theta <\/span><\/span><\/p>\n

      \u098f\u0996\u09a8,x <\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a8 <\/span>\u00a0tanx <\/span>=\u03b8<\/span> \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 \u09ac\u09b8\u09bf\u09af\u09bc\u09c7 \u09aa\u09be\u0987, <\/span>tan<\/span>-1<\/span>(tanx)=x <\/span><\/span><\/p>\n

      \u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7 \u0985\u09a8\u09cd\u09af\u09be\u09a8\u09cd\u09af \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u0997\u09c1\u09b2\u09bf \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964<\/span>\u00a0<\/span><\/p>\n

      \\sin ^{-1} x <\/span>=\\operatorname{cosec}^{-1} \\frac{1}{x} <\/span><\/span>=\\cos ^{-1} \\sqrt{1-x^{2}} <\/span><\/span>=\\sec ^{-1} \\frac{1}{\\sqrt{1-x^{2}}} <\/span><\/span>=\\cot ^{-1} \\frac{\\sqrt{1-x^{2}}}{x} <\/span><\/span>= \\tan ^{-1} \\frac{x}{\\sqrt{1-x^{2}}}<\/span><\/span><\/p>\n

      \u09aa\u09cd\u09b0\u09ae\u09be\u09a3:<\/b> \u09a7\u09b0\u09bf, \\sin ^{-1} x <\/span><\/span>=tanx <\/span><\/span>\u00a0\u2234\\sin \\theta <\/span>=x <\/span><\/span><\/p>\n

      \u098f\u0996\u09a8,<\/span><\/p>\n

      \\cos \\theta <\/span>= \\sqrt{1-\\sin ^{2} \\theta}<\/span><\/span>=\\sqrt{1-x^{2}} \\Rightarrow \\theta <\/span><\/span>\u21d2\\theta <\/span>=\\cos ^{-1} \\sqrt{1-x^{2}}\u00a0<\/span><\/span>\u0985\u09b0\u09cd\u09a5\u09be\u09ce \\sin ^{-1} x<\/span><\/span>= \\cos ^{-1} \\sqrt{1-x^{2}}<\/span><\/span>\u2026(i)<\/span><\/p>\n

      \u0986\u09ac\u09be\u09b0, \\tan \\theta <\/span><\/span>= \\frac{\\sin \\theta}{\\cos \\theta}<\/span><\/span>= <\/span><\/span>\u21d2\\theta <\/span>=\\tan ^{-1} \\frac{x}{\\sqrt{1-x^{2}}} <\/span><\/span>\u0985\u09b0\u09cd\u09a5\u09be\u09ce \\sin ^{-1} x <\/span><\/span>= \\tan ^{-1} \\frac{x}{\\sqrt{1-x^{2}}}<\/span><\/span>\u2026(ii)<\/span><\/p>\n

      \\tan \\theta <\/span>= \\frac{\\cos \\theta}{\\sin \\theta}<\/span><\/span>=\\frac{\\sqrt{1-x^{2}}}{x} <\/span><\/span>\u21d2\\theta\u00a0<\/span>= \\cot ^{-1} \\frac{\\sqrt{1-x^{2}}}{x}<\/span><\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce \\sin ^{-1} x <\/span><\/span>= \\cot ^{-1} \\frac{\\sqrt{1-x^{2}}}{x} \\ldots \\text { ( }<\/span><\/span>\u2026(iii)<\/span><\/p>\n

      \\sin \\theta <\/span>=x <\/span>\u21d2\\frac{1}{\\operatorname{cosec} \\theta} <\/span><\/span>=x <\/span>\u21d2 \\operatorname{cosec} \\theta<\/span>=\\frac{1}{x} <\/span><\/span>\u21d2\\theta <\/span>=\\operatorname{cosec}^{-1} \\frac{1}{x} <\/span><\/span>\u0985\u09b0\u09cd\u09a5\u09be\u09ce \\sin ^{-1} x <\/span><\/span>=\\operatorname{cosec}^{-1} \\frac{1}{x} <\/span><\/span>\u2026(iv)<\/span>\u00a0<\/span><\/p>\n

      \\sec \\theta <\/span>=\\frac{1}{\\cos \\theta} <\/span><\/span>=\\frac{1}{\\sqrt{1-x^{2}}} <\/span><\/span>\u21d2 \\theta<\/span>= \\frac{1}{x}<\/span><\/span>\u21d2\\theta <\/span>=\\sec ^{-1} \\frac{1}{\\sqrt{1-x^{2}}} <\/span><\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce \\sin ^{-1} x<\/span><\/span>= \\sec ^{-1} \\frac{1}{\\sqrt{1-x^{2}}}<\/span><\/span>\u2026(v)<\/span>\u00a0<\/span><\/p>\n

      (i)<\/span>, (<\/span>ii)<\/span>, (<\/span>iii)<\/span>, (<\/span>iv)<\/span> \u0993 <\/span>(v)<\/span> \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

      \\sin ^{-1} x <\/span>= \\operatorname{cosec}^{-1} \\frac{1}{x}<\/span><\/span>=\\cos ^{-1} \\sqrt{1-x^{2}} <\/span><\/span>= \\sec ^{-1} \\frac{1}{\\sqrt{1-x^{2}}}<\/span><\/span>= \\cot ^{-1} \\frac{\\sqrt{1-x^{2}}}{x}<\/span><\/span>=\\tan ^{-1} \\frac{x}{\\sqrt{1-x^{2}}} <\/span><\/span>\u00a0<\/span><\/p>\n

      \u099c\u09cd\u09af\u09be\u09ae\u09bf\u09a4\u09bf\u0995 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf <\/b>(Geometric method)<\/b>:<\/b> \u099c\u09cd\u09af\u09be\u09ae\u09bf\u09a4\u09bf\u0995 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09af\u09c7 \u0995\u09cb\u09a8 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u0995\u09c7 \u0985\u09aa\u09b0 \u09af\u09c7 \u0995\u09cb\u09a8 \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\u09c2\u09aa\u09be\u09a8\u09cd\u09a4\u09b0 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964<\/span><\/h4>\n

      \u09a7\u09b0\u09bf, \\sin ^{-1} x <\/span><\/span>= \\theta<\/span><\/span> \u09a4\u09be\u09b9\u09b2\u09c7 \\sin \\theta <\/span><\/span>=x <\/span>= \\frac{x}{1}<\/span><\/span><\/p>\n

      \u098f\u0995\u099f\u09bf \u09b8\u09ae\u0995\u09cb\u09a3\u09c0 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c <\/span>ABC<\/span>\u00a0<\/span>\u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf, \u09af\u09be\u09b0 <\/span>\u2220BAC= \\theta<\/span>, BC<\/span>\u00a0<\/span>(\u09b2\u09ae\u09cd\u09ac) <\/span>=x <\/span> AC<\/span>\u00a0<\/span>(\u0985\u09a4\u09bf\u09ad\u09c1\u099c) <\/span>=1<\/span><\/p>\n

      \"\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4-\u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995-Geometric<\/p>\n

      \u2234AB<\/span> (\u09ad\u09c2\u09ae\u09bf) <\/span>=\\sqrt{A C^{2}-B C^{2}} <\/span><\/span>=\\sqrt{1-x^{2}} <\/span><\/span>, \\theta <\/span>=<\/span>\\frac{\\text { \u09ad\u09c2\u09ae\u09bf }}{\\text { \u0985\u09a4\u09bf\u09ad\u09c1\u099c }} <\/span><\/span><\/p>\n

      \u2234\\theta <\/span>=\\cos ^{-1}\\left(\\frac{\\text { \u09ad\u09c2\u09ae\u09bf }}{\\text { \u0985\u09a4\u09bf\u09ad\u09c1\u099c }}\\right)<\/span><\/span>\u00a0<\/span>=\\cos ^{-1}\\left(\\frac{\\sqrt{1-x^{2}}}{1}\\right) <\/span><\/span>=\\cos ^{-1} \\sqrt{1-x^{2}} <\/span><\/span>=\\sec ^{-1} \\frac{1}{\\sqrt{1-x^{2}}} <\/span><\/span>\u00a0<\/span><\/p>\n

      \u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7,<\/span> \\theta <\/span><\/span>=\\tan ^{-1}\\left(\\frac{\\text { \u09b2\u09ae\u09cd\u09ac }}{\\text { \u09ad\u09c2\u09ae\u09bf}}\\right) <\/span><\/span>=\\tan ^{-1} \\frac{x}{\\sqrt{1-x^{2}}} <\/span><\/span>=\\cot ^{-1} \\frac{\\sqrt{1-x^{2}}}{x} <\/span><\/span><\/p>\n

      \u2234 \\sin ^{-1} x<\/span><\/span>=\\operatorname{cosec}^{-1} \\frac{1}{x} <\/span><\/span>=\\cos ^{-1} \\sqrt{1-x^{2}} <\/span><\/span>=\\sec ^{-1} \\frac{1}{\\sqrt{1-x^{2}}} <\/span><\/span>=\\cot ^{-1} \\frac{\\sqrt{1-x^{2}}}{x} <\/span><\/span>=\\tan ^{-1} \\frac{x}{\\sqrt{1-x^{2}}} <\/span><\/span>\u00a0<\/span><\/p>\n

      [<\/span>\u09ac\u09bf.\u09a6\u09cd\u09b0.<\/b> \\sin ^{-1} \\frac{x \\text { (\u09b2\u09ae\u09cd\u09ac) }}{y \\text { (\u09ad\u09c2\u09ae\u09bf) }} <\/span><\/span>=\\tan ^{-1} \\frac{\\text { \u09b2\u09ae\u09cd\u09ac }(x)}{\\text { \u09ad\u09c2\u09ae\u09bf }\\left(\\sqrt{y^{2}-x^{2}}\\right)} <\/span><\/span>,\\tan ^{-1} \\frac{x}{y} <\/span><\/span>=\\sin ^{-1} \\frac{x}{\\sqrt{x^{2}+y^{2}}} <\/span><\/span><\/p>\n

      \\cos ^{-1} \\frac{x \\text { (\u09b2\u09ae\u09cd\u09ac) }}{y \\text { (\u09ad\u09c2\u09ae\u09bf) }} <\/span>=\\tan ^{-1} \\frac{\\text { \u09b2\u09ae\u09cd\u09ac }(\\sqrt{y^{2}-x^{2}})}{\\text { \u09ad\u09c2\u09ae\u09bf }\\left(x\\right)} <\/span><\/span>,\\tan ^{-1} \\frac{x}{y} <\/span><\/span>=\\cos ^{-1} \\frac{y}{\\sqrt{x^{2}+y^{2}}} <\/span><\/span><\/p>\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(Inverse trigonometric functions): \u09b9\u09b2\u09c7 \u0986\u09ae\u09b0\u09be \u09ac\u09c1\u099d\u09bf \u098f\u0995\u099f\u09bf \u0995\u09cb\u09a3 \u09af\u09be\u09b0 \u098f\u09b0 \u09ae\u09be\u09a8 \u098f\u09b0 \u09b8\u09ae\u09be\u09a8\u0964 \u098f \u0995\u09a5\u09be\u099f\u09bf\u0995\u09c7 \u0989\u09b2\u09cd\u099f\u09cb\u09ad\u09be\u09ac\u09c7 \u09a6\u09cd\u09ac\u09be\u09b0\u09be\u0993 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09aa\u09cd\u09b0\u09a4\u09c0\u0995\u099f\u09bf \u098f\u09ae\u09a8 \u098f\u0995\u099f\u09bf \u0995\u09cb\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7 \u09af\u09be\u09b0 \u00a0\u0985\u09a8\u09c1\u09aa\u09be\u09a4 \u098f\u09b0 \u09b8\u09ae\u09be\u09a8\u0964 \u09a4\u09be\u0987 \u09a6\u09c7\u0996\u09be \u09af\u09be\u099a\u09cd\u099b\u09c7 \u09af\u09c7, \u098f\u0995\u099f\u09bf \u0995\u09cb\u09a3\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1<\/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":[167,2500,2503,2499,2498,2501,2502,2497],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3595"}],"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=3595"}],"version-history":[{"count":23,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3595\/revisions"}],"predecessor-version":[{"id":7865,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3595\/revisions\/7865"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=3595"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=3595"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=3595"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}