{"id":3623,"date":"2021-01-30T12:26:12","date_gmt":"2021-01-30T12:26:12","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3623"},"modified":"2022-04-17T12:02:23","modified_gmt":"2022-04-17T06:02:23","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-%e0%a6%a1%e0%a7%8b%e0%a6%ae%e0%a7%87","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-%e0%a6%a1%e0%a7%8b%e0%a6%ae%e0%a7%87\/","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 \u09a1\u09cb\u09ae\u09c7\u09a8 \u0993 \u09b0\u09c7\u099e\u09cd\u099c\u09c7\u09b0 \u099b\u0995 | Table of domains and ranges of inverse trigonometric functions"},"content":{"rendered":"\r\n\r\n\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u09ab\u09be\u0982\u09b6\u09a8<\/b><\/td>\r\n\u09a1\u09cb\u09ae\u09c7\u09a8<\/b><\/td>\r\n\u09b0\u09c7\u099e\u09cd\u099c<\/b><\/td>\r\n<\/tr>\r\n
\r\n

\\sin ^{-1} x <\/span> <\/span><\/p>\r\n<\/td>\r\n

 [-1,\u00a01] <\/span><\/span><\/td>\r\n {\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]} <\/span> <\/span><\/td>\r\n<\/tr>\r\n
\r\n

\\cos ^{-1}x <\/span><\/p>\r\n<\/td>\r\n

\u00a0 [-1,\u00a01]<\/span><\/span><\/td>\r\n {[0, \\pi]} <\/span> <\/span><\/td>\r\n<\/tr>\r\n
\\tan ^{-1}x <\/span><\/td>\r\n (-\\infty, \\infty) <\/span> \u09ac\u09be, \\mathbb{R} <\/span><\/span><\/td>\r\n\r\n

\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right) <\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n

\\cot ^{-1} x<\/span><\/td>\r\n (-\\infty, \\infty) <\/span> \u09ac\u09be, \\mathbb{R} <\/span><\/span><\/td>\r\n\r\n

(0, \\pi) <\/span> <\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n

\\sec ^{-1} x <\/span><\/td>\r\n (-\\infty, -1) <\/span> \\mathbf{U} <\/span> [1,+\\infty) <\/span>\u00a0\u09ac\u09be,<\/span>\r\n

\u00a0 \\mathbb{R} <\/span>– (-1,1)<\/span><\/span><\/p>\r\n<\/td>\r\n

\r\n

\\left[0, \\frac{\\pi}{2}\\right) <\/span> \\mathbf{U} <\/span> \\left[0, \\frac{3\\pi}{2}\\right) <\/span><\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n

\\operatorname{cosec}^{-1} x <\/span> <\/span><\/td>\r\n (-\\infty, -1) <\/span> \\mathbf{U} <\/span> [1,+\\infty) <\/span>\u00a0\u09ac\u09be,<\/span>\r\n

\\mathbb{R} <\/span>– (-1,1)<\/span><\/span><\/p>\r\n<\/td>\r\n

\r\n

\\left(0, \\frac{\\pi}{2}\\right] <\/span> \\mathbf{U} <\/span> \\left(0, \\frac{3\\pi}{2}\\right] <\/span> <\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n\r\n\r\n\r\n

\\sec ^{-1} x <\/span> \u0993 \\operatorname{cosec}^{-1} x <\/span><\/span> \u098f\u09b0 \u09b0\u09c7\u099e\u09cd\u099c \u09a8\u09bf\u09af\u09bc\u09c7 \u09a6\u09cd\u09ac\u09bf\u09ae\u09a4 \u0986\u099b\u09c7\u0964 \u0995\u09cb\u09a8\u09cb \u0995\u09cb\u09a8\u09cb \u0997\u09a3\u09bf\u09a4\u09ac\u09bf\u09a6 \u09ae\u09a8\u09c7 \u0995\u09b0\u09c7\u09a8 \\sec ^{-1} x <\/span><\/span> \u098f\u09b0 \u09b0\u09c7\u099e\u09cd\u099c \\left[0, \\frac{\\pi}{2}\\right) <\/span>\\mathbb{R} <\/span> \\left(\\frac{\\pi}{2}, \\pi\\right]<\/span><\/span>\u098f\u09ac\u0982 \\operatorname{cosec}^{-1} x <\/span><\/span>\u00a0\u098f\u09b0 \u09b0\u09c7\u099e\u09cd\u099c x <\/span> \\mathbb{R} <\/span> \\left(0, \\frac{\\pi}{2}\\right] <\/span><\/span><\/p>\r\n\r\n\r\n\r\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 \u0995\u09af\u09bc\u09c7\u0995\u099f\u09bf \u09b8\u09c2\u09a4\u09cd\u09b0 (<\/b>Some rules of inverse trigonometric functions):<\/b><\/h2>\r\n\r\n\r\n\r\n

(i) \\sin ^{-1} x <\/span><\/span><\/b>+<\/span>cos<\/span>-1<\/span>x= \\frac{\\pi}{2}<\/span>; [-1 \\leq x \\leq 1]<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

(ii) <\/b> \\tan ^{-1}x <\/span>+ \\cot ^{-1} x<\/span>= \\frac{\\pi}{2}<\/span>;[ x\u22650]<\/span><\/p>\r\n\r\n\r\n\r\n

(iii) \\operatorname{cosec}^{-1} x <\/span><\/span><\/b>+\\sec ^{-1} x <\/span><\/span>= \\frac{\\pi}{2}<\/span><\/span>;[ x\u2264-1 <\/span>, x\u22651 <\/span>]<\/span>\u00a0<\/b><\/p>\r\n\r\n\r\n\r\n

\u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b>:<\/span><\/h3>\r\n

(i) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\sin ^{-1} x <\/span>= \\theta <\/span><\/span>\u00a0\u2234 \\sin \\theta<\/span>= x<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0996\u09a8, \\cos \\left(\\frac{\\pi}{2}-\\theta\\right) <\/span><\/span>= \\sin \\theta<\/span><\/span>\u00a0\u21d2\\cos \\left(\\frac{\\pi}{2}-\\theta\\right) <\/span><\/span>= x<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\\cos ^{-1} x <\/span>=\\frac{\\pi}{2} <\/span>-\\theta <\/span><\/span>\u21d2\\theta <\/span>+\\cos ^{-1} x <\/span><\/span>=\\frac{\\pi}{2} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u2234 \\sin ^{-1} x <\/span><\/span>+\\cos ^{-1} x <\/span><\/span>=\\frac{\\pi}{2} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

(ii) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\tan ^{-1} x<\/span>=\u00a0\\theta <\/span><\/span>\u00a0\u2234\\tan \\theta <\/span>=x <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0996\u09a8, \\cot \\left(\\frac{\\pi}{2}-\\theta\\right)\u00a0<\/span><\/span>=\\tan \\theta <\/span><\/span>\u00a0\u21d2\\cot \\left(\\frac{\\pi}{2}-\\theta\\right) <\/span><\/span>=x <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\\cot ^{-1} x <\/span>=\\frac{\\pi}{2} <\/span>-\\theta <\/span><\/span>\u00a0\u21d2\\theta <\/span>+\\cot ^{-1} x <\/span>\u00a0<\/span>=\\frac{\\pi}{2} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u2234 \\tan ^{-1} x<\/span><\/span>+\\cot ^{-1} x <\/span><\/span>=\\frac{\\pi}{2} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

(iii) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\sec ^{-1} x <\/span>=\\theta <\/span><\/span>\u00a0\u2234\\sec \\theta <\/span>=x <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0996\u09a8, \\operatorname{cosec}\\left(\\frac{\\pi}{2}-\\theta\\right) <\/span><\/span>=\\sec \\theta <\/span><\/span>\u21d2\\operatorname{cosec}\\left(\\frac{\\pi}{2}-\\theta\\right) <\/span><\/span>=x <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\\operatorname{cosec}^{-1} x <\/span>=\\frac{\\pi}{2} <\/span><\/span>-\\theta <\/span><\/span>\u00a0\u21d2\\theta <\/span>+<\/span>cosec<\/span>-1<\/span>x=\\frac{\\pi}{2} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u2234\\sec ^{-1} x <\/span><\/span>+ \\operatorname{cosec}^{-1} x<\/span><\/span>=\\frac{\\pi}{2} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

(i) \\tan ^{-1} x<\/span><\/span><\/b>+ \\tan ^{-1} y<\/span><\/span>=\\tan ^{-1} \\frac{x+y}{1-x y} <\/span><\/span>; [xy\u22641] <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

(ii) \\tan ^{-1} y <\/span><\/span> -<\/span> \\tan ^{-1} y<\/span><\/span><\/b>=\\tan ^{-1} \\frac{x-y}{1+x y} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

[\u09ac\u09bf.\u09a6\u09cd\u09b0. \u09af\u09a6\u09bf \u0995\u09cb\u09a8\u09cb \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \\tan ^{-1} x<\/span><\/span> \u098f\u09ac\u0982 \\tan ^{-1} y<\/span><\/span> \u0989\u09ad\u09af\u09bc\u09c7\u0987 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u098f\u09ac\u0982 <\/span> 1-xy<\/span><\/span> \u09ac\u09be <\/span>\\frac{x+y}{1-x y} <\/span>\u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09af\u09bc\u0964 \u09a4\u0996\u09a8 \u09a1\u09be\u09a8\u09aa\u0995\u09cd\u09b7\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09ae\u09c1\u0996\u09cd\u09af\u09ae\u09be\u09a8 \u0997\u09cd\u09b0\u09b9\u09a3\u09af\u09cb\u0997\u09cd\u09af \u09a8\u09af\u09bc\u0964 \u09ac\u09bf\u09b7\u09af\u09bc\u099f\u09bf \u09aa\u09c2\u09b0\u09cd\u09ac\u09c7\u0987 \u0989\u09a6\u09be\u09b9\u09b0\u09a3\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09ac\u09cd\u09af\u09be\u0996\u09cd\u09af\u09be \u0995\u09b0\u09be \u09b9\u09af\u09bc\u09c7\u099b\u09c7\u0964]<\/span><\/p>\r\n\r\n\r\n\r\n

(iii) \\tan ^{-1} x<\/span><\/span><\/b>+ \\tan ^{-1} y<\/span>+ \\tan ^{-1} z<\/span>=\\tan ^{-1} \\frac{x+y+z-x y z}{1-y z-z x-x y} <\/span><\/span>;[xy+yz+zx\u22641] <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b>:<\/span><\/h3>\r\n

(i) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span> \\tan ^{-1} x<\/span><\/span>=<\/span>\u00a0 A<\/span>\u098f\u09ac\u0982 \\tan ^{-1} y<\/span><\/span>=B <\/span><\/span> \u2234tanA <\/span>=x <\/span><\/span> \u098f\u09ac\u0982 tanB<\/span><\/span>=y <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0996\u09a8, tan(A+B) <\/span><\/span>\u00a0=\\frac{\\tan A+\\tan B}{1-\\tan A \\tan B}<\/span><\/span>=\\frac{x+y}{1-x y} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u21d2 A+B<\/span>=\\tan ^{-1} \\frac{x+y}{1-x y} <\/span><\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u2234 \\tan ^{-1} x<\/span><\/span>+ \\tan ^{-1} y<\/span><\/span>=\\tan ^{-1} \\frac{x+y}{1-x y} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

(ii) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\tan ^{-1} x<\/span><\/span>=<\/span>\u00a0 A<\/span>\u098f\u09ac\u0982 \\tan ^{-1} y<\/span><\/span>=B <\/span><\/span> \u2234tanA <\/span>=x <\/span><\/span> \u098f\u09ac\u0982 tanB<\/span><\/span>=y <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0996\u09a8,tan(A-B) <\/span> =\\frac{\\tan A-\\tan B}{1+\\tan A \\tan B} <\/span>=\\frac{x-y}{1+x y} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u21d2 A-B<\/span>=\\tan ^{-1} \\frac{x-y}{1+x y} <\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u2234 \\tan ^{-1} x<\/span>- \\tan ^{-1} y<\/span>=\\tan ^{-1} \\frac{x-y}{1+x y} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

(iii) \\tan ^{-1} x<\/span><\/span>+ \\tan ^{-1} y<\/span><\/span>+ \\tan ^{-1} z<\/span><\/span>=\\tan ^{-1} \\frac{x+y}{1-x y} <\/span><\/span>\u00a0[(i) \u098f\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7]<\/span><\/p>\r\n\r\n\r\n\r\n

\u21d2\\tan ^{-1}\\left\\{\\frac{\\frac{x+y}{1-x y}+z}{1-\\left(\\frac{x+y}{1-x y}\\right) z}\\right\\} <\/span>=\\tan ^{-1} \\frac{(x+y) z+z(1-x y)}{1-x y-(x+y) z} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

=\\tan ^{-1} \\frac{x+y+z-x y z}{1-y z-z x-x y} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u2234 \\tan ^{-1} x<\/span>+ \\tan ^{-1} y<\/span>+ \\tan ^{-1} z<\/span><\/span>=\\tan ^{-1} \\frac{x+y+z-x y z}{1-y z-z x-x y} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u2234 \\tan ^{-1} x<\/span>+ \\tan ^{-1} y<\/span>+ \\tan ^{-1} z<\/span>=\\tan ^{-1} \\frac{x+y+z-x y z}{1-y z-z x-x y} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \r\n
  • \u099c\u09c7\u09a8\u09c7 \u09b0\u09be\u0996\u09be \u09ad\u09be\u09b2\u09cb: \\tan ^{-1} x<\/span>+ \\tan ^{-1} y<\/span><\/span>=<\/span><\/li>\r\n<\/ul>\r\n\r\n\r\n \\left\\{\\begin{array}{c} \\tan ^{-1} \\frac{x+y}{1-x y} \\text {\u09af\u0996\u09a8 } x y<1 \\pi+\\tan ^{-1} \\frac{x+y}{1-x y} \\text { \u09af\u0996\u09a8 } x y>1 \\\\ \\frac{\\pi}{2} \\text { \u09af\u0996\u09a8 } x y=1 \\end{array}\\right.<\/span>\n\r\n\r\n\r\n

    (i) <\/b>\\sin ^{-1} x <\/span>+ <\/b>\\sin ^{-1} y <\/span><\/span>=\\sin ^{-1}\\left\\{x \\sqrt{1-y^{2}}+y \\sqrt{1-x^{2}}\\right\\} <\/span><\/span>[\u09af\u0996\u09a8 x^{2}+y^{2} \\leq 1<\/span><\/b>]<\/b><\/p>\r\n\r\n\r\n\r\n

    (ii) <\/b>\\sin ^{-1} x <\/span>-\\sin ^{-1} y <\/span><\/span>=\\sin ^{-1}\\left\\{x \\sqrt{1-y^{2}}-y \\sqrt{1-x^{2}}\\right\\} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    (iii) <\/b> \\cos ^{-1} x<\/span>+ \\cos ^{-1} y<\/span><\/span>=\\cos ^{-1}\\left\\{x y-\\sqrt{\\left(1-x^{2}\\right)\\left(1-y^{2}\\right)}\\right\\} <\/span><\/span>\u00a0[<\/b>\u09af\u0996\u09a8 x+y\u22650 <\/span><\/strong>]<\/b><\/p>\r\n\r\n\r\n\r\n

    (iv) <\/b> \\cos ^{-1} x<\/span>- \\cos ^{-1} y<\/span><\/span>= \\cos ^{-1}\\left\\{x y+\\sqrt{\\left(1-x^{2}\\right)\\left(1-y^{2}\\right)}\\right\\}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b>:<\/span><\/h3>\r\n

    (i) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span> \\sin ^{-1} x <\/span><\/span>=A <\/span><\/span> \u098f\u09ac\u0982 \\sin ^{-1} y <\/span><\/span>=B <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u2234 sinA<\/span>= x<\/span><\/span> \u098f\u09ac\u0982 <\/span> sinB<\/span>=y <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, \\sin (A+B)<\/span><\/span>\u00a0= sinAcosB<\/span>+cosAsinB <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u00a0 \u00a0 <\/span>= \\sin A \\sqrt{1-\\sin ^{2} B}<\/span><\/span>+ \\sin B \\sqrt{1-\\sin ^{2} A}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u00a0 \u00a0 <\/span>= x \\sqrt{1-y^{2}}<\/span><\/span>+ y \\sqrt{1-x^{2}}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d2 A+B<\/span>= \\sin ^{-1}\\left\\{x \\sqrt{1-y^{2}}+y \\sqrt{1-x^{2}}\\right\\}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u2234\\sin ^{-1} x <\/span><\/span>+\\sin ^{-1} y <\/span><\/span>= \\sin ^{-1}\\left\\{x \\sqrt{1-y^{2}}+y \\sqrt{1-x^{2}}\\right\\}<\/span><\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

    (ii) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\sin ^{-1} x <\/span><\/span>=A <\/span><\/span> \u098f\u09ac\u0982 \\sin ^{-1} y <\/span><\/span>= B<\/span><\/span> \u2234 sinA <\/span>=<\/span> x <\/span>\u098f\u09ac\u0982 <\/span>\u00a0sinB <\/span>=y <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, sin(A-B) <\/span><\/span>\u00a0= sinAcosB<\/span> - cosAsinB<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u00a0 \u00a0 <\/span>= \\sin A \\sqrt{1-\\sin ^{2} B}<\/span><\/span>-\\sin B \\sqrt{1-\\sin ^{2} A} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u00a0 \u00a0 <\/span>= x \\sqrt{1-y^{2}}<\/span><\/span>-y \\sqrt{1-x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d2A-B <\/span>=\\sin ^{-1}\\left\\{x \\sqrt{1-y^{2}}-y \\sqrt{1-x^{2}}\\right\\} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u2234\\sin ^{-1} x <\/span><\/span>-\\sin ^{-1} y <\/span><\/span>=\\sin ^{-1}\\left\\{x \\sqrt{1-y^{2}}-y \\sqrt{1-x^{2}}\\right\\} <\/span><\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

    (iii) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\cos ^{-1} x <\/span><\/span>=A <\/span><\/span> \u098f\u09ac\u0982\\cos ^{-1} y <\/span><\/span>=B <\/span><\/span> \u2234 cosA<\/span>=<\/span>\u00a0 x<\/span>\u098f\u09ac\u0982 <\/span>\u00a0cosB <\/span>=y <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, cos(A+B)<\/span><\/span>\u00a0=cosAcosB <\/span> -sinAsinB <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u00a0 \u00a0 <\/span>=cosAcosB <\/span><\/span>-\\sqrt{1-\\cos ^{2} A} <\/span>\\sqrt{1-\\cos ^{2} B} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u00a0 \u00a0 <\/span>=x y <\/span>-\\sqrt{\\left(1-x^{2}\\right)\\left(1-y^{2}\\right)} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d2A+B <\/span>= \\cos ^{-1}<\/span>\\left\\{x y-\\sqrt{\\left(1-x^{2}\\right)\\left(1-y^{2}\\right)}\\right\\} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u2234\\cos ^{-1} x <\/span><\/span>+\\cos ^{-1} y <\/span><\/span>= \\cos ^{-1}<\/span>\\left\\{x y-\\sqrt{\\left(1-x^{2}\\right)\\left(1-y^{2}\\right)}\\right\\} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    (iv) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\cos ^{-1} x <\/span><\/span>=A <\/span><\/span> \u098f\u09ac\u0982\\cos ^{-1} y <\/span><\/span>=B <\/span><\/span> \u2234 cosA<\/span>=<\/span>\u00a0 x<\/span>\u098f\u09ac\u0982 <\/span>\u00a0cosB <\/span>=y <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, cos(A-B)<\/span><\/span>\u00a0=cosAcosB <\/span> +sinAsinB <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u00a0 \u00a0 <\/span>=cosAcosB <\/span> +\\sqrt{1-\\cos ^{2} A} <\/span>\\sqrt{1-\\cos ^{2} B} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u00a0 \u00a0 <\/span>=x y <\/span>+\\sqrt{\\left(1-x^{2}\\right)\\left(1-y^{2}\\right)} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d2A+B <\/span>= \\cos ^{-1}<\/span>\\left\\{x y+\\sqrt{\\left(1-x^{2}\\right)\\left(1-y^{2}\\right)}\\right\\} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u2234\\cos ^{-1} x <\/span>-\\cos ^{-1} y <\/span><\/span>= \\cos ^{-1}<\/span>\\left\\{x y+\\sqrt{\\left(1-x^{2}\\right)\\left(1-y^{2}\\right)}\\right\\} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u09ac\u09bf.\u09a6\u09cd\u09b0<\/b>. \\sin ^{-1} x <\/span><\/span>+\\sin ^{-1} y <\/span><\/span>= \\pi<\/span>-<\/span> \\sin ^{-1}\\left\\{x \\sqrt{1-y^{2}}+y \\sqrt{1-x^{2}}\\right\\} <\/span>[\u09af\u0996\u09a8 <\/span>\\left.x^{2}+y^{2}>1\\right] <\/span><\/span>]<\/span><\/p>\r\n\r\n\r\n\r\n

    (i)2 \\sin ^{-1} x<\/span>=\\sin ^{-1}\\left\\{2 x \\sqrt{1-x^{2}}\\right\\} <\/span><\/p>\r\n\r\n\r\n\r\n

    (ii) 2\\cos ^{-1} x <\/span>= \\cos ^{-1}\\left\\{2 x^{2}-1\\right\\}<\/span><\/p>\r\n\r\n\r\n\r\n

    (iii) 2 \\tan ^{-1} x<\/span>=\\tan ^{-1} \\frac{2 x}{1-x^{2}} <\/span><\/p>\r\n\r\n\r\n\r\n

    (iv) 3 \\sin ^{-1} x <\/span>=\\sin ^{-1}\\left\\{3 x-4 x^{3}\\right\\} <\/span><\/p>\r\n\r\n\r\n\r\n

    (v) 3 \\cos ^{-1} x <\/span>=\\cos ^{-1}\\left\\{4 x^{3}-3 x\\right\\} <\/span><\/p>\r\n\r\n\r\n\r\n

    (vi) 3\\tan ^{-1} x <\/span>=\\tan ^{-1} \\frac{3 x-x^{3}}{1-3 x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b>: <\/span><\/h3>\r\n

    (i) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\sin ^{-1} x <\/span><\/span>=A<\/span> \u2234 sinA<\/span>= x<\/span><\/span> \u098f\u09ac\u0982 <\/span>\u00a0cosA <\/span>=\\sqrt{1-\\sin ^{2} A} <\/span><\/span>=\\sqrt{1-x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, <\/span>sin 2A <\/span><\/span> = 2sinAcosA<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    =2 x \\sqrt{1-x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d2 2A<\/span>=\\sin ^{-1}\\left(2 x \\sqrt{1-x^{2}}\\right) <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u22342 \\sin ^{-1} x <\/span>= \\sin ^{-1}\\left(2 x \\sqrt{1-x^{2}}\\right)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    (ii) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\cos ^{-1} x <\/span><\/span>=<\/span> A <\/span>\u2234 cosA<\/span>= x<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, <\/span> cos 2A<\/span><\/span> = 2 \\cos ^{2} A-1<\/span><\/span>= 2 x^{2}-1<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d22A <\/span>= \\cos ^{-1}<\/span> \\left(2 x^{2}-1\\right)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u22342\\cos ^{-1} x <\/span><\/span>= \\cos ^{-1}<\/span> \\left(2 x^{2}-1\\right)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    (iii) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\tan ^{-1} x <\/span><\/span>=A <\/span><\/span> \u2234 tanA <\/span>=x <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, 2A<\/span><\/span>=\\frac{2 \\tan A}{1-\\tan ^{2} A} <\/span><\/span>=\\frac{2 x}{1-x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d2 2A<\/span>=\\tan ^{-1} \\frac{2 x}{1-x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u22342\\tan ^{-1} x <\/span><\/span>=\\tan ^{-1} \\frac{2 x}{1-x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    (iv) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\sin ^{-1} x <\/span><\/span>\u00a0\u2234 sinA <\/span>=x <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, <\/span> sin 3A<\/span><\/span> =<\/span>3 \\sin A-4 \\sin ^{3} A\u00a0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    = 3 x-4 x^{3}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d23A <\/span>= \\sin ^{-1}\\left(3 x-4 x^{3}\\right)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u22343 \\sin ^{-1} x <\/span> <\/span>= \\sin ^{-1}\\left(3 x-4 x^{3}\\right)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    (v) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span> \\cos ^{-1} x <\/span><\/span>=A <\/span><\/span> \u2234 cosA<\/span>= x<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, \\cos 3 A<\/span><\/span>\u00a0=<\/span>4 \\cos ^{3} A-3 \\cos A <\/span>=4 x^{3}-3 x <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d23A <\/span>= \\cos ^{-1}\\left(4 x^{3}-3 x\\right)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u22343\\cos ^{-1} x <\/span><\/span>= \\cos ^{-1}\\left(4 x^{3}-3 x\\right)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    (vi) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span> \\tan ^{-1} x <\/span>=A <\/span>\u2234 tanA <\/span>=x <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, <\/span>tan<\/span> 3A <\/span><\/span> =\\frac{3 \\tan A-\\tan ^{3} A}{1-3 \\tan ^{2} A} <\/span><\/span>= \\frac{3 x-x^{3}}{1-3 x^{2}}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d2 3A<\/span>= \\tan ^{-1} \\frac{3 x-x^{3}}{1-3 x^{2}}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u22343\\tan ^{-1} x <\/span><\/span>= \\tan ^{-1} \\frac{3 x-x^{3}}{1-3 x^{2}}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    2\\tan ^{-1} x <\/span><\/span>=\\sin ^{-1} \\frac{2 x}{1+x^{2}} <\/span>= \\cos ^{-1} \\frac{1-x^{2}}{1+x^{2}}<\/span>=\\tan ^{-1} \\frac{2 x}{1-x^{2}} <\/span><\/p>\r\n\r\n\r\n\r\n

    \u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b>:<\/span><\/h3>\r\n

    \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span> \\tan ^{-1} x<\/span><\/span>=\\theta <\/span><\/span>\u00a0\u2234x <\/span>=\\tan \\theta <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u0996\u09a8, <\/span> \\sin 2 \\theta <\/span>=\\frac{2 \\tan \\theta}{1+\\tan ^{2} \\theta} <\/span>=\\frac{2 x}{1+x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d22\u03b8 <\/span>=\\sin ^{-1} \\frac{2 x}{1+x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u22342\\tan ^{-1} x <\/span><\/span>=\\sin ^{-1} \\frac{2 x}{1+x^{2}} <\/span><\/span>\u2026\u2026\u2026(i)<\/span><\/p>\r\n\r\n\r\n\r\n

    \u0986\u09ac\u09be\u09b0, <\/span>cos 2\u03b8 <\/span><\/span> =\\frac{1-\\tan ^{2} \\theta}{1+\\tan ^{2} \\theta} <\/span><\/span>=\\frac{1-x^{2}}{1+x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d22\u03b8<\/span>=\\cos ^{-1} \\frac{1-x^{2}}{1+x^{2}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u22342\\tan ^{-1} x <\/span><\/span>=\\cos ^{-1} \\frac{1-x^{2}}{1+x^{2}} <\/span><\/span>\u2026\u2026\u2026(ii)<\/span><\/p>\r\n\r\n\r\n\r\n

    \u098f\u09ac\u0982, \\tan 2 \\theta<\/span><\/span>\u00a0= \\frac{2 \\tan \\theta}{1-\\tan ^{2} \\theta}<\/span><\/span>= \\frac{2 x}{1-x^{2}}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u21d2 2 \\theta<\/span>= \\tan ^{-1} \\frac{2 x}{1-x^{2}}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

    \u22342\\tan ^{-1} x <\/span><\/span>= \\tan ^{-1} \\frac{2 x}{1-x^{2}}<\/span><\/span>\u2026\u2026\u2026(iii)<\/span><\/p>\r\n\r\n\r\n\r\n

    (i)<\/span>,\u00a0(ii)<\/span> \u0993 <\/span>(iii)<\/span> \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, 2\\tan ^{-1} x <\/span>=\\sin ^{-1} \\frac{2 x}{1+x^{2}} <\/span>= \\cos ^{-1} \\frac{1-x^{2}}{1+x^{2}}<\/span>=\\tan ^{-1} \\frac{2 x}{1-x^{2}} <\/span><\/span><\/p>\r\n","protected":false},"excerpt":{"rendered":"

    \u09ab\u09be\u0982\u09b6\u09a8 \u09a1\u09cb\u09ae\u09c7\u09a8 \u09b0\u09c7\u099e\u09cd\u099c \u00a0 \u09ac\u09be, \u09ac\u09be, \u00a0\u09ac\u09be, \u00a0 – \u00a0\u09ac\u09be, – \u0993 \u098f\u09b0 \u09b0\u09c7\u099e\u09cd\u099c \u09a8\u09bf\u09af\u09bc\u09c7 \u09a6\u09cd\u09ac\u09bf\u09ae\u09a4 \u0986\u099b\u09c7\u0964 \u0995\u09cb\u09a8\u09cb \u0995\u09cb\u09a8\u09cb \u0997\u09a3\u09bf\u09a4\u09ac\u09bf\u09a6 \u09ae\u09a8\u09c7 \u0995\u09b0\u09c7\u09a8 \u098f\u09b0 \u09b0\u09c7\u099e\u09cd\u099c \u098f\u09ac\u0982 \u00a0\u098f\u09b0 \u09b0\u09c7\u099e\u09cd\u099c \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 \u0995\u09af\u09bc\u09c7\u0995\u099f\u09bf \u09b8\u09c2\u09a4\u09cd\u09b0 (Some rules of inverse trigonometric functions): (i) +cos-1x=;<\/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":[2570,882,2498,2568,2497,2564,2569],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3623"}],"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=3623"}],"version-history":[{"count":32,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3623\/revisions"}],"predecessor-version":[{"id":13292,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3623\/revisions\/13292"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=3623"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=3623"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=3623"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}