{"id":3776,"date":"2024-04-21T13:27:22","date_gmt":"2024-04-21T07:27:22","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3776"},"modified":"2024-11-05T13:01:30","modified_gmt":"2024-11-05T07:01:30","slug":"trigonometric-ratio-of-connected-angle","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/trigonometric-ratio-of-connected-angle\/","title":{"rendered":"\u09b8\u0982\u09af\u09c1\u0995\u09cd\u09a4 \u0995\u09cb\u09a3\u09c7\u09b0 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09a8\u09c1\u09aa\u09be\u09a4 (Trigonometric Ratio Of Connected Angles)"},"content":{"rendered":"

\u09b8\u0982\u09af\u09c1\u0995\u09cd\u09a4 \u0995\u09cb\u09a3\u09c7\u09b0 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u09c7\u09b0 <\/b>\u09ae\u09cc\u09b2\u09bf\u0995 \u0995\u09bf\u099b\u09c1 \u09ac\u09bf\u09b7\u09af\u09bc<\/span><\/span><\/h2>\n

\u0985\u09a8\u09c1\u09b6\u09c0\u09b2\u09a8\u09c0 – <\/b>7(A)<\/span><\/span><\/h3>\n

\u2192 \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u0995\u09cb\u09a3\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995<\/a> <\/span>\u0985\u09a8\u09c1\u09aa\u09be\u09a4 (Trigonometric Ratio) \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u099a\u09a4\u09c1\u09b0\u09cd\u09ad\u09be\u0997\u09c7 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u0964\u00a0<\/span><\/p>\n

\u2192 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u0995\u09cb\u09a3, \u098b\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u0995\u09cb\u09a3, (A \\pm \\theta)<\/span><\/span>\u00a0\u09aa\u09cd\u09b0\u09ad\u09c3\u09a4\u09bf\u09b0 \u099c\u09a8\u09cd\u09af \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u09c7\u09b0 \u09aa\u09cd\u09b0\u0995\u09c3\u09a4\u09bf \u0964\u00a0\u00a0\u00a0<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b6\u09c0\u09b2\u09a8\u09c0 – <\/b>7(B)<\/span><\/span><\/h3>\n

\u2192(A \\pm B)<\/span><\/span> \u09b8\u09c2\u0995\u09cd\u09b7\u09cd\u09ae \u0995\u09cb\u09a3\u09c7\u09b0 \u099c\u09a8\u09cd\u09af <\/span>\\sin , \\cos , \\tan , \\cot<\/span><\/span> \u09b8\u09c2\u09a4\u09cd\u09b0 \u0964\u00a0<\/span><\/p>\n

\u2192 \u09b8\u09c2\u09a4\u09cd\u09b0 \u09b8\u09ae\u09c2\u09b9\u09c7\u09b0 \u0997\u09cd\u09b0\u09be\u09ab\u09bf\u0995\u09cd\u09af\u09be\u09b2 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u0964\u00a0<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b6\u09c0\u09b2\u09a8\u09c0 – <\/b>7(C)<\/span><\/span><\/h3>\n

\u2192 \u09a6\u09c1\u099f\u09bf \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995<\/a><\/span> \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u09c7\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2\u0995\u09c7 \u0985\u09aa\u09b0 \u09a6\u09c1\u099f\u09bf \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2\/\u09ac\u09bf\u09df\u09cb\u0997\u09ab\u09b2\u09c7 <\/span>\u09aa\u09cd\u09b0\u0995\u09be\u09b6\u0964\u00a0<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b6\u09c0\u09b2\u09a8\u09c0 – <\/b>7(D)<\/span><\/span><\/h3>\n

\u2192 \u0997\u09c1\u09a3\u09bf\u09a4\u0995 \u0995\u09cb\u09a3\u09c7\u09b0\u00a0 \u0985\u09a8\u09c1\u09aa\u09be\u09a4 \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 (\u09aa\u09c1\u09b0\u09cd\u09ac\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b8\u09c2\u09a4\u09cd\u09b0\u09c7 \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09c7)\u00a0<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b6\u09c0\u09b2\u09a8\u09c0 – <\/b>7(E)<\/span><\/span><\/h3>\n

\u2192 \u099f\u09aa\u0997\u09c1\u09a3\u09bf\u09a4\u0995 \u0995\u09cb\u09a3\u09c7\u09b0\u00a0 \u09b8\u09c2\u09a4\u09cd\u09b0\u00a0<\/span><\/p>\n

\u2192 \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u0989\u09aa\u0997\u09c1\u09a3\u09bf\u09a4\u0995 \u0995\u09cb\u09a3\u09c7\u09b0\u00a0 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09a8\u09c1\u09aa\u09be\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \\left(\\frac{25^{\\circ}}{2}, 18^{\\circ}, 36^{\\circ}, 15^{\\circ}\\right)<\/span><\/span><\/p>\n

\u0985\u09a8\u09c1\u09b6\u09c0\u09b2\u09a8\u09c0 – <\/b>7(F)<\/span><\/span><\/h3>\n

\u2192 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09ad\u09c7\u09a6\u00a0<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b6\u09c0\u09b2\u09a8\u09c0 – <\/b>7(G)<\/span><\/span><\/h3>\n

\u2192 \u09aa\u09cd\u09b0\u09a4\u09c0\u0995\u09c7\u09b0 \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0\u00a0<\/span><\/p>\n

\u2192 \u0985\u09a8\u09cd\u09a4\u0983\u09b8\u09cd\u09a5 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u09a7\u09b0\u09cd\u09ae\u09be\u09ac\u09b2\u09bf\u00a0\u00a0<\/span><\/p>\n

\u2192 \u0985\u09a8\u09cd\u09a4\u0983\u09b8\u09cd\u09a5 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09b8\u09be\u0987\u09a8 \u09b8\u09c2\u09a4\u09cd\u09b0\u00a0<\/span><\/p>\n

\u2192 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 cosine <\/span>\u09b8\u09c2\u09a4\u09cd\u09b0\u00a0<\/span><\/p>\n

\u2192 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09b8\u0982\u0995\u09cd\u09b0\u09be\u09a8\u09cd\u09a4 \u09b8\u09c2\u09a4\u09cd\u09b0\u00a0<\/span><\/p>\n

\u09b8\u0982\u09af\u09c1\u0995\u09cd\u09a4 \u0995\u09cb\u09a3\u09c7\u09b0 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a7\u09be\u09a8 \u09ac\u09bf\u09b7\u09af\u09bc\u0997\u09c1\u09b2\u09cb\u00a0<\/span><\/span><\/b><\/span><\/h2>\n

\u09a7\u09b0\u09bf<\/span>, 0<\\theta<90^{\\circ}<\/span><\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce<\/span>\u00a0<\/span>\\theta<\/span>,<\/span>\u00a0<\/span>\u09b8\u09c2\u0995\u09cd\u09b7\u09cd\u09ae\u0995\u09cb\u09a3<\/span><\/i><\/p>\n

(i)<\/b> \u0995\u09cb\u09a3 \u09af\u09a6\u09bf \u09a7\u09a8\u09be\u09a4\u09cd\u09ac\u0995 \u09b9\u09df \u09a4\u09ac\u09c7 \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8\u0995\u09be\u09b0\u09c0 \u0998\u09c2\u09b0\u09cd\u09a3\u09be\u09df\u09ae\u09be\u09a8 \u09b0\u09c7\u0996\u09be \u09ea\u09b0\u09cd\u09a5 \u099a\u09a4\u09c1\u09b0\u09cd\u09ad\u09be\u0997\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n

(ii)<\/b> \u09af\u09a6\u09bf <\/span>\\pi \/ 2<\/span><\/span> \u09ac\u09be <\/span>90^{\\circ}<\/span>–<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ac\u09bf\u099c\u09cb\u09dc \u09aa\u09c2\u09b0\u09cd\u09a3\u09b8\u0982\u0996\u09cd\u09af\u09be \u0997\u09c1\u09a8 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09a5\u09be\u0995\u09c7 \u09a4\u09ac\u09c7 \u0985\u09a8\u09c1\u09aa\u09be\u09a4 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09c7\u09b0 \u09a7\u09be\u09b0\u09be \\sin \\theta \\rightleftharpoons \\cos \\theta , \\tan \\theta \\rightleftharpoons \\cot \\theta , \\operatorname{cosec} \\theta \\rightleftharpoons \\sec \\theta<\/span><\/span>
\n<\/span><\/p>\n

(iii)<\/b> \u09af\u09a6\u09bf <\/span>\\frac{\\pi}{2}<\/span><\/span> \u09ac\u09be 90^{\\circ}<\/span> <\/span>–<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u099c\u09cb\u09dc \u09aa\u09c2\u09b0\u09cd\u09a3\u09b8\u0982\u0996\u09cd\u09af\u09be \u0997\u09c1\u09a8 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09a5\u09be\u0995\u09c7 \u09a4\u09ac\u09c7 \u0985\u09a8\u09c1\u09aa\u09be\u09a4 \u0985\u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09bf\u09a4 \u09a5\u09be\u0995\u09ac\u09c7\u0964\u00a0\u00a0<\/span><\/p>\n

(iv)<\/b> \u09af\u09a6\u09bf <\/span>\\pi<\/span> <\/span>\u09ac\u09be<\/span> 180^{\\circ}<\/span>–<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u099c\u09cb\u09dc \u09aa\u09c2\u09b0\u09cd\u09a3\u09b8\u0982\u0996\u09cd\u09af\u09be \u0997\u09c1\u09a8 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09a5\u09be\u0995\u09c7 \u09a4\u09ac\u09c7 \u09ac\u09be\u09a6 \u09a6\u09c7\u0993\u09df\u09be \u09af\u09be\u09df\u0964\u00a0<\/span><\/p>\n

(v)<\/b> \u0995\u09cb\u09a3 \u09af\u09a6\u09bf <\/span>2 \\pi<\/span><\/span> \u09ac\u09be <\/span>360^{\\circ}<\/span> –<\/span> \u098f\u09b0 \u0985\u09a7\u09bf\u0995 \u09b9\u09df \u09a4\u09ac\u09c7 <\/span>2 \\pi<\/span><\/span> \u09ac\u09be <\/span>360^{\\circ}<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964\u00a0\u00a0<\/span><\/p>\n

\u00a0\u00a0\u00a0<\/span><\/p>\n

\"\u09b8\u0982\u09af\u09c1\u0995\u09cd\u09a4<\/p>\n

\u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u0995\u09c7 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u0993 (trigonometric function) \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 \u09b8\u09be\u0987\u09a8 \u098f\u09ac\u0982 \u0995\u09cb\u09b8\u09be\u0987\u09a8\u0995\u09c7 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u09c7\u09b0 \u09b8\u09b9-\u09ab\u09be\u0982\u09b6\u09a8 \u09ac\u09b2\u09c7\u0964 \u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7, \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u099f \u098f\u09ac\u0982 \u0995\u09cb\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u099f\u0995\u09c7\u0993 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u09c7\u09b0 \u09b8\u09b9-\u09ab\u09be\u0982\u09b6\u09a8 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 \u09a4\u09a6\u09cd\u09b0\u09c1\u09aa, \u099f\u09c7\u09a8\u099c\u09c7\u09a8\u09cd\u099f \u0993 \u0995\u09cb\u099f\u09c7\u09a8\u099c\u09c7\u09a8\u09cd\u099f \u09b9\u09b2 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u09c7\u09b0 \u09b8\u09b9-\u09ab\u09be\u0982\u09b6\u09a8\u0964 \u09af\u09a6\u09bf \u09a6\u09c1\u0987\u099f\u09bf \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf \u098f\u0995 \u09b8\u09ae\u0995\u09cb\u09a3 \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u098f\u0995\u099f\u09bf\u0995\u09c7 \u0985\u09aa\u09b0\u099f\u09bf\u09b0 \u09aa\u09b0\u09bf\u09aa\u09c2\u09b0\u0995 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 \u09a4\u09be\u09b9\u09b2\u09c7, <\/span>30^{\\circ}<\/span><\/span> \u098f\u09ac\u0982 <\/span>60^{\\circ}<\/span><\/span> \u0995\u09cb\u09a3\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u0985\u09aa\u09b0\u099f\u09bf\u09b0 \u09aa\u09b0\u09bf\u09aa\u09c2\u09b0\u0995\u0964\u00a0<\/span><\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u098f\u0995\u099f\u09bf \u0995\u09cb\u09a3\u09c7\u09b0 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8 <\/span>=<\/span> \u098f\u09b0 \u09aa\u09b0\u09bf\u09aa\u09c2\u09b0\u0995\u09c7\u09b0 \u09b8\u09b9-\u09ab\u09be\u0982\u09b6\u09a8\u0964<\/span><\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3\u09b8\u09cd\u09ac\u09b0\u09c2\u09aa \\sin \\left(3 \\times 90^{\\circ}+\\theta\\right)=-\\cos \\theta<\/span><\/span>\u00a0,<\/span> \u0995\u09be\u09b0\u09a3 \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 <\/span>\\left(3 \\times 90^{\\circ}+\\theta\\right)<\/span><\/span> \u0995\u09cb\u09a3\u099f\u09bf\u09a4\u09c7 <\/span>90^{\\circ}<\/span><\/span>\u00a0\u098f\u09b0 \u09ac\u09bf\u099c\u09cb\u09a1\u09bc \u0997\u09c1\u09a3\u09bf\u09a4\u0995 \u09b0\u09af\u09bc\u09c7\u099b\u09c7 \u09a4\u09be\u0987 <\/span>sine<\/span> \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09bf\u09a4 \u09b9\u09af\u09bc\u09c7 <\/span>cosine<\/span> \u09b9\u09af\u09bc\u09c7\u099b\u09c7 \u098f\u09ac\u0982 <\/span>\\left(3 \\times 90^{\\circ}+\\theta\\right)<\/span><\/span> \u0995\u09cb\u09a3\u099f\u09bf \u09ea\u09b0\u09cd\u09a5 \u099a\u09a4\u09c1\u09b0\u09cd\u09ad\u09be\u0997\u09c7 (<\/span>Quadrant) <\/span>\u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09ac\u09b2\u09c7 <\/span>(-)<\/span> \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u099a\u09bf\u09b9\u09cd\u09a8 \u09ac\u09b8\u09c7\u099b\u09c7 [\u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \u09ea\u09b0\u09cd\u09a5 \u099a\u09a4\u09c1\u09b0\u09cd\u09ad\u09be\u0997\u09c7 <\/span>sine<\/span> \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995]\u0964 \u0986\u09b0 c<\/span>osecant<\/span>, <\/span>secant<\/span> \u098f\u09ac\u0982 <\/span>cotangent<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u0985\u09a4\u09bf\u09b0\u09bf\u0995\u09cd\u09a4 \u09ad\u09be\u09ac\u09be\u09b0 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u099c\u09a8 \u09a8\u09c7\u0987, \u0995\u09be\u09b0\u09a3 \u098f\u0987 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 <\/span>\u0985\u09a8\u09c1\u09aa\u09be\u09a4\u0997\u09c1\u09b2\u09cb \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>sine,\u00a0cosine\u00a0<\/span>\u098f\u09ac\u0982 <\/span>tangent<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\n

(X) \\tan (A+B) \\tan (A-B)=\\frac{\\tan ^{2} A-\\tan ^{2} B}{1-\\tan ^{2} A \\tan ^{2} B}, \\cot (A+B) \\cot (A-B)=\\frac{\\cot ^{2} A \\cot ^{2} B-1}{\\cot ^{2} B-\\cot ^{2} A}<\/span>\u00a0 <\/span>\u00a0<\/b><\/p>\n

\u09aa\u09cd\u09b0\u09ae\u09be\u09a8 :<\/b> \\tan (A+B) \\tan (A-B)=\\frac{\\tan A+\\tan B}{1-\\tan A \\tan B} \\times \\frac{\\tan A-\\tan B}{1+\\tan A \\tan B}=\\frac{\\tan ^{2} A-\\tan ^{2} B}{1-\\tan ^{2} A \\tan ^{2} B}<\/span><\/p>\n

\\cot (A+B) \\cot (A-B)=\\frac{\\cot A \\cot B-1}{\\cot B+\\cot A} \\times \\frac{\\cot A \\cot B+1}{\\cot B-\\cot A}=\\frac{\\cot ^{2} A \\cot ^{2} B-1}{\\cot ^{2} B-\\cot ^{2} A}<\/span>\u00a0<\/span><\/p>\n

(i)<\/b> \u0993 (<\/b>ii)<\/b> \u09a6\u09c1\u0987\u099f\u09bf \u0995\u09cb\u09a3\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u0993 \u09ac\u09bf\u09df\u09cb\u0997\u09ab\u09b2 \u0995\u09c0 \u09b8\u09c2\u09a4\u09cd\u09b0 \u09a8\u09be\u09ae\u09c7 \u09aa\u09b0\u09bf\u099a\u09bf\u09a4?<\/b><\/p>\n

\u09a8\u09bf\u09ae\u09cd\u09a8\u09ac\u09b0\u09cd\u09a3\u09bf\u09a4 \u09b8\u09c2\u09a4\u09cd\u09b0\u09b8\u09ae\u09c2\u09b9 \u09b6\u09bf\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09b0\u09be \u09ac\u09bf\u09a8\u09be \u09aa\u09cd\u09b0\u09ae\u09be\u09a3\u09c7 \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09a4\u09c7 \u09aa\u09be\u09b0\u09ac\u09c7\u0964<\/span><\/p>\n

(i) <\/b>\u00a0\\sin (A+B) \\sin (A-B)=\\sin ^{2} A-\\sin ^{2} B=\\cos ^{2} B-\\cos ^{2} A<\/span><\/span><\/p>\n

(ii)<\/b> \\cos (A+B) \\cos (A-B)=\\cos ^{2} A-\\sin ^{2} B=\\cos ^{2} B-\\sin ^{2} A<\/span>\u00a0\u00a0<\/i><\/b><\/p>\n

(iii)<\/b> \\sin (A+B+C)=\\cos A \\cos B \\cos C(\\tan A+\\tan B+\\tan C-\\tan A \\tan B \\tan C)<\/span>\u00a0 \u00a0 \u00a0\u00a0<\/span>\u00a0<\/b><\/p>\n

(iv)<\/b> \\cos (A+B+C)=\\cos A \\cos B \\cos C(1-\\tan B \\tan C-\\tan C \\tan A-\\tan A \\tan B)<\/span>\u00a0 \u00a0\u00a0<\/span>\u00a0<\/span>\u00a0 <\/span><\/p>\n

(v)<\/b> \\tan (A+B+C)=\\frac{\\tan A+\\tan B+\\tan C-\\tan A \\tan B \\tan C}{1-\\tan B \\tan C-\\tan C \\tan A-\\tan A \\tan B}<\/span>\u00a0 \u00a0 \u00a0<\/span><\/p>\n

\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 :\u00a0<\/b><\/span><\/h3>\n

(i) <\/b>\\sin (A+B) \\sin (A-B)=(\\sin A \\cos B+\\cos A \\sin B)(\\sin A \\cos B-\\cos A \\sin B)<\/span><\/p>\n

\u00a0 \u00a0 =\\sin ^{2} A \\cos ^{2} B-\\cos ^{2} A \\sin ^{2} B=\\sin ^{2} A\\left(1-\\sin ^{2} A\\right)=\\sin ^{2} A-\\sin ^{2} B<\/span><\/span><\/p>\n

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

(ii)<\/b> \\cos (A+B) \\cos (A-B)=(\\cos A \\cos B-\\sin A \\sin B)(\\cos A \\cos B+\\sin A \\sin B)<\/span><\/p>\n

\u00a0 \u00a0 =\\cos ^{2} A \\cos ^{2} B-\\sin ^{2} A \\sin ^{2} B=\\cos ^{2} A\\left(1-\\sin ^{2} B\\right)-\\sin ^{2} B\\left(1-\\cos ^{2} A\\right) \\cos ^{2} A-\\sin ^{2} B<\/span><\/span><\/p>\n

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

(iii)<\/b> \\sin (A+B+C)=\\sin \\{A+B+C\\}=\\sin (A+B) \\cos C+\\cos (A+B) \\sin C<\/span>\u00a0 \u00a0\u00a0<\/span>\u00a0<\/span><\/p>\n

\u00a0 \u00a0 =(\\sin A \\cos B+\\cos A \\sin B) \\cos C+(\\cos A \\cos B-\\sin A \\sin B) \\sin C<\/span><\/span>\u00a0\u00a0<\/span><\/p>\n

\u00a0 \u00a0 =\\sin A \\cos B \\cos C+\\cos A \\sin B \\cos C+\\cos A \\cos B \\sin C-\\sin A \\sin B \\sin C<\/span><\/span><\/p>\n\\begin{array}{l}\n\n=\\cos A \\cos B \\cos C\\left(\\frac{\\sin A}{\\cos A}+\\frac{\\sin B}{\\cos B}+\\frac{\\sin C}{\\cos C}-\\frac{\\sin A \\sin B \\sin C}{\\cos A \\cos B \\cos C}\\right) \\\\\n\n=\\cos A \\cos B \\cos C(\\tan A+\\tan B+\\tan C-\\tan A \\tan B \\tan C)\n\n\\end{array}<\/span>\n

(iv)<\/b> \\cos (A+B+C)=\\cos \\{(A+B)+C\\}=\\cos (A+B) \\cos C-\\sin (A+B) \\sin C<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 =(\\cos A \\cos B-\\sin A \\sin B) \\cos C-(\\sin A \\cos B+\\cos A \\sin B) \\sin C<\/span><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 =\\cos A \\cos B \\cos C-\\sin A \\sin B \\sin C-\\sin A \\sin B \\sin C-\\cos A \\sin B \\sin C<\/span><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 =\\cos A \\cos B \\cos C\\left(1-\\frac{\\sin A}{\\cos A} \\frac{\\sin B}{\\cos B}-\\frac{\\sin C}{\\cos C} \\frac{\\sin A}{\\cos A}-\\frac{\\sin B}{\\cos B} \\frac{\\sin C}{\\cos C}\\right)<\/span><\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>=\\cos A \\cos B \\cos C(1-\\tan A \\tan B-\\tan C \\tan A-\\tan B \\tan C)<\/span><\/span>\u00a0<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>=\\cos A \\cos B \\cos C(1-\\tan B \\tan C-\\tan C \\tan A-\\tan A \\tan B)<\/span><\/span>\u00a0<\/span><\/p>\n

(v)<\/b> \\tan (A+B+C)=\\tan \\{(A+B)+C\\}=\\frac{\\tan (A+B)+\\tan C}{1-\\tan (A+B) \\tan C}=\\frac{\\frac{\\tan A+\\tan B}{1-\\tan A \\tan B}+\\tan C}{1-\\frac{\\tan A+\\tan B}{1-\\tan A \\tan B} \\tan C}<\/span>\u00a0\u00a0<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 =\\frac{\\tan A+\\tan B+\\tan C-\\tan A \\tan B \\tan C}{1-\\tan A \\tan B-\\tan C \\tan A-\\tan B \\tan C}=\\frac{\\tan A+\\tan B+\\tan C-\\tan A \\tan B \\tan C}{1-\\tan B \\tan C-\\tan C \\tan A-\\tan A \\tan B}<\/span><\/span>\u00a0\u00a0\u00a0<\/span><\/p>\n

\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af:<\/b> A+B, A-B, A+B+C, A+B-C<\/span><\/span> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0995\u09cb\u09a3\u0997\u09c1\u09b2\u09bf \u09af\u09cc\u0997\u09bf\u0995 \u0995\u09cb\u09a3 \u09a8\u09be\u09ae\u09c7 \u09aa\u09b0\u09bf\u099a\u09bf\u09a4\u0964\u00a0\u00a0\u00a0<\/span><\/p>\n

\\therefore \\cot 3 A=\\frac{3 \\cot A -\\cot ^{3} A}{1-3 \\cot ^{2} A}\\{\\tan (3 A)<\/span> \u098f\u09b0<\/span>\u00a0<\/span>\u09b8\u09c2\u09a4\u09cd\u09b0\u09c7<\/span> \\tan<\/span><\/span> \u098f\u09b0 \u099c\u09be\u09df\u0997\u09be\u09df <\/span>\\cot<\/span><\/span> \u09ac\u09cd\u09af\u09be\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09b2\u09c7\u0987 \u09b9\u09df\u0964 }<\/span><\/p>\n

\u0989\u09aa\u2013\u0997\u09c1\u09a3\u09bf\u09a4\u0995 \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0\u09b8\u09ae\u09c2\u09b9 :<\/b><\/span><\/h3>\n

(i)<\/b> \\sin \\theta=2 \\sin \\frac{\\theta}{2} \\cos \\frac{\\theta}{2}<\/span>\u00a0\u00a0<\/span>\u00a0<\/span><\/p>\n

(ii)<\/b> \\cos \\theta=\\cos ^{2} \\frac{\\theta}{2}-\\sin ^{2} \\frac{\\theta}{2}=2 \\cos ^{2} \\frac{\\theta}{2}-1=1-2 \\sin ^{2} \\frac{\\theta}{2}<\/span>\u00a0\u00a0<\/span><\/p>\n

(iii)<\/b> 1+\\cos \\theta=2 \\cos ^{2} \\frac{\\theta}{2}<\/span>\u00a0\u00a0<\/span><\/p>\n

(iv)<\/b> 1-\\cos \\theta=2 \\sin ^{2} \\frac{\\theta}{2}<\/span>\u00a0<\/span><\/p>\n

(v)<\/b> \\sin \\theta=\\frac{2 \\tan \\frac{\\theta}{2}}{1+\\tan ^{2} \\frac{\\theta}{2}}<\/span><\/p>\n

(vi)<\/b> \\cos \\theta=\\frac{1-\\tan ^{2} \\frac{\\theta}{2}}{1+\\tan 2 \\frac{\\theta}{2}}<\/span><\/p>\n

(vii)<\/b> \\tan \\theta=\\frac{2 \\tan \\frac{\\theta}{2}}{1-\\tan ^{2} \\frac{\\theta}{2}}<\/span>\u00a0\u00a0<\/span><\/p>\n

\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af: <\/b>\u0997\u09c1\u09a3\u09bf\u09a4\u0995 \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0\u09b8\u09ae\u09c2\u09b9\u09c7 A=\\frac{\\theta}{2}<\/span><\/span> \u09ac\u09b8\u09bf\u09af\u09bc\u09c7 \u09b8\u09c2\u09a4\u09cd\u09b0\u0997\u09c1\u09b2\u09bf \u09b8\u09b9\u099c\u09c7\u0987 \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964\u00a0<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a7\u09be\u09a8\u09cd\u09a4 : <\/b>\u0989\u09aa\u09b0\u09cd\u09af\u09c1\u0995\u09cd\u09a4 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, a=2 R \\sin A, b=2 R \\sin B, c=2 R \\sin C<\/span><\/span>\u00a0 \u00a0<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\\sin A=\\frac{a}{2 R}, \\sin B=\\frac{b}{2 R}, \\sin C=\\frac{c}{2 R}<\/span><\/span>\u00a0\u00a0<\/span><\/p>\n

\u098f <\/span>sine <\/span>\u0993 <\/span>cosine<\/span> \u09b8\u09c2\u09a4\u09cd\u09b0 \u09a6\u09c1\u2019\u099f\u09bf \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09aa\u09cd\u09b0\u09be\u09af\u09bc \u09b8\u0995\u09b2 \u0997\u09c1\u09a3\u09be\u09ac\u09b2\u09bf \u09ac\u09cd\u09af\u09be\u0996\u09cd\u09af\u09be \u0995\u09b0\u09c7\u0964<\/span><\/p>\n