{"id":3619,"date":"2021-01-30T12:24:20","date_gmt":"2021-01-30T12:24:20","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3619"},"modified":"2022-03-23T19:31:16","modified_gmt":"2022-03-23T19:31:16","slug":"%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%b8%e0%a6%ae%e0%a7%80%e0%a6%95%e0%a6%b0%e0%a6%a3","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/%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%b8%e0%a6%ae%e0%a7%80%e0%a6%95%e0%a6%b0%e0%a6%a3\/","title":{"rendered":"\u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (Trigonometric equations)"},"content":{"rendered":"

\u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/b>(Trigonometric equations)<\/b><\/h2>\n

(i) \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u0997\u09c1\u09b2\u09cb\u0995\u09c7 \u09b8\u09b0\u09b2\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7<\/span> tan <\/span><\/span> \u0995\u09bf\u0982\u09ac\u09be<\/span> cot <\/span><\/span> \u0995\u09bf\u0982\u09ac\u09be <\/span>cos <\/span><\/span> \u09a8\u09c7\u0993\u09af\u09bc\u09be \u09b8\u09c1\u09ac\u09bf\u09a7\u09be\u099c\u09a8\u0995\u0964<\/span><\/p>\n

(ii) \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0989\u09ad\u09af\u09bc\u09aa\u0995\u09cd\u09b7\u0995\u09c7 \u09ac\u09b0\u09cd\u0997 \u0995\u09b0\u09c7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u098f\u09a1\u09bc\u09bf\u09af\u09bc\u09c7 \u09af\u09be\u0993\u09af\u09bc\u09be \u09ad\u09be\u09b2\u0964 \u098f\u0995\u09be\u09a8\u09cd\u09a4 \u09b8\u09ae\u09cd\u09ad\u09ac \u09a8\u09be \u09b9\u09b2\u09c7 \u09a6\u09c7\u0996\u09a4\u09c7 \u09b9\u09ac\u09c7 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 \u0989\u09a4\u09cd\u09a4\u09b0\u0997\u09c1\u09b2\u09cb \u09af\u09a5\u09be\u09af\u09a5 \u0995\u09bf\u09a8\u09be \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09b6\u09c1\u09a6\u09cd\u09a7\u09bf \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7 \u098f\u09ac\u0982 \u0985\u09ac\u09be\u09a8\u09cd\u09a4\u09b0 \u09ae\u09c2\u09b2\u0997\u09c1\u09b2\u09cb \u09ac\u09be\u09a6 \u09a6\u09bf\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

(iii) \u0989\u09b0\u09cd\u09a6\u09cd\u09a7\u0998\u09be\u09a4\u0997\u09c1\u09b2\u09cb\u0995\u09c7 \u0997\u09c1\u09a3\u09bf\u09a4\u0995 \u0995\u09cb\u09a3\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be\u09b0 \u099a\u09c7\u09b7\u09cd\u099f\u09be \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

(iv) \u0997\u09c1\u09a3\u09ab\u09b2\u0997\u09c1\u09b2\u09cb\u0995\u09c7 \u09af\u09cb\u0997\u09ab\u09b2 \u09ac\u09be \u09af\u09cb\u0997\u09ab\u09b2\u0997\u09c1\u09b2\u09cb\u0995\u09c7 \u0997\u09c1\u09a3\u09ab\u09b2\u09c7 \u09a8\u09bf\u09af\u09bc\u09c7 \u09af\u09be\u0993\u09af\u09bc\u09be \u0985\u09a8\u09c7\u0995 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09b8\u09c1\u09ac\u09bf\u09a7\u09be\u099c\u09a8\u0995\u0964<\/span><\/p>\n

(v)<\/span> n <\/span>=0,\u00a0\u00b11,\u00a0\u00b12,\u00a0\u2026\u2026<\/span> \u0995\u09bf\u0982\u09ac\u09be<\/span> n <\/span><\/span>\u00a0\u09af\u09c7 \u0995\u09cb\u09a8 \u09aa\u09c2\u09b0\u09cd\u09a3 \u09b8\u0982\u0996\u09cd\u09af\u09be \u0995\u09bf\u0982\u09ac\u09be <\/span>n\u2208Z<\/span> \u09b2\u09bf\u0996\u09a4\u09c7 \u09af\u09c7\u09a8 \u09ad\u09c1\u09b2 \u09a8\u09be \u09b9\u09af\u09bc \u098f\u09ac\u0982 \u09aa\u09cd\u09b0\u09b6\u09cd\u09a8\u09c7 n <\/span><\/span>\u00a0 \u09a5\u09be\u0995\u09b2\u09c7<\/span> n <\/span><\/span> \u098f\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 m <\/span><\/span>,k <\/span> <\/span>\u00a0\u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

(vi) \u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09c2\u09a4\u09cd\u09b0\u099f\u09bf\u09a4\u09c7 <\/span> \u03c0<\/b><\/span><\/span> \u09ac\u09cd\u09af\u09ac\u09b9\u09c3\u09a4 \u09b9\u09af\u09bc \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ac\u09bf\u09b6\u09c7\u09b7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u03c0<\/b><\/span><\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \\sin \\theta <\/span><\/span>=<\/span>1<\/span>2<\/span>\u00a0<\/span>\u09b9\u09b2\u09c7 \\sin 30^{\\circ} <\/span><\/span>\u00a0\u09a8\u09be \u09b2\u09bf\u0996\u09c7<\/span> \\sin \\frac{\\pi}{6} <\/span><\/span>\u00a0\u09b2\u09c7\u0996\u09be\u0987 \u09b6\u09cd\u09b0\u09c7\u09af\u09bc\u0964<\/span><\/p>\n

(vii) \u09b8\u09c0\u09ae\u09be \u09b0\u09c7\u09a1\u09bf\u09af\u09bc\u09be\u09a8\u09c7 \u0985\u09b0\u09cd\u09a5\u09be\u09ce <\/span> \u03c0<\/b><\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09a6\u09c7\u09af\u09bc\u09be \u09a5\u09be\u0995\u09b2\u09c7 \u0989\u09a4\u09cd\u09a4\u09b0\u0997\u09c1\u09b2\u09cb <\/span> \u03c0<\/b><\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09a6\u09bf\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n

(viii) \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0989\u09a4\u09cd\u09a4\u09b0 \u09aa\u0995\u09cd\u09b7\u09c7\u09b0 \u09b2\u09ac sinx <\/span><\/span>,cosx\u00a0 <\/span> <\/span>\u099c\u09be\u09a4\u09c0\u09af\u09bc \u0989\u09ce\u09aa\u09be\u09a6\u0995 \u09a5\u09be\u0995\u09b2\u09c7 \u09b8\u09c7\u099f\u09bf\u0995\u09c7 \u0985\u09aa\u09b8\u09be\u09b0\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09ac\u09c7 \u09a8\u09be\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u0989\u09ad\u09af\u09bc \u09aa\u0995\u09cd\u09b7\u09c7\u09b0 \u09b9\u09b0\u09c7 \u09a5\u09be\u0995\u09b2\u09c7 \u09b8\u09c7\u099f\u09bf\u0995\u09c7 \u0985\u09aa\u09b8\u09be\u09b0\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09ac\u09c7\u0964 \u0995\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09b9\u09b0 <\/span>0<\/span> (\u09b6\u09c2\u09a8\u09cd\u09af) \u09b9\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7 \u09a8\u09be\u0964<\/span><\/p>\n

\u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 <\/b>(General solution of trigonometric equations)<\/b><\/h3>\n

\u09a6\u09c1\u0987 \u09ac\u09be \u09a4\u09a4\u09cb\u09a7\u09bf\u0995 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09a8\u09c1\u09aa\u09be\u09a4 \u09b8\u09ae\u09cd\u09ac\u09b2\u09bf\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0995\u09c7 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 \u09af\u09c7 \u09b8\u0995\u09b2 \u0995\u09cb\u09a3 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0995\u09c7 \u09b8\u09bf\u09a6\u09cd\u09a7 \u0995\u09b0\u09c7 \u09a4\u09be\u09a6\u09c7\u09b0\u0995\u09c7 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

\u09aa\u09c2\u09b0\u09cd\u09ac\u09c7 \u0986\u09ae\u09b0\u09be \u099c\u09c7\u09a8\u09c7\u099b\u09bf, \u098f\u0995\u09be\u09a7\u09bf\u0995 \u0995\u09cb\u09a3\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u09c7\u09b0 \u09ae\u09be\u09a8 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7\u0964\"\u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995<\/span><\/p>\n

\u09af\u09c7\u09ae\u09a8: <\/span>\u03b8=30\u00b0,\u00a0150\u00b0,\u00a030\u00b0,\u00a0360\u00b0,\u00a0\u00b1360<\/span> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0995\u09cb\u09a3\u09c7\u09b0 \u099c\u09a8\u09cd\u09af,<\/span> \\sin \\theta <\/span><\/span>=\\frac{1}{2} <\/span><\/span>\u00a0\u09b9\u09ac\u09c7\u0964<\/span>\u00a0<\/span><\/p>\n

\\sin \\theta <\/span><\/span>=0, \\cos \\theta <\/span><\/span>=0, \\tan \\theta <\/span><\/span>=0, \\cot \\theta <\/span><\/span>=0, \\sin \\theta <\/span><\/span>=1<\/span> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8:<\/b><\/p>\n

(i) <\/b>\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \\sin \\theta <\/span><\/span>=\\frac{y}{r} <\/span><\/span>\u0964 \u098f\u0996\u09a8 \\sin \\theta <\/span><\/span>=0<\/span>\u00a0<\/span>\u09b9\u09b2\u09c7<\/span>\u00a0<\/span>y=0<\/span> \u09b9\u09ac\u09c7\u0964 \u0986\u09ac\u09be\u09b0,<\/span>\u00a0<\/span>y=0<\/span> \u09b9\u09ac\u09c7, \u09af\u09a6\u09bf \u0998\u09c2\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u09ae\u09be\u09a8 \u09b0\u09b6\u09cd\u09ae\u09bf \u09a4\u09be\u09b0 \u0986\u09a6\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 (<\/span>x<\/span>-\u0985\u0995\u09cd\u09b7) \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ae\u09bf\u09b6\u09c7 \u09af\u09be\u09af\u09bc\u0964 \u09aa\u09c1\u09a8\u09b0\u09be\u09af\u09bc \u0998\u09c2\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u09ae\u09be\u09a8 \u09b0\u09b6\u09cd\u09ae\u09bf \u0986\u09a6\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 (<\/span>x<\/span>-\u0985\u0995\u09cd\u09b7) \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 <\/span>0(<\/span>0\u00b0)<\/span>, \u00b1\u03c0(<\/span>\u00b1180)<\/span>,\u00b12\u03c0(<\/span>\u00b1360)<\/span>,\u00b13\u03c0(<\/span>\u00b1540)<\/span>\u2026\u2026<\/span> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09be\u09b0 \u09aa\u09b0 \u09a4\u09be\u09b0 \u0986\u09a6\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ae\u09bf\u09b6\u09c7 \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\n

\\sin \\theta <\/span>=0<\/span> \u09b9\u09b2\u09c7, \\theta <\/span><\/span>=n \\pi <\/span><\/span>\u00a0\u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7<\/span> n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

(ii) <\/b>\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \\cos \\theta <\/span><\/span>=\\frac{x}{r} <\/span><\/span>\u00a0<\/span>\u098f\u09ac\u0982 \\cos \\theta <\/span><\/span>=0<\/span> \u09b9\u09b2\u09c7<\/span> x <\/span>=0<\/span> \u09b9\u09ac\u09c7\u0964 \u0986\u09ac\u09be\u09b0<\/span> x <\/span>=0<\/span> \u09b9\u09ac\u09c7, \u09af\u09a6\u09bf \u0998\u09c2\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u09ae\u09be\u09a8 \u09b0\u09b6\u09cd\u09ae\u09bf <\/span>y <\/span><\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ae\u09bf\u09b6\u09c7 \u09af\u09be\u09af\u09bc\u0964 \u09aa\u09c1\u09a8\u09b0\u09be\u09af\u09bc \u0998\u09c2\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u09ae\u09be\u09a8 \u09b0\u09b6\u09cd\u09ae\u09bf \u0986\u09a6\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 (x <\/span><\/span>-\u0985\u0995\u09cd\u09b7) \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \\frac{\\pi}{2} <\/span><\/span>, \u00b13.\\frac{\\pi}{2} <\/span><\/span>, \u00b15.\\frac{\\pi}{2} <\/span><\/span> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09be\u09b0 \u09aa\u09b0 y <\/span><\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ae\u09bf\u09b6\u09c7 \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\n

\\cos \\theta <\/span>=0<\/span> \u09b9\u09b2\u09c7, \\theta <\/span><\/span>=(2n+1) <\/span>\\frac{\\pi}{2} <\/span><\/span>\u00a0\u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7<\/span> n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

(iii) \\tan \\theta <\/span><\/span><\/b>=0<\/span>\u00a0<\/span>\u09b9\u09b2\u09c7 \\frac{\\sin \\theta}{\\cos \\theta} <\/span><\/span>=0<\/span>\u00a0<\/span>\u0985\u09b0\u09cd\u09a5\u09be\u09ce \\sin \\theta <\/span><\/span>=0<\/span> \u09ab\u09b2\u09c7<\/span>\u00a0<\/span> \\theta <\/span><\/span>=n \\pi <\/span><\/span> \u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

\\tan \\theta <\/span>=0<\/span> \u09b9\u09b2\u09c7,\\theta <\/span><\/span>=n \\pi <\/span><\/span>\u00a0\u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

\u09a4\u09a6\u09cd\u09b0\u09c2\u09aa, <\/span>\\cot \\theta <\/span>=0<\/span>\u00a0<\/span>\u09b9\u09b2\u09c7 \\frac{\\cos \\theta}{\\sin \\theta} <\/span><\/span>=0<\/span>\u00a0<\/span>\u0985\u09b0\u09cd\u09a5\u09be\u09ce \\cos \\theta <\/span><\/span>=0<\/span> \u09ab\u09b2\u09c7<\/span> \\theta <\/span><\/span>=(2n+1) <\/span>\\frac{\\pi}{2} <\/span><\/span> \u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

\\cot \\theta <\/span>=0<\/span> \u09b9\u09b2\u09c7, \\theta <\/span><\/span>=(2n+1) <\/span>\\frac{\\pi}{2} <\/span><\/span>\u00a0\u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

\u0995\u09bf\u09a8\u09cd\u09a4\u09c1 |cosec\u03b8| <\/span><\/span>\u22651<\/span> \u098f\u09ac\u0982 |sec\u03b8| <\/span><\/span>\u22651<\/span> \u09ac\u09b2\u09c7cosec\u03b8 <\/span><\/span>\u00a0\u098f\u09ac\u0982<\/span> sec\u03b8 <\/span><\/span>\u00a0\u098f\u09b0 \u09ae\u09be\u09a8 \u0995\u0996\u09a8\u0993 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7 \u09a8\u09be \u0964<\/span><\/p>\n

(iv) \\sin \\theta <\/span><\/span><\/b>=1<\/span>\u00a0<\/span>\u09b9\u09b2\u09c7 \u0998\u09c2\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u09ae\u09be\u09a8 \u09b0\u09c7\u0996\u09be <\/span>y <\/span><\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09a6\u09bf\u0995<\/span>\u00a0<\/span>OY<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ae\u09bf\u09b6\u09c7 \u09af\u09be\u09af\u09bc \u098f\u09ac\u0982 \u098f\u09ac\u0982 \u09a4\u09be \u09a4\u09be\u09b0 \u0986\u09a6\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 <\/span>OX<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \\frac{\\pi}{2} <\/span><\/span>, -3\\frac{\\pi}{2} <\/span><\/span>, 2\u03c0+\\frac{\\pi}{2} <\/span><\/span>=5.\\frac{\\pi}{2} <\/span><\/span>, -2\u03c0-3\\frac{\\pi}{2} <\/span><\/span>=-7.\\frac{\\pi}{2} <\/span><\/span>, 4\u03c0+\\frac{\\pi}{2} <\/span><\/span>=9.\\frac{\\pi}{2} <\/span><\/span>\u2026\u2026<\/span> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

\\sin \\theta <\/span>=1<\/span> \u09b9\u09b2\u09c7,\\theta <\/span><\/span>=(4n+1) <\/span>\\frac{\\pi}{2} <\/span><\/span>\u00a0\u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

(v) \\sin \\theta <\/span><\/span><\/b>=-1<\/span>\u00a0<\/span>\u09b9\u09b2\u09c7 \u0998\u09c2\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u09ae\u09be\u09a8 \u09b0\u09c7\u0996\u09be <\/span>y<\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u098b\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09a6\u09bf\u0995<\/span>\u00a0<\/span>OY’<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ae\u09bf\u09b6\u09c7 \u09af\u09be\u09af\u09bc \u098f\u09ac\u0982 \u098f\u09ac\u0982 \u09a4\u09be \u09a4\u09be\u09b0 \u0986\u09a6\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 <\/span>OX<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 <\/span>3.\\frac{\\pi}{2} <\/span><\/span>, –\\frac{\\pi}{2} <\/span><\/span>-7.\\frac{\\pi}{2} <\/span><\/span>, -5.\\frac{\\pi}{2} <\/span><\/span>\u2026\u2026<\/span> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

\\sin \\theta <\/span>=-1<\/span> \u09b9\u09b2\u09c7, \\theta <\/span><\/span>=(4n-1)\\frac{\\pi}{2} <\/span><\/span>\u00a0\u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

(vi) \\cos \\theta <\/span><\/span><\/b>=1<\/span>\u00a0<\/span>\u09b9\u09b2\u09c7 \u0998\u09c2\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u09ae\u09be\u09a8 \u09b0\u09c7\u0996\u09be <\/span>x<\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09a6\u09bf\u0995<\/span>\u00a0<\/span>OX<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ae\u09bf\u09b6\u09c7 \u09af\u09be\u09af\u09bc \u098f\u09ac\u0982 \u098f\u09ac\u0982 \u09a4\u09be \u09a4\u09be\u09b0 \u0986\u09a6\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 <\/span>OX<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 <\/span>0,\u00a0\u00b12<\/span>,\u00a0\u00b14<\/span>,\u00a0\u00b16<\/span>\u2026\u2026<\/span> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

\\cos \\theta <\/span>=1<\/span> \u09b9\u09b2\u09c7, \\theta <\/span><\/span>=2n <\/span><\/span> \u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

(vii) \\cos \\theta <\/span><\/span><\/b>=-1<\/span>\u00a0<\/span>\u09b9\u09b2\u09c7 \u0998\u09c2\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u09ae\u09be\u09a8 \u09b0\u09c7\u0996\u09be <\/span>x<\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u098b\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09a6\u09bf\u0995<\/span>\u00a0<\/span>OX’<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ae\u09bf\u09b6\u09c7 \u09af\u09be\u09af\u09bc \u098f\u09ac\u0982 \u098f\u09ac\u0982 \u09a4\u09be \u09a4\u09be\u09b0 \u0986\u09a6\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 <\/span>OX<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 <\/span>\u00a0\u00b1{\\Pi} <\/span>,\u00b13{\\Pi} <\/span><\/span>, \u00b15{\\Pi} <\/span><\/span>\u2026\u2026<\/span> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

\\cos \\theta <\/span>=-1<\/span> \u09b9\u09b2\u09c7, \\theta <\/span><\/span>=(2n+1){\\Pi} <\/span><\/span> \u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

(viii) <\/b>\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,\\sin \\theta <\/span><\/span>=\\sin \\alpha <\/span>\u21d2\\sin \\theta <\/span>–\\sin \\alpha <\/span>=0<\/span><\/p>\n

\u21d22\\sin \\frac{1}{2}(\\theta-\\alpha) <\/span><\/span>–\\cos \\frac{1}{2}(\\theta+\\alpha) <\/span><\/span>=0<\/span>\u00a0<\/span><\/p>\n

\u2234\\sin \\frac{1}{2}(\\theta-\\alpha) <\/span><\/span>=0<\/span> \u09b9\u09b2\u09c7 \\frac{1}{2} <\/span>(\\theta-\\alpha) <\/span>=m \\pi <\/span><\/span>, \u09af\u09c7\u0996\u09be\u09a8\u09c7 m \\in \\mathbb{Z} <\/span><\/span><\/p>\n

\u21d2\\theta <\/span>=2m \\pi <\/span>+\\alpha <\/span>=2m \\pi <\/span>+(-1)^{2 m} [katex] <\/span>\\alpha [\/katex]<\/span>\u2026\u2026(<\/span>i)<\/span>;<\/span> \u098f\u0996\u09be\u09a8\u09c7 <\/span>2m <\/span><\/span> \u099c\u09cb\u09a1\u09bc \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964\u00a0<\/span><\/p>\n

\u0985\u09a5\u09ac\u09be, \\cos \\frac{1}{2}(\\theta+\\alpha) <\/span><\/span>=0<\/span> \u09b9\u09b2\u09c7 \\frac{1}{2} <\/span>(\\theta-\\alpha) <\/span><\/span>=(2 m+1) <\/span>\\frac{\\pi}{2} <\/span><\/span>,<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 m \\in \\mathbb{Z} <\/span><\/span><\/p>\n

\u21d2\\theta <\/span>=(2 m+1) \\pi <\/span>–\\alpha <\/span>=(2 m+1) \\pi <\/span>+(-1)^{2 m+1} <\/span>\\alpha <\/span><\/span>\u2026\u2026<\/span>ii<\/span>;<\/span> \u098f\u0996\u09be\u09a8\u09c7 <\/span>(2m+1) <\/span><\/span> \u099c\u09cb\u09a1\u09bc \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964\u00a0<\/span><\/p>\n

\u2234(i)<\/span> \u0993 <\/span>(ii)<\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, \\theta <\/span><\/span>=2 m \\pi <\/span>+(-1)^{2 m} <\/span>\\alpha <\/span><\/span>+(2m+1) <\/span>\\pi <\/span><\/span>+(2 m+1) \\pi+(-1)^{2 m+1} <\/span><\/span> \\alpha <\/span>\u09af\u09c7\u0996\u09be\u09a8\u09c7 m \\in \\mathbb{Z} <\/span><\/span> \u09b8\u09c1\u09a4\u09b0\u09be\u0982, <\/span>n <\/span><\/span> \u099c\u09cb\u09a1\u09bc \u09ac\u09be \u09ac\u09bf\u099c\u09cb\u09a1\u09bc \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u0985\u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09ac\u09be \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u099c\u09a8\u09cd\u09af \u0985\u09b0\u09cd\u09a5\u09be\u09ce n \\in \\mathbb{Z} <\/span><\/span>\u00a0\u098f\u09b0 \u099c\u09a8\u09cd\u09af \u09b2\u09bf\u0996\u09be \u09af\u09be\u09af\u09bc,<\/span> \\theta <\/span><\/span>=n \\pi <\/span>+(-1)^{n} \u00a0<\/span>\\alpha <\/span><\/span><\/p>\n

cosec\u03b8 <\/span>=cosec\u03b1 <\/span><\/span> \u09ac\u09be, \\sin \\theta <\/span><\/span>=\\sin \\alpha <\/span><\/span> \u09b9\u09b2\u09c7, \\theta <\/span><\/span>=n \\pi <\/span>+(-1)^{n} \u00a0<\/span>\\alpha <\/span><\/span>\u00a0\u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

(ix) <\/b>\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \\cos \\theta <\/span><\/span>=cos\u03b1 <\/span>\u21d2\\cos \\theta <\/span>–cos\u03b1 <\/span>=0<\/span><\/p>\n

\u21d22\\sin \\frac{1}{2}(\\theta+\\alpha) <\/span>\\sin \\frac{1}{2}(\\theta-\\alpha) <\/span><\/span>=0<\/span>\u00a0<\/span><\/p>\n

\u2234\\sin \\frac{1}{2}(\\theta+\\alpha) <\/span><\/span>=0\u21d2\\frac{1}{2}(\\theta+\\alpha) <\/span><\/span>=m\u03c0 <\/span>\u21d2\\theta <\/span>=2m\u03c0 <\/span>–\\alpha <\/span>\u2026\u2026(<\/span>i)<\/span>,<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 m \\in \\mathbb{Z} <\/span><\/span><\/p>\n

\u0985\u09a5\u09ac\u09be, \\sin \\frac{1}{2}(\\theta-\\alpha) <\/span><\/span>=0\u21d2\\frac{1}{2}(\\theta-\\alpha) <\/span><\/span>=m\u03c0 <\/span>\u21d2\\theta <\/span>=2m\u03c0 <\/span>+\\alpha <\/span>\u2026\u2026<\/span>ii<\/span>,<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 m \\in \\mathbb{Z} <\/span><\/span><\/p>\n

\u2234(i)<\/span> \u0993 <\/span>(ii)<\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, \\theta <\/span><\/span>=2m\u03c0 <\/span>\u00b1\\alpha <\/span>,<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 m \\in \\mathbb{Z} <\/span><\/span><\/p>\n

sec\u03b8 <\/span>=sec\u03b1 <\/span><\/span> \u09ac\u09be, \\cos \\theta <\/span><\/span>=cos\u03b1\u00a0<\/span>\u09b9\u09b2\u09c7, \\theta <\/span><\/span>=2m\u03c0 <\/span>\u00b1\\alpha <\/span><\/span>\u00a0\u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n

(x) <\/b>\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>tan\u03b8 <\/span>=tan\u03b1 <\/span>\u21d2<\/span>\\frac{\\sin \\theta}{\\cos \\theta} <\/span><\/span>–\\frac{\\sin \\alpha}{\\cos \\alpha} <\/span><\/span>=0<\/span><\/p>\n

\u21d2\\frac{\\sin \\theta \\cos \\alpha-\\sin \\alpha \\cos \\theta}{\\cos \\theta \\cos \\alpha} <\/span><\/span>=0<\/span>\u00a0<\/span><\/p>\n

\u21d2\\frac{\\sin (\\theta-\\alpha)}{\\cos \\theta \\cos \\alpha} <\/span>=0\u21d2<\/span>sin <\/span>(\\theta-\\alpha) <\/span><\/span> =0\u21d2(\\theta-\\alpha) <\/span>=n\u03c0 <\/span>\u21d2\\theta <\/span>=n\u03c0 <\/span>+\\alpha <\/span>,<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 m \\in \\mathbb{Z} <\/span><\/span><\/p>\n

cot\u03b8 <\/span>=cot\u03b1 <\/span><\/span> \u09ac\u09be, <\/span>tan\u03b8 <\/span>=tan\u03b1 <\/span>\u00a0<\/span>\u09b9\u09b2\u09c7, \\theta <\/span><\/span>=n\u03c0 <\/span>+\\alpha <\/span><\/span>\u00a0\u09b9\u09ac\u09c7, \u09af\u09c7\u0996\u09be\u09a8\u09c7 n \\in \\mathbb{Z} <\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

\u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (Trigonometric equations) (i) \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u0997\u09c1\u09b2\u09cb\u0995\u09c7 \u09b8\u09b0\u09b2\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u0995\u09bf\u0982\u09ac\u09be \u0995\u09bf\u0982\u09ac\u09be \u09a8\u09c7\u0993\u09af\u09bc\u09be \u09b8\u09c1\u09ac\u09bf\u09a7\u09be\u099c\u09a8\u0995\u0964 (ii) \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0989\u09ad\u09af\u09bc\u09aa\u0995\u09cd\u09b7\u0995\u09c7 \u09ac\u09b0\u09cd\u0997 \u0995\u09b0\u09c7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u098f\u09a1\u09bc\u09bf\u09af\u09bc\u09c7 \u09af\u09be\u0993\u09af\u09bc\u09be \u09ad\u09be\u09b2\u0964 \u098f\u0995\u09be\u09a8\u09cd\u09a4 \u09b8\u09ae\u09cd\u09ad\u09ac \u09a8\u09be \u09b9\u09b2\u09c7 \u09a6\u09c7\u0996\u09a4\u09c7 \u09b9\u09ac\u09c7 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 \u0989\u09a4\u09cd\u09a4\u09b0\u0997\u09c1\u09b2\u09cb \u09af\u09a5\u09be\u09af\u09a5 \u0995\u09bf\u09a8\u09be \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09b6\u09c1\u09a6\u09cd\u09a7\u09bf \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7 \u098f\u09ac\u0982 \u0985\u09ac\u09be\u09a8\u09cd\u09a4\u09b0 \u09ae\u09c2\u09b2\u0997\u09c1\u09b2\u09cb<\/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":[2498,2505,1592,2504,2506],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3619"}],"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=3619"}],"version-history":[{"count":13,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3619\/revisions"}],"predecessor-version":[{"id":7884,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3619\/revisions\/7884"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=3619"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=3619"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=3619"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}