{"id":100698,"date":"2022-02-12T19:11:21","date_gmt":"2022-02-12T13:41:21","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=100698"},"modified":"2024-02-21T17:00:15","modified_gmt":"2024-02-21T11:30:15","slug":"trigonometric-equations","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/study-material\/trigonometric-equations\/","title":{"rendered":"Trigonometric Equations"},"content":{"rendered":"

Common Trigonometric equations and inequalities<\/h2>\n
    \n
  1. If sin \u03b8 = sin \u03b1 \u21d2 \u03b8 = n \u03c0 + (\u22121)n<\/sup> \u03b1 where \u03b1 \u2208 \\[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\] , n \u2208 I.<\/li>\n
  2. If cos \u03b8 = cos \u03b1 \u21d2 \u03b8 = 2 n \u03c0 \u00b1 \u03b1 where \u03b1 \u2208 [0 , \u03c0] , n \u2208 I .<\/li>\n
  3. If tan \u03b8 = tan \u03b1 \u21d2 \u03b8 = n \u03c0 + \u03b1 where \u03b1 \u2208 (\\(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\)) , n \u2208 I<\/li>\n
  4. If sin\u00b2 \u03b8 = sin\u00b2 \u03b1 \u21d2 \u03b8 = n \u03c0 \u00b1 \u03b1.<\/li>\n
  5. cos\u00b2 \u03b8 = cos\u00b2 \u03b1 \u21d2 \u03b8 = n \u03c0 \u00b1 \u03b1.<\/li>\n
  6. tan\u00b2 \u03b8 = tan\u00b2 \u03b1 \u21d2 \u03b8 = n \u03c0 \u00b1 \u03b1. [ Note: \u03b1 is called the principal angle]<\/li>\n<\/ol>\n

    Types Of Trigonometric Equations:<\/strong><\/p>\n

    1. Solutions of equations by factorising<\/strong><\/p>\n

    Consider the equation,
    \n(2 sin x \u2212 cos x) (1 + cos x) = sin\u00b2 x<\/p>\n

    cotx \u2013 cosx = 1 \u2013 cotx cosx<\/p>\n

    2. Solutions of equations reducible to quadratic equations<\/strong><\/p>\n

    Consider the equation,
    \n3 cos\u00b2 x \u2212 10 cos x + 3 = 0 and 2 sin2x + \\(\\sqrt {3}\\) sinx + 1 = 0<\/p>\n

    3. Solving equations by introducing an Auxilliary argument<\/strong><\/p>\n

    Consider the equation,<\/p>\n

    sin x + cos x = \\(\\sqrt{2} ; \\sqrt{3}\\) cos x + sin x = 2 ; secx \u2013 1 = (\\(\\sqrt {2}\\)-1) tanx<\/p>\n

    4. Solving equations by Transforming a sum of Trigonometric functions into a product<\/strong><\/p>\n

    Consider the example,<\/p>\n

    cos 3 x + sin 2 x \u2212 sin 4 x = 0
    \nsin\u00b2x + sin\u00b22x + sin\u00b23x + sin\u00b24x = 2<\/p>\n

    sinx + sin5x = sin2x + sin4x<\/p>\n

    5. Solving equations by transforming a product of trigonometric functions into a sum<\/strong><\/p>\n

    Consider the equation,<\/p>\n

    sin 5 x . cos 3 x = sin 6x .cos 2x<\/p>\n

    8cosx cos2x cos4x = \\(\\frac{\\sin 6 x}{\\sin x}\\) ; sin3\u03b8 = 4sin\u03b8 sin2\u03b8 sin4\u03b8<\/p>\n

    6. Solving equations by a change of variable:<\/strong><\/p>\n

    (i) Equations of the form of a . sin x + b . cos x + d = 0 , where a , b & d are real numbers & a , b \u2260 0 can be solved by changing sin x <\/i>& cos x <\/i>into their corresponding tangent of half the angle. Consider the equation 3 cos x + 4 sin x = 5.
    \n(ii) Many equations can be solved by introducing a new variable . eg. the equation sin4 2x + cos4 2x = sin 2x . cos 2x changes to 2 (y + 1) (y \u2212 \\(\\frac {1}{2}\\)) = 0 by substituting , sin 2 x <\/i>. cos 2 x <\/i>= y.<\/p>\n

      \n
    1. \n
        \n
      1. Solving equations with the use of the Boundness of the functions sin x & cos x or by making two perfect squares. Consider the equations:
        \nTrigonometric Equestion
        \n\\(\\begin{array}{l}{\\sin x\\left(\\cos \\frac{x}{4}-2 \\sin x\\right)+\\left(1+\\sin \\frac{x}{4}-2 \\cos x\\right) \\cdot \\cos x=0} \\\\ {\\sin ^{2} x+2 \\tan ^{2} x+\\frac{4}{\\sqrt{3}} \\tan x-\\sin x+\\frac{11}{12}=0}\\end{array}\\)<\/li>\n<\/ol>\n<\/li>\n
      2. Trigonometric Inequalities:<\/strong>
        \nThere is no general rule to solve Trigonometric inequations and the same rules of algebra are valid except the domain and range of trigonometric functions should be kept in mind.
        \nConsider the examples:
        \n\\(\\log _{2}\\left(\\sin \\frac{x}{2}\\right)<-1 ; \\sin x\\left(\\cos x+\\frac{1}{2}\\right) \\leq 0 ; \\sqrt{5-2 \\sin 2 x} \\geq 6 \\sin x-1\\)<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

        Common Trigonometric equations and inequalities If sin \u03b8 = sin \u03b1 \u21d2 \u03b8 = n \u03c0 + (\u22121)n \u03b1 where […]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"trigonometric equations","_yoast_wpseo_title":"Trigonometric solutions of equations and inequalities class 11 Maths","_yoast_wpseo_metadesc":"Trigonometric equations are mathematical expressions that involve trigonometric functions, such as sine, cosine, and tangent.","custom_permalink":"study-material\/trigonometric-equations\/"},"categories":[55,13,21],"tags":[],"table_tags":[],"acf":[],"yoast_head":"\nTrigonometric solutions of equations and inequalities class 11 Maths<\/title>\n<meta name=\"description\" content=\"Trigonometric equations are mathematical expressions that involve trigonometric functions, such as sine, cosine, and tangent.\" \/>\n<meta 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