{"id":13747,"date":"2018-06-14T23:49:09","date_gmt":"2018-06-14T23:49:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/finding-coterminal-angles\/"},"modified":"2021-11-22T21:10:34","modified_gmt":"2021-11-22T21:10:34","slug":"coterminal-angles","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/coterminal-angles\/","title":{"raw":"Coterminal Angles","rendered":"Coterminal Angles"},"content":{"raw":"Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of 0\u00b0 to 360\u00b0, or 0 to [latex]2\\pi [\/latex]. It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.\r\n\r\nIt is possible for more than one angle to have the same terminal side. Look at Figure 16. The angle of 140\u00b0 is a positive angle<\/strong>, measured counterclockwise. The angle of \u2013220\u00b0 is a negative angle<\/strong>, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles<\/strong>. Every angle greater than 360\u00b0 or less than 0\u00b0 is coterminal with an angle between 0\u00b0 and 360\u00b0, and it is often more convenient to find the coterminal angle within the range of 0\u00b0 to 360\u00b0 than to work with an angle that is outside that range.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Figure 16.<\/b> An angle of 140\u00b0 and an angle of \u2013220\u00b0 are coterminal angles.[\/caption]\r\n\r\nThis video shows examples of how to determine if two angles are coterminal.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=TuyF8fFg3B0\r\n\r\nAny angle has infinitely many coterminal angles<\/strong> because each time we add 360\u00b0 to that angle\u2014or subtract 360\u00b0 from it\u2014the resulting value has a terminal side in the same location. For example, 100\u00b0 and 460\u00b0 are coterminal for this reason, as is \u2212260\u00b0. Recognizing that any angle has infinitely many coterminal angles explains the repetitive shape in the graphs of trigonometric functions.\r\n\r\nAn angle\u2019s reference angle is the measure of the smallest, positive, acute angle [latex]t[\/latex] formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See Figure 17\u00a0for examples of reference angles for angles in different quadrants.\r\n<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Four Figure 17<\/b>[\/caption]\r\n\r\n \r\n
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A General Note: Coterminal and Reference Angles<\/h3>\r\nCoterminal angles are two angles in standard position that have the same terminal side.\r\n\r\nAn angle\u2019s reference angle<\/strong> is the size of the smallest acute angle, [latex]{t}^{\\prime }[\/latex], formed by the terminal side of the angle [latex]t[\/latex]\u00a0and the horizontal axis.\r\n\r\n<\/div>\r\n
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How To: Given an angle greater than 360\u00b0, find a coterminal angle between 0\u00b0 and 360\u00b0.<\/h3>\r\n
    \r\n \t
  1. Subtract 360\u00b0 from the given angle.<\/li>\r\n \t
  2. If the result is still greater than 360\u00b0, subtract 360\u00b0 again till the result is between 0\u00b0 and 360\u00b0.<\/li>\r\n \t
  3. The resulting angle is coterminal with the original angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Example 5: Finding an Angle Coterminal with an Angle of Measure Greater Than 360\u00b0<\/h3>\r\nFind the least positive angle [latex]\\theta [\/latex] that is coterminal with an angle measuring 800\u00b0, where [latex]0^\\circ \\le \\theta <360^\\circ [\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nAn angle with measure 800\u00b0 is coterminal with an angle with measure 800 \u2212 360 = 440\u00b0, but 440\u00b0 is still greater than 360\u00b0, so we subtract 360\u00b0 again to find another coterminal angle: 440 \u2212 360 = 80\u00b0.\r\n\r\nThe angle [latex]\\theta =80^\\circ [\/latex] is coterminal with 800\u00b0. To put it another way, 800\u00b0 equals 80\u00b0 plus two full rotations, as shown in Figure 18.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Figure 18<\/b>[\/caption]\r\n\r\n<\/div>\r\n
    \r\n

    Try It 5<\/h3>\r\nFind an angle [latex]\\alpha [\/latex] that is coterminal with an angle measuring 870\u00b0, where [latex]0^\\circ \\le \\alpha <360^\\circ [\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n
    \r\n

    How To: Given an angle with measure less than 0\u00b0, find a coterminal angle having a measure between 0\u00b0 and 360\u00b0.<\/h3>\r\n
      \r\n \t
    1. Add 360\u00b0 to the given angle.<\/li>\r\n \t
    2. If the result is still less than 0\u00b0, add 360\u00b0 again until the result is between 0\u00b0 and 360\u00b0.<\/li>\r\n \t
    3. The resulting angle is coterminal with the original angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Example 6: Finding an Angle Coterminal with an Angle Measuring Less Than 0\u00b0<\/h3>\r\nShow the angle with measure \u221245\u00b0 on a circle and find a positive coterminal angle [latex]\\alpha [\/latex] such that 0\u00b0 \u2264 \u03b1<\/em> < 360\u00b0.\r\n\r\n<\/div>\r\n
      \r\n

      Solution<\/h3>\r\nSince 45\u00b0 is half of 90\u00b0, we can start at the positive horizontal axis and measure clockwise half of a 90\u00b0 angle.\r\n\r\nBecause we can find coterminal angles by adding or subtracting a full rotation of 360\u00b0, we can find a positive coterminal angle here by adding 360\u00b0:\r\n
      [latex]-45^\\circ +360^\\circ =315^\\circ [\/latex]<\/div>\r\nWe can then show the angle on a circle, as in Figure 19.\r\n<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Figure 19<\/strong>[\/caption]\r\n\r\n<\/div>\r\n
      \r\n\r\nWatch this video for another example of how to determine positive and negative coterminal angles.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=m7jTGVVzb0s\r\n\r\n<\/div>\r\n
      \r\n

      Try It 6<\/h3>\r\nFind an angle [latex]\\beta [\/latex] that is coterminal with an angle measuring \u2212300\u00b0 such that [latex]0^\\circ \\le \\beta <360^\\circ [\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n

      Finding Coterminal Angles Measured in Radians<\/h2>\r\nWe can find coterminal angles<\/strong> measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.\r\n
      \r\n\r\nGiven an angle greater than<\/strong> [latex]2\\pi [\/latex], find a coterminal angle between 0 and<\/strong> [latex]2\\pi [\/latex].\r\n
        \r\n \t
      1. Subtract [latex]2\\pi [\/latex] from the given angle.<\/li>\r\n \t
      2. If the result is still greater than [latex]2\\pi [\/latex], subtract [latex]2\\pi [\/latex] again until the result is between [latex]0[\/latex] and [latex]2\\pi [\/latex].<\/li>\r\n \t
      3. The resulting angle is coterminal with the original angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
        \r\n

        Example 7: Finding Coterminal Angles Using Radians<\/h3>\r\nFind an angle [latex]\\beta [\/latex] that is coterminal with [latex]\\frac{19\\pi }{4}[\/latex], where [latex]0\\le \\beta <2\\pi [\/latex].\r\n\r\n<\/div>\r\n
        \r\n

        Solution<\/h3>\r\nWhen working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of [latex]2\\pi [\/latex] radians:\r\n
        [latex]\\begin{array}{l}\\frac{19\\pi }{4}-2\\pi =\\frac{19\\pi }{4}-\\frac{8\\pi }{4}\\hfill \\\\ =\\frac{11\\pi }{4}\\hfill \\end{array}[\/latex]<\/div>\r\nThe angle [latex]\\frac{11\\pi }{4}[\/latex] is coterminal, but not less than [latex]2\\pi [\/latex], so we subtract another rotation:\r\n
        [latex]\\begin{array}{l}\\frac{11\\pi }{4}-2\\pi =\\frac{11\\pi }{4}-\\frac{8\\pi }{4}\\hfill \\\\ =\\frac{3\\pi }{4}\\hfill \\end{array}[\/latex]<\/div>\r\nThe angle [latex]\\frac{3\\pi }{4}[\/latex] is coterminal with [latex]\\frac{19\\pi }{4}[\/latex], as shown in Figure 20.\r\n<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Figure 20<\/b>[\/caption]\r\n\r\n<\/div>\r\n
        \r\n

        Try It 7<\/h3>\r\nFind an angle of measure [latex]\\theta [\/latex] that is coterminal with an angle of measure [latex]-\\frac{17\\pi }{6}[\/latex] where [latex]0\\le \\theta <2\\pi [\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

        Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of 0\u00b0 to 360\u00b0, or 0 to [latex]2\\pi[\/latex]. It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.<\/p>\n

        It is possible for more than one angle to have the same terminal side. Look at Figure 16. The angle of 140\u00b0 is a positive angle<\/strong>, measured counterclockwise. The angle of \u2013220\u00b0 is a negative angle<\/strong>, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles<\/strong>. Every angle greater than 360\u00b0 or less than 0\u00b0 is coterminal with an angle between 0\u00b0 and 360\u00b0, and it is often more convenient to find the coterminal angle within the range of 0\u00b0 to 360\u00b0 than to work with an angle that is outside that range.<\/p>\n

        \"A<\/p>\n

        Figure 16.<\/b> An angle of 140\u00b0 and an angle of \u2013220\u00b0 are coterminal angles.<\/p>\n<\/div>\n

        This video shows examples of how to determine if two angles are coterminal.<\/p>\n