{"id":1793,"date":"2025-07-28T19:39:46","date_gmt":"2025-07-28T19:39:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1793"},"modified":"2025-10-08T15:58:09","modified_gmt":"2025-10-08T15:58:09","slug":"angles-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/angles-learn-it-4\/","title":{"raw":"Angles: Learn It 5","rendered":"Angles: Learn It 5"},"content":{"raw":"
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Finding Coterminal Angles<\/h2>\r\nConverting 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\u00a0positive angle<\/strong>, measured counterclockwise. The angle of \u2013220\u00b0 is a\u00a0negative 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\u00a0coterminal 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 An angle of 140\u00b0 and an angle of \u2013220\u00b0 are coterminal angles.[\/caption]\r\n\r\n
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coterminal angles<\/h3>\r\nTwo angles in standard position have the same terminal side.\r\n\r\n<\/section>Any angle has infinitely many\u00a0coterminal angles<\/strong>\u00a0because 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.\r\n\r\n\"Four\r\n\r\n
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reference angles<\/h3>\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] and the horizontal axis.\r\n\r\n<\/section>
How To: Given an angle greater than 360\u00b0, find a coterminal angle between 0\u00b0 and 360\u00b0<\/strong>\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<\/section>
    Find 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].[reveal-answer q=\"964169\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"964169\"]An 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.The 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.\"A[\/hidden-answer]\r\n\r\n<\/section>
    \r\n
    \r\n\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\n[reveal-answer q=\"363809\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"363809\"]\r\n\r\n[latex]\\alpha =150^\\circ [\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>
    How To: Given an angle with measure less than 0\u00b0, find a coterminal angle having a measure between 0\u00b0 and 360\u00b0.<\/b>\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<\/section>
      Show 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.[reveal-answer q=\"713003\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"713003\"]Since 45\u00b0 is half of 90\u00b0, we can start at the positive horizontal axis and measure clockwise half of a 90\u00b0 angle.Because 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]<\/p>\r\n\"A\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

      Find an angle [latex]\\beta [\/latex] that is coterminal with an angle measuring \u2212300\u00b0 such that [latex]0^\\circ \\le \\beta <360^\\circ [\/latex].[reveal-answer q=\"113901\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"113901\"][latex]\\beta =60^\\circ [\/latex][\/hidden-answer]<\/section><\/div>\r\n
      How To: Finding a reference angle<\/b>\r\n
        \r\n \t
      1. First find the coterminal angle between 0\u00b0 and 360\u00b0<\/li>\r\n \t
      2. Find the angle between the terminal side and the nearest [latex]x[\/latex]-axis.\r\n
          \r\n \t
        • For angles in the second quadrant: subtract the angle from 180\u00b0<\/li>\r\n \t
        • For angles in the third quadrant: subtract 180\u00b0<\/li>\r\n \t
        • For angles in the fourth quadrant: subtract the angle from 360<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section><\/section>\r\n
          [ohm_question hide_question_numbers=1]147466[\/ohm_question]<\/section>\r\n

          Finding Coterminal Angles Measured in Radians<\/h2>\r\n
          Given 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<\/section>\r\n
            Find an angle [latex]\\beta [\/latex] that is coterminal with [latex]\\frac{19\\pi }{4}[\/latex], where [latex]0\\le \\beta <2\\pi [\/latex].[reveal-answer q=\"742225\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"742225\"]When 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{align}\\frac{19\\pi }{4}-2\\pi =\\frac{19\\pi }{4}-\\frac{8\\pi }{4} =\\frac{11\\pi }{4} \\end{align}[\/latex]<\/p>\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{align}\\frac{11\\pi }{4}-2\\pi =\\frac{11\\pi }{4}-\\frac{8\\pi }{4} =\\frac{3\\pi }{4} \\end{align}[\/latex]<\/p>\r\n\"A\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

            Find 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].[reveal-answer q=\"436085\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"436085\"][latex]\\frac{7\\pi }{6}[\/latex][\/hidden-answer]<\/section>
            [ohm_question hide_question_numbers=1]99894[\/ohm_question]<\/section><\/div>\r\n<\/div>","rendered":"
            \n

            Finding Coterminal Angles<\/h2>\n

            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\u00a0positive angle<\/strong>, measured counterclockwise. The angle of \u2013220\u00b0 is a\u00a0negative 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\u00a0coterminal 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
            An angle of 140\u00b0 and an angle of \u2013220\u00b0 are coterminal angles.<\/figcaption><\/figure>\n
            \n

            coterminal angles<\/h3>\n

            Two angles in standard position have the same terminal side.<\/p>\n<\/section>\n

            Any angle has infinitely many\u00a0coterminal angles<\/strong>\u00a0because 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.<\/p>\n

            An 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.<\/p>\n

            \"Four<\/p>\n

            \n

            reference angles<\/h3>\n

            An 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] and the horizontal axis.<\/p>\n<\/section>\n

            How To: Given an angle greater than 360\u00b0, find a coterminal angle between 0\u00b0 and 360\u00b0<\/strong><\/p>\n
              \n
            1. Subtract 360\u00b0 from the given angle.<\/li>\n
            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>\n
            3. The resulting angle is coterminal with the original angle.<\/li>\n<\/ol>\n<\/section>\n
              Find 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].\n\n