{"id":1792,"date":"2025-07-28T19:39:44","date_gmt":"2025-07-28T19:39:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1792"},"modified":"2026-04-13T16:46:46","modified_gmt":"2026-04-13T16:46:46","slug":"angles-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/angles-learn-it-3\/","title":{"raw":"Angles: Learn It 3","rendered":"Angles: Learn It 3"},"content":{"raw":"

Identifying Special Angles Measured in Radians<\/h2>\r\nIn addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. Memorizing these angles will be very useful as we study the properties associated with angles.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Commonly encountered angles measured in degrees[\/caption]\r\n\r\n
[ohm_question hide_question_numbers=1]322639[\/ohm_question]<\/section>\r\n

Converting between Radians and Degrees<\/h2>\r\nBecause degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion.\r\n
\r\n

converting between radians and degrees<\/h3>\r\nTo convert between degrees and radians, use the proportion [latex]\\dfrac{\\theta }{180}=\\frac{{\\theta }^{R}}{\\pi }[\/latex]\r\n
    \r\n \t
  • radian [latex]\\times \\dfrac{180}{\\pi}[\/latex]<\/li>\r\n \t
  • degree [latex]\\times \\dfrac{\\pi}{180}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n
    Convert each radian measure to degrees.\r\n

    a. [latex]\\frac{\\pi }{6}[\/latex]<\/p>\r\n

    b. 3<\/p>\r\n[reveal-answer q=\"620590\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"620590\"]\r\n

    To convert radians to degrees, multiply by [latex]\\frac{180}{\\pi}[\/latex].<\/p>\r\n

    a. [latex]\\begin{gathered} \\frac{\\pi}{6} \\times \\frac{180}{\\pi} \\ =\\frac{180\\pi}{6\\pi} \\ =\\frac{180}{6} \\ =30^{\\circ} \\end{gathered}[\/latex]<\/p>\r\n

    b. [latex]\\begin{gathered} 3 \\times \\frac{180}{\\pi} \\ =\\frac{540}{\\pi} \\ \\approx 172^{\\circ} \\end{gathered}[\/latex]<\/p>\r\n

    In part (a), the [latex]\\pi[\/latex] values cancel and the fraction reduces to a whole number. In part (b), since 3 is not a rational multiple of [latex]\\pi[\/latex], we divide to get an approximate decimal answer.<\/p>\r\n[\/hidden-answer]<\/span>\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]322643[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]322645[\/ohm_question]<\/section>
    Convert [latex]15[\/latex] degrees to radians.[reveal-answer q=\"867109\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867109\"]\r\n

    To convert degrees to radians, multiply by [latex]\\frac{\\pi}{180}[\/latex].<\/p>\r\n

    [latex]\\begin{gathered} 15 \\times \\frac{\\pi}{180} \\ =\\frac{15\\pi}{180} \\ =\\frac{\\pi}{12} \\end{gathered}[\/latex]<\/p>\r\n

    Both 15 and 180 share a common factor of 15, so dividing numerator and denominator by 15 gives the final answer of [latex]\\frac{\\pi}{12}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]322647[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]322648[\/ohm_question]<\/section><\/div>\r\n
    \r\n
    [ohm_question hide_question_numbers=1]322649[\/ohm_question]<\/section><\/div>\r\n<\/div>","rendered":"

    Identifying Special Angles Measured in Radians<\/h2>\n

    In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. Memorizing these angles will be very useful as we study the properties associated with angles.<\/p>\n

    \"A
    Commonly encountered angles measured in degrees<\/figcaption><\/figure>\n