{"id":1817,"date":"2025-07-28T20:27:24","date_gmt":"2025-07-28T20:27:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1817"},"modified":"2025-08-13T03:05:30","modified_gmt":"2025-08-13T03:05:30","slug":"the-other-trigonometric-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-other-trigonometric-functions-learn-it-2\/","title":{"raw":"The Other Trigonometric Functions: Learn It 2","rendered":"The Other Trigonometric Functions: Learn It 2"},"content":{"raw":"

Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent<\/h2>\r\nWe can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle<\/strong> formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x<\/em>- and y<\/em>-values in the original quadrant.\r\n\r\n
To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase \"A Smart Trig Class.\" Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is \"A<\/strong>,\" a<\/u><\/strong>ll of the six trigonometric functions are positive. In quadrant II, \"S<\/strong>mart,\" only s<\/u><\/strong>ine and its reciprocal function, cosecant, are positive. In quadrant III, \"T<\/strong>rig,\" only t<\/u><\/strong>angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, \"C<\/strong>lass,\" only c<\/u><\/strong>osine and its reciprocal function, secant, are positive.\"Graph\r\n\r\n<\/section>
How To: Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions.\r\n<\/strong>\r\n
    \r\n \t
  1. Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle.<\/li>\r\n \t
  2. Evaluate the function at the reference angle.<\/li>\r\n \t
  3. Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.<\/li>\r\n<\/ol>\r\n<\/section>
    Use reference angles to find all six trigonometric functions of [latex]-\\frac{5\\pi }{6}[\/latex].[reveal-answer q=\"720177\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"720177\"]\r\n\r\nThe angle between this angle\u2019s terminal side and the x<\/em>-axis is [latex]\\frac{\\pi }{6}[\/latex], so that is the reference angle. Since [latex]-\\frac{5\\pi }{6}[\/latex] is in the third quadrant, where both [latex]x[\/latex] and [latex]y[\/latex] are negative, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive.\r\n

    [latex]\\begin{gathered}\\cos \\left(-\\frac{5\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2} \\\\ \\sin \\left(-\\frac{5\\pi }{6}\\right)=-\\frac{1}{2} \\\\ \\tan\\left(-\\frac{5\\pi }{6}\\right)=\\frac{\\sqrt{3}}{3} \\\\ \\sec\\left(-\\frac{5\\pi }{6}\\right)=-\\frac{2\\sqrt{3}}{3}\\\\ \\csc\\left(-\\frac{5\\pi }{6}\\right)=-2\\\\ \\cot \\left(-\\frac{5\\pi }{6}\\right)=\\sqrt{3} \\end{gathered}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    \r\n
    \r\n\r\nUse reference angles to find all six trigonometric functions of [latex]-\\frac{7\\pi }{4}[\/latex].\r\n\r\n[reveal-answer q=\"621482\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"621482\"]\r\n\r\n[latex]\\sin \\left(-\\frac{7\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2},\\cos \\left(-\\frac{7\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2},\\tan \\left(-\\frac{7\\pi }{4}\\right)=1[\/latex],\r\n[latex]\\sec \\left(-\\frac{7\\pi }{4}\\right)=\\sqrt{2},\\csc \\left(-\\frac{7\\pi }{4}\\right)=\\sqrt{2},\\cot \\left(-\\frac{7\\pi }{4}\\right)=1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>
    [ohm_question hide_question_numbers=1]100617[\/ohm_question]<\/section>","rendered":"

    Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent<\/h2>\n

    We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle<\/strong> formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x<\/em>– and y<\/em>-values in the original quadrant.<\/p>\n

    To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “A<\/strong>,” a<\/u><\/strong>ll of the six trigonometric functions are positive. In quadrant II, “S<\/strong>mart,” only s<\/u><\/strong>ine and its reciprocal function, cosecant, are positive. In quadrant III, “T<\/strong>rig,” only t<\/u><\/strong>angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “C<\/strong>lass,” only c<\/u><\/strong>osine and its reciprocal function, secant, are positive.\"Graph<\/p>\n<\/section>\n
    How To: Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions.
    \n<\/strong><\/p>\n
      \n
    1. Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle.<\/li>\n
    2. Evaluate the function at the reference angle.<\/li>\n
    3. Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.<\/li>\n<\/ol>\n<\/section>\n
      Use reference angles to find all six trigonometric functions of [latex]-\\frac{5\\pi }{6}[\/latex].<\/p>\n