{"id":1815,"date":"2025-07-28T20:27:38","date_gmt":"2025-07-28T20:27:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1815"},"modified":"2025-08-13T03:06:29","modified_gmt":"2025-08-13T03:06:29","slug":"the-other-trigonometric-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-other-trigonometric-functions-learn-it-4\/","title":{"raw":"The Other Trigonometric Functions: Learn It 4","rendered":"The Other Trigonometric Functions: Learn It 4"},"content":{"raw":"

Recognize and Use Fundamental Identities<\/h2>\r\nWe have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.\r\n\r\n
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fundamental identities<\/h3>\r\nWe can derive some useful identities<\/strong> from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:\r\n

[latex]\\tan t=\\frac{\\sin t}{\\cos t}[\/latex]<\/p>\r\n

[latex]\\sec t=\\frac{1}{\\cos t}[\/latex]<\/p>\r\n

[latex]\\csc t=\\frac{1}{\\sin t}[\/latex]<\/p>\r\n

[latex]\\cot t=\\frac{1}{\\tan t}=\\frac{\\cos t}{\\sin t}[\/latex]<\/p>\r\n\r\n<\/section>

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  1. Given [latex]\\sin \\left(45^\\circ \\right)=\\frac{\\sqrt{2}}{2},\\cos \\left(45^\\circ \\right)=\\frac{\\sqrt{2}}{2}[\/latex], evaluate [latex]\\tan \\left(45^\\circ \\right)[\/latex].<\/li>\r\n \t
  2. Given [latex]\\sin \\left(\\frac{5\\pi }{6}\\right)=\\frac{1}{2},\\cos\\left(\\frac{5\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex], evaluate [latex]\\sec \\left(\\frac{5\\pi }{6}\\right)[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"581515\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"581515\"]\r\n\r\nBecause we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.\r\n
      \r\n \t
    1. \r\n
      [latex]\\begin{align}\\tan \\left(45^\\circ \\right)=\\frac{\\sin \\left(45^\\circ \\right)}{\\cos \\left(45^\\circ \\right)} =\\frac{\\frac{\\sqrt{2}}{2}}{\\frac{\\sqrt{2}}{2}} =1 \\end{align}[\/latex]<\/div><\/li>\r\n \t
    2. \r\n
      [latex]\\begin{align}\\sec \\left(\\frac{5\\pi }{6}\\right)=\\frac{1}{\\cos \\left(\\frac{5\\pi }{6}\\right)} =\\frac{1}{-\\frac{\\sqrt{3}}{2}} =\\frac{-2\\sqrt{3}}{1} =\\frac{-2}{\\sqrt{3}} =-\\frac{2\\sqrt{3}}{3} \\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
      Evaluate [latex]\\csc\\left(\\frac{7\\pi }{6}\\right)[\/latex].[reveal-answer q=\"608392\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"608392\"][latex]-2[\/latex][\/hidden-answer]<\/section>
      Simplify [latex]\\frac{\\sec t}{\\tan t}[\/latex].[reveal-answer q=\"803642\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"803642\"]We can simplify this by rewriting both functions in terms of sine and cosine.\r\n

      [latex]\\begin{align} \\frac{\\sec t}{\\tan t}&=\\frac{\\frac{1}{\\cos t}}{\\frac{\\sin t}{\\cos t}}&& \\text{To divide the functions, we multiply by the reciprocal.} \\\\&=\\frac{1}{\\cos t}\\frac{\\cos t}{\\sin t}&&\\text{Divide out the cosines.} \\\\ &=\\frac{1}{\\sin t}&&\\text{Simplify and use the identity.}\\\\ &=\\csc t \\end{align}[\/latex]<\/p>\r\nBy showing that [latex]\\frac{\\sec t}{\\tan t}[\/latex] can be simplified to [latex]\\csc t[\/latex], we have, in fact, established a new identity.\r\n

      [latex]\\frac{\\sec t}{\\tan t}=\\csc t[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

      Simplify [latex]\\tan t\\left(\\cos t\\right)[\/latex].[reveal-answer q=\"232307\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232307\"][latex]\\sin t[\/latex][\/hidden-answer]<\/section>","rendered":"

      Recognize and Use Fundamental Identities<\/h2>\n

      We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.<\/p>\n

      \n

      fundamental identities<\/h3>\n

      We can derive some useful identities<\/strong> from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:<\/p>\n

      [latex]\\tan t=\\frac{\\sin t}{\\cos t}[\/latex]<\/p>\n

      [latex]\\sec t=\\frac{1}{\\cos t}[\/latex]<\/p>\n

      [latex]\\csc t=\\frac{1}{\\sin t}[\/latex]<\/p>\n

      [latex]\\cot t=\\frac{1}{\\tan t}=\\frac{\\cos t}{\\sin t}[\/latex]<\/p>\n<\/section>\n

      \n
        \n
      1. Given [latex]\\sin \\left(45^\\circ \\right)=\\frac{\\sqrt{2}}{2},\\cos \\left(45^\\circ \\right)=\\frac{\\sqrt{2}}{2}[\/latex], evaluate [latex]\\tan \\left(45^\\circ \\right)[\/latex].<\/li>\n
      2. Given [latex]\\sin \\left(\\frac{5\\pi }{6}\\right)=\\frac{1}{2},\\cos\\left(\\frac{5\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex], evaluate [latex]\\sec \\left(\\frac{5\\pi }{6}\\right)[\/latex].<\/li>\n<\/ol>\n