{"id":1986,"date":"2025-07-31T23:16:48","date_gmt":"2025-07-31T23:16:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1986"},"modified":"2025-10-14T18:22:52","modified_gmt":"2025-10-14T18:22:52","slug":"sum-and-difference-identities-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sum-and-difference-identities-learn-it-2\/","title":{"raw":"Sum and Difference Identities: Learn It 2","rendered":"Sum and Difference Identities: Learn It 2"},"content":{"raw":"

Use sum and difference formulas for sine<\/h2>\r\nThe sum and difference formulas for sine<\/strong> can be derived in the same manner as those for cosine, and they resemble the cosine formulas.\r\n\r\n
\r\n

sum and difference formula for sine<\/h3>\r\n

[latex]\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta [\/latex]<\/p>\r\n

[latex]\\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta [\/latex]<\/p>\r\n\r\n<\/section>

How To: Given two angles, find the sine of the difference between the angles.\r\n<\/strong>\r\n
    \r\n \t
  1. Write the difference formula for sine.<\/li>\r\n \t
  2. Substitute the given angles into the formula.<\/li>\r\n \t
  3. Simplify.<\/li>\r\n<\/ol>\r\n<\/section>
    Use the sum and difference identities to evaluate the difference of the angles and show that part a<\/em> equals part b.<\/em>\r\n
      \r\n \t
    1. [latex]\\sin \\left({45}^{\\circ }-{30}^{\\circ }\\right)[\/latex]<\/li>\r\n \t
    2. [latex]\\sin \\left({135}^{\\circ }-{120}^{\\circ }\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"191680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"191680\"]\r\n
        \r\n \t
      1. Let\u2019s begin by writing the formula and substitute the given angles.\r\n
        [latex]\\begin{align}\\sin \\left(\\alpha -\\beta \\right)&=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\\\ \\sin \\left({45}^{\\circ }-{30}^{\\circ }\\right)&=\\sin \\left({45}^{\\circ }\\right)\\cos \\left({30}^{\\circ }\\right)-\\cos \\left({45}^{\\circ }\\right)\\sin \\left({30}^{\\circ }\\right) \\end{align}[\/latex]<\/div>\r\nNext, we need to find the values of the trigonometric expressions.\r\n
        [latex]\\sin \\left({45}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\text{ }\\cos \\left({30}^{\\circ }\\right)=\\frac{\\sqrt{3}}{2},\\text{ }\\cos \\left({45}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\text{ }\\sin \\left({30}^{\\circ }\\right)=\\frac{1}{2}[\/latex]<\/div>\r\nNow we can substitute these values into the equation and simplify.\r\n
        [latex]\\begin{align} \\sin \\left({45}^{\\circ }-{30}^{\\circ }\\right)&=\\frac{\\sqrt{2}}{2}\\left(\\frac{\\sqrt{3}}{2}\\right)-\\frac{\\sqrt{2}}{2}\\left(\\frac{1}{2}\\right) \\\\ &=\\frac{\\sqrt{6}-\\sqrt{2}}{4}\\hfill \\end{align}[\/latex]<\/div><\/li>\r\n \t
      2. Again, we write the formula and substitute the given angles.\r\n
        [latex]\\begin{align}\\sin \\left(\\alpha -\\beta \\right)&=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta\\\\ \\sin \\left({135}^{\\circ }-{120}^{\\circ }\\right)&=\\sin \\left({135}^{\\circ }\\right)\\cos \\left({120}^{\\circ }\\right)-\\cos \\left({135}^{\\circ }\\right)\\sin \\left({120}^{\\circ }\\right)\\end{align}[\/latex]<\/div>\r\nNext, we find the values of the trigonometric expressions.\r\n
        [latex]\\sin \\left({135}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\cos \\left({120}^{\\circ }\\right)=-\\frac{1}{2},\\cos \\left({135}^{\\circ }\\right)=\\frac{\\sqrt{2}}{2},\\sin \\left({120}^{\\circ }\\right)=\\frac{\\sqrt{3}}{2}[\/latex]<\/div>\r\nNow we can substitute these values into the equation and simplify.\r\n
        [latex]\\begin{align}\\sin \\left({135}^{\\circ }-{120}^{\\circ }\\right)&=\\frac{\\sqrt{2}}{2}\\left(-\\frac{1}{2}\\right)-\\left(-\\frac{\\sqrt{2}}{2}\\right)\\left(\\frac{\\sqrt{3}}{2}\\right) \\\\ &=\\frac{-\\sqrt{2}+\\sqrt{6}}{4} \\\\ &=\\frac{\\sqrt{6}-\\sqrt{2}}{4} \\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
        Find the exact value of [latex]\\sin \\left({\\cos }^{-1}\\frac{1}{2}+{\\sin }^{-1}\\frac{3}{5}\\right)[\/latex].[reveal-answer q=\"954448\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"954448\"]The pattern displayed in this problem is [latex]\\sin \\left(\\alpha +\\beta \\right)[\/latex]. Let [latex]\\alpha ={\\cos }^{-1}\\frac{1}{2}[\/latex] and [latex]\\beta ={\\sin }^{-1}\\frac{3}{5}[\/latex]. Then we can write\r\n

        [latex]\\begin{align} \\cos \\alpha &=\\frac{1}{2},0\\le \\alpha \\le \\pi\\\\ \\sin \\beta &=\\frac{3}{5},-\\frac{\\pi }{2}\\le \\beta \\le \\frac{\\pi }{2} \\end{align}[\/latex]<\/p>\r\nWe will use the Pythagorean identities to find [latex]\\sin \\alpha [\/latex] and [latex]\\cos \\beta [\/latex].\r\n

        [latex]\\begin{align}\\sin \\alpha &=\\sqrt{1-{\\cos }^{2}\\alpha } \\\\ &=\\sqrt{1-\\frac{1}{4}} \\\\ &=\\sqrt{\\frac{3}{4}} \\\\ &=\\frac{\\sqrt{3}}{2} \\\\ \\cos \\beta &=\\sqrt{1-{\\sin }^{2}\\beta } \\\\ &=\\sqrt{1-\\frac{9}{25}} \\\\ &=\\sqrt{\\frac{16}{25}} \\\\ &=\\frac{4}{5}\\end{align}[\/latex]<\/p>\r\nUsing the sum formula for sine,\r\n

        [latex]\\begin{align}\\sin \\left({\\cos }^{-1}\\frac{1}{2}+{\\sin }^{-1}\\frac{3}{5}\\right)&=\\sin \\left(\\alpha +\\beta \\right) \\\\ &=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta \\\\ &=\\frac{\\sqrt{3}}{2}\\cdot \\frac{4}{5}+\\frac{1}{2}\\cdot \\frac{3}{5} \\\\ &=\\frac{4\\sqrt{3}+3}{10}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

        [ohm_question hide_question_numbers=1]173443[\/ohm_question]<\/section>","rendered":"

        Use sum and difference formulas for sine<\/h2>\n

        The sum and difference formulas for sine<\/strong> can be derived in the same manner as those for cosine, and they resemble the cosine formulas.<\/p>\n

        \n

        sum and difference formula for sine<\/h3>\n

        [latex]\\sin \\left(\\alpha +\\beta \\right)=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta[\/latex]<\/p>\n

        [latex]\\sin \\left(\\alpha -\\beta \\right)=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta[\/latex]<\/p>\n<\/section>\n

        How To: Given two angles, find the sine of the difference between the angles.
        \n<\/strong><\/p>\n
          \n
        1. Write the difference formula for sine.<\/li>\n
        2. Substitute the given angles into the formula.<\/li>\n
        3. Simplify.<\/li>\n<\/ol>\n<\/section>\n
          Use the sum and difference identities to evaluate the difference of the angles and show that part a<\/em> equals part b.<\/em><\/p>\n
            \n
          1. [latex]\\sin \\left({45}^{\\circ }-{30}^{\\circ }\\right)[\/latex]<\/li>\n
          2. [latex]\\sin \\left({135}^{\\circ }-{120}^{\\circ }\\right)[\/latex]<\/li>\n<\/ol>\n