Bài 3. Một số phương trình lượng giác thường gặp
Hướng dẫn giải Hoạt động 5 (Trang 35 SGK Toán Đại số & Giải tích 11)
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<p>Dựa vào các công thức cộng đã học <span id="MathJax-Element-1-Frame" class="mjx-full-width mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: table-cell !important; line-height: 0; text-indent: 0px; text-align: center; text-transform: none; font-style: normal; font-weight: normal; font-size: 21.78px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 14.724em; min-height: 0px; border: 0px; width: 10000em; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>b</mi><mo>+</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>b</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>a</mi><mo>;</mo><mspace linebreak="newline" /><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>a</mi><mo>&#x2013;</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>b</mi><mo>&#x2212;</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>b</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>a</mi><mo>;</mo><mspace linebreak="newline" /><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>a</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>b</mi><mo>&#x2013;</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>b</mi><mo>;</mo><mspace linebreak="newline" /><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo>&#x2013;</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>a</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>b</mi><mo>+</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>b</mi></math>"></span></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>=</mo><mi>sin</mi><mi>a</mi><mi>cos</mi><mi>b</mi><mo>+</mo><mi>sin</mi><mi>b</mi><mi>cos</mi><mi>a</mi><mo>;</mo><mspace linebreak="newline"></mspace><mi>sin</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow></mfenced><mo>=</mo><mi>sin</mi><mi>a</mi><mi>cos</mi><mi>b</mi><mo>-</mo><mi>sin</mi><mi>b</mi><mi>cos</mi><mi>a</mi><mo>;</mo><mspace linebreak="newline"></mspace><mi>cos</mi><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>=</mo><mi>cos</mi><mi>a</mi><mi>cos</mi><mi>b</mi><mo>-</mo><mi>sin</mi><mi>a</mi><mi>sin</mi><mi>b</mi><mo>;</mo><mspace linebreak="newline"></mspace><mi>cos</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow></mfenced><mo>=</mo><mi>cos</mi><mi>a</mi><mi>cos</mi><mi>b</mi><mo>+</mo><mi>sin</mi><mi>a</mi><mi>sin</mi><mi>b</mi></math></p>
<p>và kết quả <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac></math>, hãy chứng minh rằng:</p>
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<div id="sub-question-1" class="box-question top20">
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p><strong>Lời giải chi tiết:</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mfenced><mrow><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mi>sin</mi><mi>x</mi><mo>+</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mi>cos</mi><mi>x</mi></mrow></mfenced><mspace linebreak="newline"></mspace><mo>=</mo><msqrt><mn>2</mn></msqrt><mfenced><mrow><mi>sin</mi><mi>x</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mi>cos</mi><mi>x</mi></mrow></mfenced><mspace linebreak="newline"></mspace><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p><strong>Cách khác:</strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced><mspace linebreak="newline"></mspace><mo>=</mo><msqrt><mn>2</mn></msqrt><mfenced><mrow><mi>cos</mi><mi>x</mi><mo>.</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>+</mo><mi>sin</mi><mi>x</mi><mo>.</mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced><mspace linebreak="newline"></mspace><mo>=</mo><msqrt><mn>2</mn></msqrt><mfenced><mrow><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>cos</mi><mi>x</mi><mo>+</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mi>sin</mi><mi>x</mi></mrow></mfenced><mspace linebreak="newline"></mspace><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>cos</mi><mi>x</mi><mo>+</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>sin</mi><mi>x</mi></math></strong><span id="MathJax-Element-8-Frame" class="mjx-full-width mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: table-cell !important; line-height: 0; text-indent: 0px; text-align: center; text-transform: none; font-style: normal; font-weight: normal; font-size: 21.78px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 13.316em; min-height: 0px; border: 0px; width: 10000em; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mo stretchy="false">(</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi><mo>+</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi><mo stretchy="false">)</mo><mspace linebreak="newline" /><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi><mo>+</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi><mspace linebreak="newline" /><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi><mo>+</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi> </mi></math></span></span></p>
<div><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>cos</mi><mi>x</mi><mo>+</mo><mi>sin</mi><mi>x</mi></math> (đpcm)</div>
<div>b) <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>x</mi><mo>-</mo><mi>cos</mi><mi>x</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math></strong></div>
<div><strong>Lời giải chi tiết:</strong></div>
<div><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>x</mi><mo>-</mo><mi>cos</mi><mi>x</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mfenced><mrow><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mi>sin</mi><mi>x</mi><mo>-</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mi>cos</mi><mi>x</mi></mrow></mfenced><mspace linebreak="newline"></mspace><mo>=</mo><msqrt><mn>2</mn></msqrt><mfenced><mrow><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mi>sin</mi><mi>x</mi><mo>-</mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mi>cos</mi><mi>x</mi></mrow></mfenced><mspace linebreak="newline"></mspace><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math></strong></div>
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<div id="sub-question-2" class="box-question top20">
<p><strong>Cách khác:</strong><span id="MathJax-Element-13-Frame" class="mjx-full-width mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: table-cell !important; line-height: 0; text-indent: 0px; text-align: center; text-transform: none; font-style: normal; font-weight: normal; font-size: 21.78px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 14.042em; min-height: 0px; border: 0px; width: 10000em; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt><mo>.</mo><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>x</mi><mo>&#x2212;</mo><mfrac><mi>&#x03C0;</mi><mn>4</mn></mfrac><mo stretchy="false">)</mo><mspace linebreak="newline" /><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mo stretchy="false">(</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi><mo>.</mo><mi>c</mi><mi>o</mi><mi>s</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>&#x03C0;</mi><mn>4</mn></mfrac></mrow><mo>&#x2212;</mo><mi>s</mi><mi>i</mi><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>&#x03C0;</mi><mn>4</mn></mfrac></mrow><mo>.</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi><mo stretchy="false">)</mo><mspace linebreak="newline" /><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mo stretchy="false">(</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi><mo>&#x2212;</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi><mo stretchy="false">)</mo><mspace linebreak="newline" /><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi><mo>&#x2212;</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi><mspace linebreak="newline" /><mo>=</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi><mo>&#x2013;</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi></math>"> </span></p>
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<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced><mspace linebreak="newline"></mspace><mo>=</mo><msqrt><mn>2</mn></msqrt><mfenced><mrow><mi>sin</mi><mi>x</mi><mo>.</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>-</mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>.</mo><mi>cos</mi><mi>x</mi></mrow></mfenced><mspace linebreak="newline"></mspace><mo>=</mo><msqrt><mn>2</mn></msqrt><mfenced><mrow><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>sin</mi><mi>x</mi><mo>-</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>cos</mi><mi>x</mi></mrow></mfenced><mspace linebreak="newline"></mspace><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>sin</mi><mi>x</mi><mo>-</mo><msqrt><mn>2</mn></msqrt><mo>.</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac><mo>.</mo><mi>cos</mi><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>sin</mi><mi>x</mi><mo>-</mo><mi>cos</mi><mi>x</mi></math> (đpcm)</p>
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