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how-to-calc-sin10sin50sin70
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sin10°sin50°sin70°=？ | Yi's Blog
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<h2>sin10°sin50°sin70°=？</h2>
<p class="post-date">2009-11-28</p>
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<strong>作者: </strong>Yi Chow
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<p>这是一道关于三角函数的高中数学题，看似简单，却有很多人在计算中走了弯路。</p>
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<p>很多人看到这道题首先就会想到这样化简：原式=sin(30-20)·sin(30+20)·sin(90-20)</p>
<p>之后利用公式：</p>
<p><code>sin(α+β)=sinα·cosβ+sinβ·cosα</code></p>
<p>进行计算，在经过繁琐的计算之后，才发现仍无法得出的三角函数。</p>
<p>其实，可以利用二倍角公式来化简：</p>
<p>二倍角公式：sin（2α）=2sinα·cosα</p>
<figure class="highlight gcode"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br></pre></td><td class="code"><pre><span class="line"></span><br><span class="line">解：</span><br><span class="line"></span><br><span class="line"> 原式 = si<span class="symbol">n10</span>°si<span class="symbol">n50</span>°si<span class="symbol">n70</span>°cos<span class="number">10</span>°/cos<span class="number">10</span>°</span><br><span class="line"> = si<span class="symbol">n20</span>°si<span class="symbol">n50</span>°si<span class="symbol">n70</span>°/<span class="number">2</span>cos<span class="number">10</span>°</span><br><span class="line"> = si<span class="symbol">n20</span>°si<span class="symbol">n50</span>°cos<span class="number">20</span>°/<span class="number">2</span>cos<span class="number">10</span>°</span><br><span class="line"> = si<span class="symbol">n40</span>°si<span class="symbol">n50</span>°/<span class="number">4</span>cos<span class="number">10</span>°</span><br><span class="line"> = si<span class="symbol">n40</span>°cos<span class="number">40</span>°/<span class="number">4</span>cos<span class="number">10</span>°</span><br><span class="line"> = si<span class="symbol">n80</span>°/<span class="number">8</span>cos<span class="number">10</span>°</span><br><span class="line"> = cos<span class="number">10</span>°/<span class="number">8</span>cos<span class="number">10</span>°</span><br><span class="line"> = <span class="number">1</span>/<span class="number">8</span></span><br></pre></td></tr></table></figure>
<p>通过这样的方法，计算过程简单明了，即使口算也可以轻易的出结果。<br>你学会了吗？</p>
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