https://mathmanstudio.com/categories/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8/Recent content in 三角函數 on MathManStudio.comHugozh-TWSun, 19 Jul 2026 19:25:37 +0800https://mathmanstudio.com/posts/114%E9%AB%98%E4%B8%AD%E6%95%B8%E5%AD%B8%E5%AD%B8%E6%B8%ACa-%E5%A4%9A%E9%81%B817%E9%80%A3%E4%B8%89%E9%A4%98%E5%BC%A6%E5%AE%9A%E7%90%86/Sun, 19 Jul 2026 19:25:37 +0800https://mathmanstudio.com/posts/114%E9%AB%98%E4%B8%AD%E6%95%B8%E5%AD%B8%E5%AD%B8%E6%B8%ACa-%E5%A4%9A%E9%81%B817%E9%80%A3%E4%B8%89%E9%A4%98%E5%BC%A6%E5%AE%9A%E7%90%86/<p><img loading="lazy" src="https://mathmanstudio.com/wp-content/uploads/2026/07/%E6%88%AA%E5%9C%96-2026-07-19-%E4%B8%8B%E5%8D%884.18.28-1024x192.png"></p> <p>先畫個示意圖吧</p> <p><img loading="lazy" src="https://mathmanstudio.com/wp-content/uploads/2026/07/%E6%88%AA%E5%9C%96-2026-07-19-%E4%B8%8B%E5%8D%884.38.48-1024x820.png"></p> <p>第一次餘弦，先利用題目已知求AB長</p> <p>$$ cos\angle{ABC}=\frac{9+9-\overline{AC}^2}{2\times3\times3}=\frac{-1}{8} $$ $$ \overline{AC}=\frac{9}{2} $$</p> <p>第二次餘弦</p> <p>$$ cos\phi=\frac{9+\frac{81}{4}-9}{2\times3\times\frac{9}{2}}=\frac{3}{4} $$</p> <p>利用等弧對等圓周角</p> <p>$$ \angle{BAC}=\angle{BDC}=\phi $$</p> <p>第三次餘弦</p> <p>$$ cos\phi=\frac{16+\overline{CD}^2-9}{2\times4\times\overline{CD}}=\frac{3}{4} $$</p> <p>解得 <details class="answer-box"> <summary> <span class="answer-show">查看答案</span> <span class="answer-hide">收起答案</span> </summary> <div class="answer-content"> $$ \overline{CD}=3\pm\sqrt{2}(取正) $$ </div> </details></p>https://mathmanstudio.com/posts/114%E9%AB%98%E4%B8%AD%E6%95%B8%E5%AD%B8%E5%AD%B8%E6%B8%AC%E6%95%B8a%E5%A4%9A%E9%81%B810/Sun, 12 Jul 2026 19:55:57 +0800https://mathmanstudio.com/posts/114%E9%AB%98%E4%B8%AD%E6%95%B8%E5%AD%B8%E5%AD%B8%E6%B8%AC%E6%95%B8a%E5%A4%9A%E9%81%B810/<p><img loading="lazy" src="https://mathmanstudio.com/wp-content/uploads/2026/07/%E6%88%AA%E5%9C%96-2026-07-12-%E4%B8%8B%E5%8D%886.45.05-1024x366.png"> <img loading="lazy" src="https://mathmanstudio.com/wp-content/uploads/2026/07/%E6%88%AA%E5%9C%96-2026-07-12-%E4%B8%8B%E5%8D%886.55.33.png"></p> <p>（1）若k&lt;0，無法滿足題目要求</p> <p>$$ x_1&lt;x_2&lt;1&lt;x_3 $$</p> <p>(2)有4個交點</p> <p>（3）由圖形對稱性可得</p> <p>$$ x_1+x_2=1 $$</p> <p>(4)</p> <p>$$ 2\overline{PQ}=2(x_2-x_1)=2(1-x_1-x_1)=2-4x_1 $$ $$ \overline{QR}=x_3-x_2=(2+x_1)-(1-x_1)=1+2x_1 $$ $$ 解得x_1=\frac{1}{6} $$ $$ 驗算一下，sin\frac{1}{6}\pi=\frac{1}{2}，原式成立。 $$</p> <p>R點也可能在最右邊，此時</p> <p>$$ \overline{QR}=(x_3-x_2)=(3-x_1)-(1-x_1)=2 $$ $$ 解得x=0，不合。 $$</p> <p>（5）</p> <p>$$ x_1+x_2+x_3+x_4=x_1+(1-x_1)+(2+x_1)+(3-x_1)=6 $$</p> <p>Ans:(1)(4)(5)</p>https://mathmanstudio.com/posts/114%E9%AB%98%E4%B8%AD%E6%95%B8%E5%AD%B8%E5%AD%B8%E6%B8%AC%E6%95%B8a%E5%96%AE%E9%81%B845%E5%B0%8D%E6%95%B8%E5%87%BD%E6%95%B8%E8%88%87%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8%E7%9A%84/Sun, 26 Oct 2025 19:25:53 +0800https://mathmanstudio.com/posts/114%E9%AB%98%E4%B8%AD%E6%95%B8%E5%AD%B8%E5%AD%B8%E6%B8%AC%E6%95%B8a%E5%96%AE%E9%81%B845%E5%B0%8D%E6%95%B8%E5%87%BD%E6%95%B8%E8%88%87%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8%E7%9A%84/<p><img loading="lazy" src="https://mathmanstudio.com/wp-content/uploads/2025/10/%E6%88%AA%E5%9C%96-2025-10-26-%E4%B8%8B%E5%8D%883.33.08-1024x775.png"> <img loading="lazy" src="https://mathmanstudio.com/wp-content/uploads/2025/10/%E6%88%AA%E5%9C%96-2025-10-26-%E4%B8%8B%E5%8D%883.33.58-1024x204.png"></p> <p>先畫個圖吧</p> <p>利用底數大於1的對數函數圖形是<strong>嚴格遞增</strong>的特性</p> <p>比如說Y座標為1的X座標就要從3數到30（因為(2,1)不能算）</p> <p>總共有</p> <p>(3,1)&hellip;..(30,1)共28個</p> <p>(5,2)&hellip;.(30,2)共26個</p> <p>(9,3)&hellip;.(30,3)共22個</p> <p>(17,4)&hellip;.(30,4)共14個</p> <p>一共90個</p> <p><strong>Ans:(3)</strong></p> <p><img loading="lazy" src="https://mathmanstudio.com/wp-content/uploads/2025/10/%E6%88%AA%E5%9C%96-2025-10-26-%E4%B8%8B%E5%8D%883.55.29-1024x171.png"> <img loading="lazy" src="https://mathmanstudio.com/wp-content/uploads/2025/10/%E6%88%AA%E5%9C%96-2025-10-26-%E4%B8%8B%E5%8D%886.52.22-1024x439.png"></p> <p>$0&lt;\theta&lt;\cfrac{\pi}{3} \bigcup \pi&lt;\theta&lt;\cfrac{5\pi}{3}$</p> <p><img loading="lazy" src="https://mathmanstudio.com/wp-content/uploads/2025/10/%E6%88%AA%E5%9C%96-2025-10-26-%E4%B8%8B%E5%8D%886.54.00-1024x432.png"></p> <p>$\cfrac{2\pi}{3}&lt;\theta&lt;\cfrac{4\pi}{3}$</p> <p>最後取兩圖形的交集</p> <p>$\pi&lt;\theta&lt;\cfrac{4\pi}{3}$</p> <p>$b-a=\cfrac{4}{3}-1=\cfrac{1}{3}$</p> <p>Ans:(1)</p> <p>🙀這題夭壽難，需要知道如何判斷函數的週期畫出圖形，還要找出圖形交點的X座標，最後記得三個範圍取交集啊！！！！</p> <p><strong>有沒有都學會啊～想問陳偉數學問題讓自己越來越厲害的話請立即加入</strong></p> <p><a href="https://mathmanstudio.com/line-%E6%95%B8%E5%AD%B8%E8%A7%A3%E9%A1%8C%E5%AE%A4">LINE數學解題室</a></p>https://mathmanstudio.com/posts/113%E9%AB%98%E4%B8%AD%E6%95%B8%E5%AD%B8%E5%88%86%E7%A7%91%E6%B8%AC%E9%A9%97%E7%B7%B4%E5%8A%9F%E5%9D%8A-%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8%E7%9A%84%E5%9C%96%E5%BD%A2-%E5%80%92%E6%95%B8136%E5%A4%A9/Tue, 27 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