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三角函数诱导公式02
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2022-10-13 19:21
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<p><strong>诱导公式</strong></p> <p>诱导公式是指三角函数中,利用三角函数周期性的特点将角度比较大的三角函数转换为角度比较小的三角函数的公式。全国高考大纲中只考 .。</p> <p><strong>公式一:关于 2π</strong></p> <p><span class="math-tex">\( \sin (2k\pi +\alpha )=\sin \alpha,k\in \mathbb{Z} \) </span></p> <p><span class="math-tex">\( \cos (2k\pi +\alpha )=\cos \alpha,k\in \mathbb{Z} \) </span></p> <p><span class="math-tex">\( \tan (2k\pi +\alpha )=\tan \alpha,k\in \mathbb{Z} \) </span></p> <p><span class="math-tex">\( \cot (2k\pi +\alpha )=\cot \alpha,k\in \mathbb{Z} \) </span></p> <p><span class="math-tex">\( \sec (2k\pi +\alpha )=\sec \alpha,k\in \mathbb{Z} \) </span></p> <p><span class="math-tex">\( \csc (2k\pi +\alpha )=\csc \alpha,k\in \mathbb{Z} \) </span></p> <p> </p> <p><strong> 公式二 关于π</strong></p> <p><span class="math-tex">\( \sin (\pi +\alpha )=-\sin \alpha \) </span></p> <p><span class="math-tex">\( \cos (\pi +\alpha )=-\cos \alpha \) </span></p> <p><span class="math-tex">\( \tan (\pi +\alpha )=\tan \alpha \) </span></p> <p><span class="math-tex">\( \cot (\pi +\alpha )=\cot \alpha \) </span></p> <p><span class="math-tex">\( \sec (\pi +\alpha )=-\sec \alpha \) </span></p> <p><span class="math-tex">\( \csc (\pi +\alpha )=-\csc \alpha \) </span></p> <p> </p> <p><strong>公式三 关于-a</strong></p> <p><span class="math-tex">\( \sin (-\alpha )=-\sin \alpha \) </span></p> <p><span class="math-tex">\( \cos (-\alpha )=\cos \alpha \) </span></p> <p><span class="math-tex">\( \tan (-\alpha )=-\tan \alpha \) </span></p> <p><span class="math-tex">\( \cot (-\alpha )=-\cot \alpha \) </span></p> <p><span class="math-tex">\( \sec (-\alpha )=\sec \alpha \) </span></p> <p><span class="math-tex">\( \csc (-\alpha )=-\csc \alpha \) </span></p> <p> </p> <p><strong>公式四 关于 π-a</strong></p> <p><span class="math-tex">\( \sin (\pi -\alpha )=\sin \alpha \) </span></p> <p><span class="math-tex">\( \cos (\pi -\alpha )=-\cos \alpha \) </span></p> <p><span class="math-tex">\( \tan (\pi -\alpha )=-\tan \alpha \) </span></p> <p><span class="math-tex">\( \cot (\pi -\alpha )=-\cot \alpha \) </span></p> <p><span class="math-tex">\( \sec (\pi -\alpha )=-\sec \alpha \) </span></p> <p><span class="math-tex">\( \csc (\pi -\alpha )=\csc \alpha \) </span></p> <p> </p> <p><strong>公式五 关于 π/2-a</strong></p> <p><strong><span class="math-tex">\( \sin (\frac{\pi }{2}-\alpha )=\cos\alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \cos (\frac{\pi }{2} -\alpha )=\sin\alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \tan (\frac{\pi }{2}-\alpha )=\cot \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \cot (\frac{\pi }{2} -\alpha )=\tan \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \sec (\frac{\pi }{2} -\alpha )=\csc \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \csc (\frac{\pi }{2} -\alpha )=\sec \alpha \) </span></strong></p> <p> </p> <p><strong>公式六 π/2+a</strong></p> <p><strong><span class="math-tex">\( \sin \left(\frac{\pi }{2}+\alpha \right)=\cos \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \cos \left(\frac{\pi }{2}+\alpha \right)=-\sin \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \tan \left(\frac{\pi }{2}+\alpha \right)=-\cot \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \cot \left(\frac{\pi }{2}+\alpha \right)=-\tan \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \sec \left(\frac{\pi }{2}+\alpha \right)=-\csc \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \csc \left(\frac{\pi }{2}+\alpha \right)=\sec \alpha \) </span></strong></p> <p> </p> <p><strong>公式 关于 3π/2-a</strong></p> <p><strong><span class="math-tex">\( \sin \left(\frac{3\pi}{2}-\alpha \right)=-\cos \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \cos \left(\frac{3\pi}{2}-\alpha \right)=-\sin \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \tan \left(\frac{3\pi}{2}-\alpha \right)=\cot \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \cot \left(\frac{3\pi}{2}-\alpha \right)=\tan \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \sec \left(\frac{3\pi}{2}-\alpha \right)=-\csc \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \csc \left(\frac{3\pi}{2}-\alpha \right)=-\sec \alpha \) </span></strong></p> <p> </p> <p><strong>公式 关于 3π/2+a</strong></p> <p><strong><span class="math-tex">\( \sin \left(\frac{3\pi}{2}+\alpha \right)=-\cos \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \cos \left(\frac{3\pi}{2}+\alpha \right)=\sin \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \tan \left(\frac{3\pi}{2}+\alpha \right)=-\cot \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \cot \left(\frac{3\pi}{2}+\alpha \right)=-\tan \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \sec \left(\frac{3\pi}{2}+\alpha \right)=\csc \alpha \) </span></strong></p> <p><strong><span class="math-tex">\( \csc \left(\frac{3\pi}{2}+\alpha \right)=-\sec \alpha \) </span></strong></p> <p> </p> <p><strong>公式一~八的内在联系</strong></p> <p><strong><span class="math-tex">\( \sin \left(\frac{k\pi }{2}\pm \alpha \right), k\in \mathbb{Z} \) </span></strong></p> <p><strong><span class="math-tex">\( \cos \left(\frac{k\pi }{2}\pm \alpha \right), k\in \mathbb{Z} \) </span></strong></p> <p><strong><span class="math-tex">\( \tan \left(\frac{k\pi }{2}\pm \alpha \right), k\in \mathbb{Z} \) </span></strong></p> <p><strong><span class="math-tex">\( \cot \left(\frac{k\pi }{2}\pm \alpha \right), k\in \mathbb{Z} \) </span></strong></p> <p><strong><span class="math-tex">\( \sec \left(\frac{k\pi }{2}\pm \alpha \right), k\in \mathbb{Z} \) </span></strong></p> <p><strong><span class="math-tex">\( \csc \left(\frac{k\pi }{2}\pm \alpha \right), k\in \mathbb{Z} \) </span></strong></p> <p>在实际教学中,老师通常使用如下口诀让学生记忆:奇变偶不变,符号看象限”。 意思为,当k为奇数时,sin 变cos,cos变sin,tan变cot,cot变tan,sec变csc,csc变sec。当k为偶数时,三角函数则不变。对于正负号,要看最后角所在的象限。</p>
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