三角函数公式大全
2023-08-18 14:31:58 11 举报AI智能生成
三角函数公式是数学中属于初等函数中的超越函数的一类函数公式。它们的本质是任意角的集合与一个比值的集合的变量之间的映射，通常的三角函数是在平面直角坐标系中定义的。三角函数公式包括和差角公式、和差化积公式、积化和差公式、倍角公式、诱导公式等。
作者其他创作
大纲/内容
定义式
单位圆与坐标系之间关系
将运用扩大为实数集
弧度制
长度等于半径的弧度所对圆心角为1弧度（rad）
同角函数公式
倒数关系
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tan\alpha *cot\alpha=1"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin\alpha*csc\alpha=1"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos\alpha*sec\alpha=1"><span></span><span></span></span>
商数关系
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{tanα}=\frac{sinα}{comα}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{ctgα}=\frac{cosα}{sinα}"><span></span><span></span></span>
平方关系
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin\alpha^2+cos\alpha^2=1"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1+tan\alpha^2=sec\alpha^2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1+cot\alpha^2=csc\alpha^2"><span></span><span></span></span>
诱导公式
奇变偶不变符号看象限
<span class="equation-text" data-index="0" data-equation="sin(n\frac{π}{2}+\alpha)=f(\alpha)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\frac{π}{2}-\alpha)与（\alpha）关于y=x对称"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="（π-\alpha）与\alpha关于y轴对称"><span></span><span></span></span>
两角和差公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{sin(α+β)}={sinα}{cosβ}+{cosα}{sinβ}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{sin(α-β)}={sinα}{cosβ}-{cosα}{sinβ}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{cos(α+β)}={cosα}{cosβ}+{sinα}{sinβ}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{cos(α-β)}={cosα}{cosβ}-{sinα}{sinβ}"><span></span><span></span></span>
tan（α+β）=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{{tanα}+{tanβ}}{1-{tanα}{tanβ}}"><span></span><span></span></span>
tan（α-β）=<span class="equation-text" data-index="0" data-equation="\frac{{tanα}-{tanβ}}{1+{tanα}{tanβ}}"><span></span><span></span></span>
二倍角公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin2\alpha=2sin\alpha cos\alpha"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos2\alpha=cos^2\alpha-sin^2\alpha=1-2sin^2\alpha=2cos^2\alpha-1"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tan2\alpha=\frac{2tan\alpha}{1-tan^2\alpha}"><span></span><span></span></span>
半角公式
一般公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin\frac{\alpha}{2}=_-^+\sqrt{\frac{1-cos\alpha}{2}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos\frac{\alpha}{2}=_-^+\sqrt{\frac{1+cos\alpha}{2}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tan\frac{\alpha}{2}=_-^+\sqrt{\frac{1-cos\alpha}{1+cos\alpha}}"><span></span><span></span></span>
正切补充公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tan\frac{α}{2}={sin\frac{α}{2}}/{cos\frac{α}{2}}=2sin\frac{α}{2}cos\frac{α}{2}/2cos^2\frac{α}{2}=\frac{sinα}{1+cosα}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tan\frac{α}{2}=sin\frac{α}{2}/cos\frac{α}{2}={2sin^2\frac{α}{2}}/{2sin\frac{α}{2}cos\frac{α}{2}}=\frac{1-cosα}{sinα}"><span></span><span></span></span>
和差化积公式
公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin\alpha+sin\beta=2sin\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin\alpha-sin\beta=2cos\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos\alpha+cos\beta=2cos\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos\alpha-cos\beta=-2sin\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2}"><span></span><span></span></span>
推导
将<span class="equation-text" contenteditable="false" data-index="0" data-equation="\alpha=\frac{\alpha+\beta}{2}+\frac{\alpha-\beta}{2};\beta=\frac{\alpha+\beta}{2}-\frac{\alpha-\beta}{2}"><span></span><span></span></span>带入二倍角公式
口诀
正加正，正在前；正减正，余在前；余加余，余并肩；余减余，负正弦
积化和差公式
公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin\alpha\cos\beta=\frac{sin(\alpha+\beta)+sin(\alpha-\beta)}{2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos\alpha\sin\beta=\frac{sin(\alpha+\beta)-sin(\alpha-\beta)}{2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin\alpha\sin\beta=-\frac{cos(\alpha+\beta)-cos(\alpha-\beta)}{2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos\alpha\cos\beta=\frac{cos(\alpha+\beta)+cos(\alpha-\beta)}{2}"><span></span><span></span></span>
推导
和差化积公式逆推导；从右推左
口诀
正余余正，正加正减；余余正正，余加负余减
万能公式
<span class="equation-text" data-index="0" data-equation="sinα=2sin\frac{α}{2}cos\frac{α}{2}/(sin^2\frac{α}{2}+cos^2\frac{α}{2})=2tan\frac{α}{2}/(1+tan^2\frac{α}{2})={2t}/(1+t^2)" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="(令tan\frac{α}{2}=t)"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="cosα=(cos^2\frac{α}{2}-sin^2\frac{α}{2})/(sin^2\frac{α}{2}+cos^2\frac{α}{2})=(1-tan^2\frac{α}{2})/(1+tan^2\frac{α}{2})=(1-t^2)/(1+t^2)" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="1" data-equation="(令tan\frac{α}{2}=t)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="tanα={2tan\frac{α}{2}}/({1-tan^2\frac{α}{2}})={2t}/({1-t^2})" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="(令tan\frac{α}{2}=t)"><span></span><span></span></span>
辅助角公式
<span class="equation-text" data-index="0" data-equation="a\sinα+b\sin\beta=\sqrt{a^2+b^2}sin(α+\Phi)" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="tan\Phi=\frac{b}{a}"><span></span><span></span></span>
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