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  <div class="t">

         <h5 class="t1">和差角公式</h5>


      
<table class="table ">
   
    <tr  >
        <th    width="180px">　</th>
        <th  width="200px" >和</th>
        <th width="200px" >差</th>
   
    </tr>

     <tr  >
        <th  > 正弦　</th>
        <td   >
            
         <math   >  
          \sin \left(x+y\right)=\sin x \cos y + \cos x \sin y  
        </math>

        </td>
        <td   >
             <math   >  
           \sin \left(x-y\right)=\sin x \cos y - \cos x \sin y 
        </math>
        </td>
   
    </tr>



    <tr>
        <th>
            余弦
        </th>

        <td>
  <math   >  
          \cos \left(x+y\right)=\cos x \cos y - \sin x \sin y  
        </math>

        </td>

        <td>
            
         <math   >  
           \cos \left(x-y\right)=\cos x \cos y + \sin x \sin y   
        </math>
        </td>
    </tr>


    <tr>
        <th>
            正切
        </th>

        <td>
            
       <math   >  
           \tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x\tan y}  
      </math>
        </td>
        

           <td>

                  <math   >  
            \tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x\tan y}
        </math>

        </td>
    </tr>


    <tr>
        <th>
            余切
        </th>

        <td>
                  <math   >  
           \cot(x + y) = \frac{\cot x   \cot y - 1 }{ \cot y + \cot x}  
      </math>

        </td>

          <td>

              <math   >  
           \cot(x - y) = \frac{\cot x   \cot y + 1 }{ \cot y - \cot x}  
      </math>
        </td>

    </tr>


    <tr>
        <th>
正割
        </th>

        <td>
            
       <math   >  
           \sec(x + y) = \frac{\sec x   \sec y  }{ 1 - \tan x  \tan y}  
      </math>

        </td>

         <td>
                
       <math   >  
           \sec(x - y) = \frac{\sec x   \sec y  }{ 1 + \tan x  \tan y}  
      </math>
        </td>

    </tr>


    <tr>
        <th>
            余割
        </th>

        <td>
             
       <math   >  
           \csc(x + y) = \frac{\csc x   \csc y  }{ \cot y + \cot x}  
      </math>
        </td>
        <td>
              <math   >  
           \csc(x - y) = \frac{\csc x   \csc y  }{ \cot y - \cot x}  
      </math>
        </td>
    </tr>
    </table>





 

         
 
        



       
   



      

          
     


      </div>



  

        
          

  

        
    

<div class="t">

    <h5 class="t1">倍角公式</h5>





    <table class="table ">

        <tr>
            <th colspan="2" width="100px"></th>
            <th width="260px">弦</th>
            <th width="260px">切</th>
            <th width="260px">割</th>
        </tr>
        <tr>
            <th rowspan="2">二倍角公式</th>
            <th>正</th>
            <td>

                <math >
                    \sin 2\theta = 2 \sin \theta \cos \theta 
                </math>

                <math >
                    \sin 2\theta = \frac{2 \tan \theta} {1 + \tan^2 \theta}
                </math>

            </td>
            <td>
                <math >
                    \tan 2\theta  = \frac{2 \tan \theta} {1 - \tan^2 \theta} 
                </math>

                <math >
                    \tan 2\theta  = \frac{1}{1-\tan\theta}-\frac{1}{1+\tan\theta}
                </math>

            </td>
            <td>
                <math >
                    \sec 2\theta  = \frac{\sec^2\theta}{1-\tan^2\theta} 
                </math>

                <math >
                    \sec 2\theta  = \frac{\sec^2\theta}{2-\sec^2\theta}
                </math>
            </td>
        </tr>
        <tr>
            <th>余</th>
            <td>
                <math >
                    \cos 2\theta  = \cos^2 \theta - \sin^2 \theta
                </math>

                <math >
                    \cos 2\theta  = 2 \cos^2 \theta - 1
                </math>


                <math >
                    \cos 2\theta  =  1 - 2 \sin^2 \theta
                </math>

                <math >
                    \cos 2\theta  = \frac{1 - \tan^2 \theta} {1 + \tan^2 \theta}
                </math>





            </td>
            <td>

                <math >
                    \cot 2\theta = \frac{\cot^2 \theta-1} {2 \cot \theta}
                </math>

                <math >
                    \cot 2\theta =  \frac{1 - \tan^2 \theta} {1 + \tan^2 \theta}
                </math>

            </td>
            <td>

                <math >
                    \csc 2\theta  = \frac{\csc^2\theta}{2\cot\theta}
                </math>

                <math >
                    \csc 2\theta  = \frac{\sec\theta\csc\theta}{2}
                </math>

            </td>
        </tr>
        <tr>
            <th rowspan="2">三倍角公式</th>
            <th>正</th>
            <td>
                <math >
                    \sin 3\theta = 3 \sin \theta- 4 \sin^3\theta  
                </math>



            </td>
            <td>
                <math >
                    \tan 3\theta = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta}
                </math>

            </td>
            <td>
                <math >
                    \sec 3\theta = \frac{\sec^3\theta}{4-3 \sec^2\theta} 
                </math>
            </td>
        </tr>
        <tr>
            <th>余</th>
            <td>
                <math >
                    \cos 3\theta = 4 \cos^3\theta - 3 \cos \theta
                </math>
            </td>
            <td>
                <math >\cot 3\theta =\frac{\cot^3\theta - 3\cot\theta}{3 \cot^2\theta-1} </math>
            </td>
            <td>
                <math >\csc 3\theta =\frac{\csc^3\theta}{3\csc^2\theta-4} </math>
            </td>
        </tr>
        <tr>
            <th rowspan="2">半角公式</th>
            <th>正</th>
            <td>


                <math >
                    \sin \frac{\theta}{2} = \pm\, \sqrt{\frac{1 - \cos \theta}{2}}
                </math>
            </td>
            <td>
                <math >
                    \begin{align} \tan \frac{\theta}{2} &= \csc \theta - \cot \theta \\ &= \pm\, \sqrt{1 - \cos \theta \over 1 + \cos \theta} \\ &= \frac{\sin \theta}{1 + \cos \theta} \\ &= \frac{1-\cos \theta}{\sin \theta} \\ &= \frac{\cos \theta+\sin \theta-1}{\cos \theta-\sin \theta+1} \end{align} 
                </math>

            </td>
            <td>
                <math >
                    \sec \frac{\theta}{2} = \pm\, \sqrt{\frac{2\sec\theta}{\sec\theta + 1}}
                </math>

            </td>
        </tr>
        <tr>
            <th>余</th>
            <td>

                <math >
                    \cos \frac{\theta}{2} = \pm\, \sqrt{\frac{1 + \cos\theta}{2}}
                </math>

            </td>
            <td>

                <math >
                    \begin{align} \cot \frac{\theta}{2} &= \csc \theta + \cot \theta \\ &= \pm\, \sqrt{1 + \cos \theta \over 1 - \cos \theta} \\ &= \frac{\sin \theta}{1 - \cos \theta} \\ &= \frac{1 + \cos \theta}{\sin \theta}\\ &= \frac{\cos \theta-\sin \theta+1}{\cos \theta+\sin \theta-1} \end{align} 


                </math>


            </td>
            <td>
                <math >
                    \csc \frac{\theta}{2} = \pm\, \sqrt{\frac{2\sec\theta}{\sec\theta - 1}}
                </math>

            </td>
        </tr>
    </table>

</div>



 
         
    



  <div class="t">

         <h5 class="t1">积化和差与和差化积</h5>


      
<table class="table ">
   
    <tr  >
       
        <th  width="200px" >积化和差</th>
        <th width="200px" >和差化积</th>
   
    </tr>

     <tr  >
        
        <td   > 
        <math> 
         \sin\alpha\cos\beta={\sin(\alpha+\beta)+\sin(\alpha-\beta)\over2} 
        </math>

        </td>
        <td   >
         <math>\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}</math>
        </td>
   
    </tr>



    <tr>
       

        <td>
<math>\cos\alpha\sin\beta={\sin(\alpha+\beta)-\sin(\alpha-\beta)\over2}</math>

        </td>

        <td>
            
       <math>\sin\alpha-\sin\beta=2\cos{\alpha+\beta\over2}\sin{\alpha-\beta\over2}</math>
        </td>
    </tr>


    <tr>
       

        <td>
            
      <math>\cos\alpha\cos\beta={\cos(\alpha+\beta)+\cos(\alpha-\beta)\over2}</math>
        </td>
        

           <td>
<math>\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}</math>

        </td>
    </tr>


    <tr>
        

        <td>
               <math>\sin\alpha\sin\beta=-{\cos(\alpha+\beta)-\cos(\alpha-\beta)\over2}</math>

        </td>

          <td>

         <math>\cos\alpha-\cos\beta=-2\sin{\alpha+\beta\over2}\sin{\alpha-\beta\over2}</math>
        </td>

    </tr>


     


     
    </table>

       


      </div>




 

          

<div class="t">

    <h5 class="t1">降幂公式</h5>





    <table class="table ">

        <tr>
           
            <th width="260px">正弦</th>
            <th width="260px">余弦</th>
            <th width="260px">其它</th>
        </tr>
        <tr>
            <td>
<math>\sin^2\theta = \frac{1 - \cos 2\theta}{2}</math>
            </td>

              <td>
                  <math>\cos^2\theta = \frac{1 + \cos 2\theta}{2}</math>
            </td>

              <td>
                  <math>\sin^2\theta \cos^2\theta = \frac{1 - \cos 4\theta}{8}</math>
            </td>

           </tr>




        <tr>
            <td>
               <math>\sin^3\theta = \frac{3 \sin\theta - \sin 3\theta}{4}</math>

            </td>

             <td>

<math>\cos^3\theta = \frac{3 \cos\theta + \cos 3\theta}{4}</math>
            </td>


             <td>
<math>\sin^3\theta \cos^3\theta = \frac{3\sin 2\theta - \sin 6\theta}{32}</math>

            </td>

        </tr>




        
        <tr>
            <td>
                <math>\sin^4\theta = \frac{3 - 4 \cos 2\theta + \cos 4\theta}{8}</math>

            </td>

             <td>
                 <math>\cos^4\theta = \frac{3 + 4 \cos 2\theta + \cos 4\theta}{8}</math>

            </td>


             <td>

<math>\sin^4\theta \cos^4\theta = \frac{3-4\cos 4\theta + \cos 8\theta}{128}</math>
            </td>

        </tr>




        
        <tr>
            <td>
<math>\sin^5\theta = \frac{10 \sin\theta - 5 \sin 3\theta + \sin 5\theta}{16}</math>

            </td>

             <td>
<math>\cos^5\theta = \frac{10 \cos\theta + 5 \cos 3\theta + \cos 5\theta}{16}</math>

            </td>


             <td>
                 <math>\sin^5\theta \cos^5\theta = \frac{10\sin 2\theta - 5\sin 6\theta + \sin 10\theta}{512}</math>

            </td>

        </tr>
        </table>
    </div>


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