[<< wikibooks] Trigonometry/Sum into Product
These are exercises on the formulae derived in Book 1 for converting the sum or difference of two sines or two cosines into a product.
[1] 
  
    
      
        
          
            
              sin
              ⁡
              7
              θ
              −
              sin
              ⁡
              5
              θ
            
            
              cos
              ⁡
              7
              θ
              +
              cos
              ⁡
              5
              θ
            
          
        
        =
        tan
        ⁡
        θ
      
    
    {\displaystyle {\frac {\sin 7\theta -\sin 5\theta }{\cos 7\theta +\cos 5\theta }}=\tan \theta }
  

[2] 
  
    
      
        
          
            
              cos
              ⁡
              6
              α
              −
              cos
              ⁡
              4
              α
            
            
              sin
              ⁡
              6
              α
              +
              sin
              ⁡
              4
              α
            
          
        
        =
        −
        tan
        ⁡
        α
      
    
    {\displaystyle {\frac {\cos 6\alpha -\cos 4\alpha }{\sin 6\alpha +\sin 4\alpha }}=-\tan \alpha }
  
[3] 
  
    
      
        
          
            
              sin
              ⁡
              A
              +
              sin
              ⁡
              3
              A
            
            
              cos
              ⁡
              A
              +
              cos
              ⁡
              3
              A
            
          
        
        =
        tan
        ⁡
        2
        A
      
    
    {\displaystyle {\frac {\sin A+\sin 3A}{\cos A+\cos 3A}}=\tan 2A}
  
[4] 
  
    
      
        
          
            
              sin
              ⁡
              7
              A
              −
              sin
              ⁡
              A
            
            
              sin
              ⁡
              8
              A
              −
              sin
              ⁡
              2
              A
            
          
        
        =
        cos
        ⁡
        4
        A
        sec
        ⁡
        5
        A
      
    
    {\displaystyle {\frac {\sin 7A-\sin A}{\sin 8A-\sin 2A}}=\cos 4A\sec 5A}
  
[5] 
  
    
      
        
          
            
              cos
              ⁡
              2
              ϕ
              +
              cos
              ⁡
              2
              θ
            
            
              cos
              ⁡
              2
              ϕ
              −
              cos
              ⁡
              2
              θ
            
          
        
        =
        cot
        ⁡
        
          (
          
            ϕ
            +
            θ
          
          )
        
        cot
        ⁡
        
          (
          
            ϕ
            −
            θ
          
          )
        
      
    
    {\displaystyle {\frac {\cos 2\phi +\cos 2\theta }{\cos 2\phi -\cos 2\theta }}=\cot \left(\phi +\theta \right)\cot \left(\phi -\theta \right)}
  
[6] 
  
    
      
        
          
            
              sin
              ⁡
              2
              A
              +
              sin
              ⁡
              2
              B
            
            
              sin
              ⁡
              2
              A
              −
              sin
              ⁡
              2
              B
            
          
        
        =
        tan
        ⁡
        
          (
          
            A
            −
            B
          
          )
        
        cot
        ⁡
        
          (
          
            A
            −
            B
          
          )
        
      
    
    {\displaystyle {\frac {\sin 2A+\sin 2B}{\sin 2A-\sin 2B}}=\tan \left(A-B\right)\cot \left(A-B\right)}
  
[7] 
  
    
      
        
          
            
              sin
              ⁡
              A
              +
              sin
              ⁡
              2
              A
            
            
              cos
              ⁡
              A
              −
              cos
              ⁡
              2
              A
            
          
        
        =
        cot
        ⁡
        
          (
          
            
              A
              2
            
          
          )
        
      
    
    {\displaystyle {\frac {\sin A+\sin 2A}{\cos A-\cos 2A}}=\cot \left({\frac {A}{2}}\right)}
  
[8] 
  
    
      
        
          
            
              sin
              ⁡
              5
              λ
              −
              sin
              ⁡
              3
              λ
            
            
              cos
              ⁡
              5
              λ
              +
              cos
              ⁡
              3
              λ
            
          
        
        =
        tan
        ⁡
        λ
      
    
    {\displaystyle {\frac {\sin 5\lambda -\sin 3\lambda }{\cos 5\lambda +\cos 3\lambda }}=\tan \lambda }
  
[9] 
  
    
      
        
          
            
              cos
              ⁡
              2
              B
              −
              cos
              ⁡
              2
              A
            
            
              sin
              ⁡
              2
              B
              +
              sin
              ⁡
              2
              A
            
          
        
        =
        tan
        ⁡
        
          (
          
            A
            −
            B
          
          )
        
      
    
    {\displaystyle {\frac {\cos 2B-\cos 2A}{\sin 2B+\sin 2A}}=\tan \left(A-B\right)}
  
[10] 
  
    
      
        cos
        ⁡
        
          (
          
            ϕ
            +
            θ
          
          )
        
        +
        sin
        ⁡
        
          (
          
            ϕ
            −
            θ
          
          )
        
        =
        2
        sin
        ⁡
        
          (
          
            
              45
              
                o
              
            
            +
            ϕ
          
          )
        
        cos
        ⁡
        
          (
          
            
              45
              
                o
              
            
            +
            θ
          
          )
        
      
    
    {\displaystyle \cos \left(\phi +\theta \right)+\sin \left(\phi -\theta \right)=2\sin \left(45^{o}+\phi \right)\cos \left(45^{o}+\theta \right)}
  
[11] 
  
    
      
        
          
            
              sin
              ⁡
              α
              +
              sin
              ⁡
              β
            
            
              sin
              ⁡
              α
              −
              sin
              ⁡
              β
            
          
        
        =
        tan
        ⁡
        
          (
          
            
              
                α
                +
                β
              
              2
            
          
          )
        
        cot
        ⁡
        
          (
          
            
              
                α
                −
                β
              
              2
            
          
          )
        
      
    
    {\displaystyle {\frac {\sin \alpha +\sin \beta }{\sin \alpha -\sin \beta }}=\tan \left({\frac {\alpha +\beta }{2}}\right)\cot \left({\frac {\alpha -\beta }{2}}\right)}
  
[12] 
  
    
      
        
          
            
              cos
              ⁡
              ψ
              +
              cos
              ⁡
              ω
            
            
              cos
              ⁡
              ω
              −
              cos
              ⁡
              ψ
            
          
        
        =
        cot
        ⁡
        
          (
          
            
              
                ψ
                +
                ω
              
              2
            
          
          )
        
        cot
        ⁡
        
          (
          
            
              
                ψ
                −
                ω
              
              2
            
          
          )
        
      
    
    {\displaystyle {\frac {\cos \psi +\cos \omega }{\cos \omega -\cos \psi }}=\cot \left({\frac {\psi +\omega }{2}}\right)\cot \left({\frac {\psi -\omega }{2}}\right)}
  
[13] 
  
    
      
        
          
            
              sin
              ⁡
              ϕ
              +
              sin
              ⁡
              θ
            
            
              cos
              ⁡
              ϕ
              +
              cos
              ⁡
              θ
            
          
        
        =
        tan
        ⁡
        
          (
          
            
              
                ϕ
                +
                θ
              
              2
            
          
          )
        
      
    
    {\displaystyle {\frac {\sin \phi +\sin \theta }{\cos \phi +\cos \theta }}=\tan \left({\frac {\phi +\theta }{2}}\right)}
  
[14] 
  
    
      
        
          
            
              sin
              ⁡
              A
              −
              sin
              ⁡
              B
            
            
              cos
              ⁡
              B
              −
              cos
              ⁡
              A
            
          
        
        =
        cot
        ⁡
        
          (
          
            
              
                A
                +
                B
              
              2
            
          
          )
        
      
    
    {\displaystyle {\frac {\sin A-\sin B}{\cos B-\cos A}}=\cot \left({\frac {A+B}{2}}\right)}
  
[15] 
  
    
      
        
          
            
              cos
              ⁡
              3
              A
              −
              cos
              ⁡
              A
            
            
              sin
              ⁡
              3
              A
              −
              sin
              ⁡
              A
            
          
        
        +
        
          
            
              cos
              ⁡
              2
              A
              −
              cos
              ⁡
              4
              A
            
            
              sin
              ⁡
              4
              A
              −
              sin
              ⁡
              2
              A
            
          
        
        =
        
          
            
              sin
              ⁡
              A
            
            
              cos
              ⁡
              2
              A
              cos
              ⁡
              3
              A
            
          
        
      
    
    {\displaystyle {\frac {\cos 3A-\cos A}{\sin 3A-\sin A}}+{\frac {\cos 2A-\cos 4A}{\sin 4A-\sin 2A}}={\frac {\sin A}{\cos 2A\cos 3A}}}
  
[16] 
  
    
      
        a
        cos
        ⁡
        ϕ
        +
        b
        sin
        ⁡
        ϕ
        =
        
          
            
              a
              
                2
              
            
            +
            
              b
              
                2
              
            
          
        
        cos
        ⁡
        [
        ϕ
        −
        
          tan
          
            −
            1
          
        
        ⁡
        (
        
          
            b
            a
          
        
        )
        ]
      
    
    {\displaystyle a\cos \phi +b\sin \phi ={\sqrt {a^{2}+b^{2}}}\cos[\phi -\tan ^{-1}({\frac {b}{a}})]}