MathematicsIf sin(A−B)= 1 2    and cos(A+B)=0   0 ° B ,   then which of the following are the measures of the angles A and B  ?

If sin(AB)= 1 2    and cos(A+B)=0   0 ° <A+B 90 ° ,A>B ,   then which of the following are the measures of the angles A and B  ?


  1. A
    A= 40 ° ,B= 50 °  
  2. B
    A= 50 ° ,B= 40 °  
  3. C
    A= 60 ° ,B= 30 °  
  4. D
    A= 70 ° ,B= 30 °   

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    Solution:

    The given trigonometric equations are,
    sin(AB)= 1 2  
    cos(A+B)=0   0 ° <A+B 90 ° ,A>B  
    We have,
    sin(AB)= 1 2  
    sin(AB)=sin 30 °  
    AB= 30 °                          (1)  
    Also,
    cos(A+B)=0  
    cos(A+B)=cos 90 °  
    A+B= 90 °                          (2)  
    By adding equations (1) and (2), we obtain,
    2A= 120 °  
    A= 60 °  
    Therefore, from equation (1), we have,
    60 ° B= 30 °  
    B= 30 °  
    Therefore, the measures of the angles A= 60 °  & B= 30 ° .  
    Hence, option (3) is correct.
     
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