MathematicsIn △ABC, the median AD divides ∠BAC  such that ∠BAD:∠CAD=2:1. Then cos(A3)  is equal to

In △ABC, the median AD divides ∠BAC  such that ∠BAD:∠CAD=2:1. Then cos(A3)  is equal to


  1. A
    sinB2sinC
  2. B
    sinC2sinB
  3. C
    2sinBsinC
  4. D
    None of These 

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    Solution:

    seoWe are given that the median AD divides  ∠BAC such that  ∠BAD:∠CAD=2:1
    BADCAD=21
    ⇒∠BAD = 2∠CAD...................… (1)
    Since we know ∠BAC=∠BAD+∠CAD
    Substitute the values in equation (1)
    ⇒∠BAC = 2∠CAD+∠CAD
    ⇒∠BAC = 3∠CAD
    We write  ∠BAC=A
    ⇒A = 3∠CAD
    Divide both sides by 3
    A3=CAD.................. (2)
    Substitute value from equation (2) in (1)
     BAD=2A3.................. (3)
    Now we apply the law of sine in both triangles made by the median separately.
    In △BAD
    BDsinBAD=ADsinB
    a2sin2A3=ADsinB  
    a2sinBsin2A3=AD     ................. (4)
    In △CAD
    CDsinCAD=ADsinC    
    a2sinA3=ADsinC   
    a2sinCsinA3=AD .................... (5)
    Since AD is the median and will have fixed length, we equate values from (4) and (5)
    a2sinBsin2A3=a2sinCsinA3   
    sinBsin2A3=sinCsinA3    
    Since, sin2x=2sinxcosx  sinB2sinA3cosA3=sinCsinA3    
    sinB2cosA3=sinC1   sinB2sinC=cosA3     The value of  cosA3 is equal to sinB2sinC
    ∴ Option 1 is correct.
     
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