Let V → 1 and V → 2 are two vectors such that V → 1 = 2 ( sin ⁡ α + cos ⁡ α ) i ^ + j ^ and V → 2 = sin ⁡ β i ^ + cos ⁡ β j ^ , where α and β satisfy the relation 2 ( sin ⁡ α + cos ⁡ α ) sin ⁡ β = 3 − cos ⁡ β , Find Value of ( 3 tan 2 ⁡ α + 4 tan 2 ⁡ β )

Question

Let                     V        →              1   and                     V        →              2   are two vectors such that                     V        →              1    =  2  (  sin  ⁡  α  +  cos  ⁡  α  )            i      ^        +            j      ^       and                     V        →              2    =  sin  ⁡  β              i      ^        +  cos  ⁡  β              j      ^      , where   α and   β satisfy the relation   2  (  sin  ⁡  α    +  cos  ⁡  α  )  sin  ⁡  β  =  3  −  cos  ⁡  β  , Find Value of   (  3      tan    2    ⁡  α    +  4      tan    2    ⁡  β  )

Quinn Alvarez · Accepted Answer

Doesn&#039;t really have anything to do with vectors.Note that the maximum value of   a  sin  ⁡  θ  +  b  cos  ⁡  θ is             a      2        +          b      2      , which occurs when   tan  ⁡  θ  =      a    b  . Here, we&#039;ve been given that:  2  (  sin  ⁡  α  +  cos  ⁡  α  )  sin  ⁡  β  +  cos  ⁡  β  =  3Now, maximum value of LHS is      4    (    sin    ⁡    α    +    cos    ⁡    α          )      2        +    1    =      5    +    4    sin    ⁡    2    α  Notice that       5    +    4    sin    ⁡    2    α    ≤  3, reaching maximum value at   sin  ⁡  2  α  =  1 Thus, maximum value of LHS, is 3, which is reached when   sin  ⁡  2  α  =  1 and   tan  ⁡  β  =  2  (  sin  ⁡  α  +  cos  ⁡  α  ). Now,   sin  ⁡  2  α  =  1    ⟹        tan    2    ⁡  α  =  1, and       tan    2    ⁡  β  =  4  (  1  +  sin  ⁡  2  α  )  =  8. Thus,   3      tan    2    ⁡  α  +  4      tan    2    ⁡  β  =  3  ⋅  1  +  4  ⋅  8  =  35

Let vec(V)_1 and vec(V)_2 are two vectors such that vec(V)_1=2(sin alpha+cos alpha)hat(i)+hat(j) and vec(V)_2=sin beta hat(i)+cos beta hat(j), where alpha and beta satisfy the relation 2(sin alpha+cos alpha)sin beta=3-cos beta, Find Value of (3 tan^2 alpha+4 tan^2 beta)

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