The question is to find x ∈ ( 0 , π 2 ) 3 − 1 sin ⁡ x + 3 + 1 cos ⁡ x = 4 2

Question

The question is to find   x  ∈      (    0    ,          π      2        )                      3            −      1              sin      ⁡      x        +                    3            +      1              cos      ⁡      x        =  4      2

Morisio64moc · Accepted Answer

Another approach is to write the equation as  (      3    −  1  )  cos  ⁡  x  +  (      3    +  1  )  sin  ⁡  x  =  4      2    sin  ⁡  x  cos  ⁡  xthen rearranging gives                    3            −      1              2              2              cos  ⁡  x  +                    3            +      1              2              2              sin  ⁡  x  =  2  sin  ⁡  x  cos  ⁡  xthen rearranging gives                    3            −      1              2              2              cos  ⁡  x  +                    3            +      1              2              2              sin  ⁡  x  =  2  sin  ⁡  x  cos  ⁡  xNow note that   (                    3            −      1              2              2                  )    2    +  (                    3            +      1              2              2                  )    2    =      1    8    (  (  3  −  2      3    +  1  )  +  (  3  +  2      3    +  1  )  =  1 so that we can write                     3            −      1              2              2             as the   sin of some angle, say   α, and identify                     3            +      1              2              2             as   cos  ⁡  αThen you have to solve   sin  ⁡  (  α  +  x  )  =  sin  ⁡  (  2  x  ), with   tan  ⁡  α  =                    3            −      1                      3            +      1

Kade Ibarra · Accepted Answer

Replace       3    =  tan  ⁡      60    ∘    ,  1  =  tan  ⁡      45    ∘  to find  sin  ⁡  (      15    ∘    +  x  )  =  2  sin  ⁡  x  cos  ⁡  x  =  sin  ⁡  2  xNow if   sin  ⁡  y  =  sin  ⁡  A  ,  y  =  n      180    ∘    +  (  −  1      )    n    A where n is any integer      3    −  1  =  tan  ⁡      60    ∘    −  tan  ⁡      45    ∘    =                    sin        ⁡        (        60        −        45                  )          ∘                            cos        ⁡                  60          ∘                cos        ⁡                  45          ∘                      =  2      2    sin  ⁡      15    ∘  Similarly,       3    +  1  =  2      2    sin  ⁡      75    ∘    =  2      2    cos  ⁡      15    ∘

The question is to find x ∈ ( 0 , π

Answered question

Answer & Explanation

New Questions in Precalculus