Find the number of solutions to the equation sin ⁡ ( π x 4 ) + cos ⁡ ( π 2 − x 4 ) = 2

Question

Find the number of solutions to the equation   sin  ⁡      (                  π                  x                    4        )    +  cos  ⁡      (                  π                  2          −          x                    4        )    =      2

Stella Andrade · Accepted Answer

From the comments, the domain is   [  0  ,  2  ]Let   f  (  x  )  =  sin  ⁡      (                  π                  x                    4        )    +  cos  ⁡      (                  π                  2          −          x                    4        )    −      2        f    ′    (  x  )  =      1          2              x                  π    4    cos  ⁡      (                  π                  x                    4        )    +      1          2              2        −        x                  π    4    sin  ⁡      (          π      4              2      −      x        )  Angles of sine and cosine are varying from 0 to       π          2              2            . That means       f    ′    (  x  )  &amp;gt;  0. Thus it crosses x-axis either once or never.Also, since f(x) is increasing, we only need to check values at domain end points.  f  (  0  )  =  cos  ⁡      (          π              2                  2                      )    −      2    &amp;lt;  0  f  (  2  )  =  sin  ⁡      (          π              2                  2                      )    +  1  −      2    4  &amp;gt;  2      2      ⟹        π    4    &amp;lt;      π          2              2                ⟹    sin  ⁡      (          π              2                  2                      )    &amp;gt;      1          2        &amp;gt;  0.7Thus,   f  (  2  )  &amp;gt;  0Thus, the given equation has one and only one solution.

Find the number of solutions to the equation sin((pi sqrt(x))/4)+cos((pi sqrt(2−x))/4)=sqrt 2

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