Let f ( x ) = p cos ⁡ x + q sin ⁡ x , | p | + | q | ≠ 0 where p , q ∈ R and | f ( x ) | ≤ 1.Let α , β be the roots of the equation f ( x ) = 1 , | α − β | = k π , k ∈ R , then the find the possible values of k.

Question

Let   f  (  x  )  =  p  cos  ⁡  x  +  q  sin  ⁡  x  ,      |    p      |    +      |    q      |    ≠  0 where   p  ,  q  ∈  R and       |    f  (  x  )      |    ≤  1.Let   α  ,  β be the roots of the equation   f  (  x  )  =  1  ,      |    α  −  β      |    =  k  π  ,  k  ∈  R  , then the find the possible values of k.

ciflantsi1pt9 · Accepted Answer

f  (  x  )  =            p      2        +          q      2        sin  ⁡  (  x  +  arctan  ⁡  (  p      q          −      1        )  )      ⟹    1  =  f  (  α  )  ≤            p      2        +          q      2        =  f  (            π      2        −  arctan  ⁡  (  p      q          −      1        )  ≤  1      ⟹    f  (  x  )  =  sin  ⁡  (  x  +  arctan  ⁡  (  p      q          −      1        )  )f(x) is just a translated sine function. Therefore α is a solution to f(x)=1 iff so is   α  +  2  n  π where   n  ∈      Z  . So the set we&#039;re looking for is       2    N    ∪  {  0  }

Let f ( x ) = p cos ⁡ x + q sin ⁡<!-

Answered question

Answer & Explanation

New Questions in Trigonometry