Period of \sin^2 x+\cos^4 x?

p3sa1bynn

p3sa1bynn

Answered question

2022-04-25

Period of sin2x+cos4x?

Answer & Explanation

Jayla Matthews

Jayla Matthews

Beginner2022-04-26Added 18 answers

Since
sin(x+π2)=cos(x)  and  cos(x+π2)=sin(x)
it is clear that sin2+cos4 is periodic with period π2. But is π2 the smallest period? Yes, because
sin2+cos4)(x)=2sin(x)cos(x)4cos3(x)sin(x)
=2sin(x)cos(x)×(12cos2(x))
=2sin(x)cos(x)×(sin2(x)cos2(x))
=sin(2x)cos(2x)
=sin(4x)2
and so obviously π2 is the smallest period of (sin2+cos4)
bailaretzy33

bailaretzy33

Beginner2022-04-27Added 15 answers

=1cos2(x)+cos4(x)=1cos2(x)(1cos2(x))=1(cos(x)sin(x))2=1(14)sin2(2x)=(18)(7+cos(4x))

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