How to find the smallest value of sin 4 ⁡ ( α ) + cos...

Theresa Archer

Theresa Archer

Answered

2022-06-30

How to find the smallest value of sin 4 ( α ) + cos 4 ( α ) ?

Answer & Explanation

America Barrera

America Barrera

Expert

2022-07-01Added 23 answers

Because actually
sin 4 ( α ) + cos 4 ( α ) = ( sin 2 ( α ) + cos 2 ( α ) ) 2 2 sin 2 ( α ) cos 2 ( α ) = 1 1 2 sin 2 ( 2 α ) ,
Ayanna Trujillo

Ayanna Trujillo

Expert

2022-07-02Added 13 answers

sin 4 x + cos 4 x = ( sin 2 x + cos 2 x ) 2 2 sin 2 x cos 2 x = 1 ( sin 2 x ) 2 2
Now 0 sin 2 2 x 1
Alternatively using cos 2 x = 1 2 sin 2 x = 2 cos 2 x 1,
( 2 sin 2 x ) 2 + ( 2 cos 2 x ) 2 = ( 1 cos 2 x ) 2 + ( 1 + cos 2 x ) 2 = 2 ( 1 + cos 2 2 x ) = 2 + 1 + cos 4 x
Now 1 cos 4 x 1

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