Write the expression in terms of the 1st power of cosine: sin^4x

Zachariah Norris

Zachariah Norris

Answered question

2022-09-20

Write the expression in terms of the 1st power of cosine:
sin 4 x

Answer & Explanation

espovilham7

espovilham7

Beginner2022-09-21Added 10 answers

We know that cos 2 x = 1 2 sin 2 x sin 2 = 1 2 ( 1 cos 2 x )
So sin 4 x = ( sin 2 x ) 2 = [ 1 2 ( 1 cos 2 x ) ] 2 sin 4 x = 1 4 ( 1 2 cos 2 x + cos 2 x ]
Now use the identity cos 2 x = 1 2 ( 1 + cos 2 x )
So we obtain
sin 4 x = 1 4 [ 1 2 cos 2 x + cos 2 2 x ] = 1 4 [ 1 2 cos 2 x + 1 2 ( 1 + cos 4 x ) ] = 1 4 1 2 cos 2 x + 1 9 cos 4 x sin 4 x = 1 8 cos 4 x 1 2 cos 2 x + 3 8

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