If \sin^8(x)+\cos^8(x)=\frac{48}{128} then find the value of x?

Jazmine Sweeney

Jazmine Sweeney

Answered question

2022-04-22

If sin8(x)+cos8(x)=48128 then find the value of x?

Answer & Explanation

Brennen Davies

Brennen Davies

Beginner2022-04-23Added 25 answers

Hint:
Use the identity:
(ab)2+(a+b)2=2(a2+b2)
So:
22(sin8(x)+cos8(x))=12((sin4(x)cos4(x))2+(sin4(x)+cos4(x))2)
You can repeat this for all the terms with the form a2+b2, for the other terms simplify using trigonometric identities.
hoppledhsy

hoppledhsy

Beginner2022-04-24Added 13 answers

If you define s=sin2x your equation becomes
s4+(1s)4=48128
2s44s3+6s24s+1=48128
s42s3+3s22s+516=0
for which Alpha finds the ugly real solutions
s=121±113
and the complex solutions
s=121±i11+3
but that doesn't seem very enlightening to me.

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