Given cos⁡(t)=(12)(eit+e−it) solve for cos−1(x) The hint says to let x=cos⁡(t) and z=eit. So I...

Ethen Wong

Ethen Wong

Answered

2022-01-28

Given cos(t)=(12)(eit+eit) solve for cos1(x)
The hint says to let x=cos(t) and z=eit. So I started first by substituting:
x=12z+1z --> multiply both sides by 2z
2xz=z2+1
z22zx+1=0 --> quadratic formula, solve for z
z=x±x21
Im

Answer & Explanation

Ydaxq

Ydaxq

Expert

2022-01-29Added 12 answers

You've done the hard part. Choosing the positive square root, you have it=log(x+x21) (where log=ln). It's easy enough to solve for t by multiplying by -i.
Note that xx21=1x+x21,so indeed,
it=log(x+x21)=log(xx21)

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