Question

Separate into real and imaginary parts cos^-1(3i/4)​

Answer 1

Answer:

Check: cos(π/2 - i ln 2) = cos π/2 cos (i ln 2) + sin π/2 sin (i ln 2)

= sin (i ln 2) = [exp(i · i ln 2) - exp(-i · i ln 2)]/(2i)

= [exp(- ln 2) - exp(ln 2)]/(2i) = (-i/2)[1/2 - 2]

= (-i/2)(-3/2) = 3i/4.

Step-by-step explanation:

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Answer 2

Given : cos⁻¹(3i/4)

To Find: Real and Imaginary Parts

Step-by-step explanation:

• This is a complex number cos⁻¹(3i/4)
• It can be separated into real and imaginary parts
• Real part consists of a numerical value and imaginary part does not have any numerical value.

• Let cos⁻¹ 3i/4 =x + iy
• ∴3i/4=cos(x + iy)=cosx  coshy − isinxsinhy
• Comparing real and imaginary parts,
• cosxcoshy=0 ∴cosx=0 ∴x=π/2
• −sinxsinhy=3i/4  but sinx=sin(π/2)=1
• ∴sinhy=−3/4
• ∴y=sinh⁻¹ −3/4=log(−3/4+$\sqrt{1+ 9/16}$)=log(1/2)=−log2

Hence,

real part =π/2 and

imaginary parts = -log 2