Separate into Real and Imaginary Parts of Cos − 1 ( 3 I 4 ) - Applied Mathematics 1

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

Solution

Let a+ib= cos"^-1((3i)/4)         .................(1)

∴ cos(a+ib)=(3i)/4

∴ cos (a) cos(ib)-sin (a) sin(ib)=(3i)/4

cos(a)cosh(b) – isin(a)sinh(b) =0+(3i)/4  {∵ cos (ix)=cosh(ix)=sinh(x)}

Comparing Real and Imaginary terms on both sides,

cos(a)cosh(b)=0           ...............(2)    &   -sin(a)sinj(b)=3/4......(3)

From (2), cos(a)=0 or cosh(b)=0,

∴ a=pi/2                  ....................(4)

"From" (3) & (4), -sin(pi/2)sin(b)=3/4

∴ 1.sinh(b)=(-3)/4

∴ b= sinh"^1((-3)/4)

= log[((-3)/4)+sqrt(((-3)/4)^2+1)]  {∵ sinh"^-1z=log (z+sqrt(z^2+1))}

= Log [((-3)/4)+sqrt(9/16+1)]

= log[((-3)/4)+5/4]

=log  1/2

= log2^-1

∴ b=-log2                            ........(5)

Substituting (4) & (5) in (1), cos^-1((3i)/4)=pi/2-i log2

Comparing Real and Imaginary terms on both sides,

Real part=a= pi/2

Imaginary part=b=-log2

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions
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