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Given that ‘z’ is in set of complex number and ‘a’ is any real numbers. Solve the trigonometric equation sin(z) = a for all general solutions.

User Saqibahmad
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2 Answers

22 votes
22 votes

We are given with:


{\quad \qquad \longrightarrow \sin (z)={\sf a}\:,\:z\in \mathbb{C}}

Recall the identity what we have for the sine function of complex numbers


  • {\boxed{\bf{\sin (z)=(e^(\iota z)-e^(-\iota z))/(2\iota)}}}

Put the values to thus obtain:


{:\implies \quad \sf (e^(\iota z)-e^(-\iota z))/(2\iota)=a}


{:\implies \quad \sf e^(\iota z)-e^(-\iota z)=2a\iota}

Multiply both sides by
{\sf e^(\iota z)}


{:\implies \quad \sf e^(\iota z)\cdot e^(\iota z)-e^(-\iota z)\cdot e^(\iota z)=2a\iota e^(\iota z)}


{:\implies \quad \sf (e^(\iota z))^(2)-2a\iota e^(\iota z)-1=0}

Put x =
{\sf e^(\iota z)}:


{:\implies \quad \sf x^(2)-2a\iota x-1=0}

Find the discriminant, here D will be, D = (-2ai)² - 4 × 1 × (-1) = 4 - 4a² = 4(1-a²)

Now, By quadratic formula:


{:\implies \quad \sf x=\frac{-(-2a\iota)\pm \sqrt{4(1-a^(2))}}{2}}


{:\implies \quad \sf x=\frac{a\iota \pm \sqrt{1-a^(2)}}{2}}


{:\implies \quad \sf e^(\iota z)=\frac{a\iota \pm \sqrt{1-a^(2)}}{2}}


{:\implies \quad \sf \iota z=log\bigg(\frac{a\iota \pm \sqrt{1-a^(2)}}{2}\bigg)}

Using the formula for logarithms, we have:


{:\implies \quad \sf \iota z=log(a\iota \pm \sqrt{1-a^(2)})-log(2)}


{:\implies \quad \sf z=(1)/(\iota)log(a\iota \pm \sqrt{1-a^(2)})-(1)/(\iota)log(2)}

The sine function is periodic on 2πn and zero on (π/2), and the logarithmic expression becomes undefined for all ia±√(1-a²) < 0, so we will take modulus of it


{:\implies \quad \boxed{\bf{z=(1)/(\iota)log\bigg|a\iota \pm \sqrt{1-a^(2)}\bigg|-(1)/(\iota)log(2)+(\pi)/(2)+2\pi n\:\:\forall \:n\in \mathbb{Z}}}}

User Lumii
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3.1k points
22 votes
22 votes

Recall that for all
z\in\Bbb C,


\sin(z) = (e^(iz) - e^(-iz))/(2i)

so that


\sin(z) = a \iff e^(iz) - e^(-iz) = 2ia

Multiply both sides by
e^(iz) to get a quadratic equation,


e^(2iz) - 2iae^(iz) - 1 = 0

Solve for
e^(iz). By completing the square,


e^(2iz) - 2ia e^(iz) + i^2a^2 = 1 + i^2a^2


\left(e^(iz) - ia\right)^2 = 1 - a^2


e^(iz) - ia = \pm √(1-a^2)


e^(iz) = ia \pm √(1-a^2)


iz = \log\left(ia \pm √(1-a^2)\right)


iz = \ln\left|ia \pm √(1-a^2)\right| + i \left(\arg\left(ia \pm √(1-a^2)\right) + 2\pi n\right)


\boxedz = -i \ln\left

where n is any integer.

User Isset
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2.5k points