solve sin z 2: A Comprehensive Guide to Solving the Equation \(\sin z = 2\)
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Introduction
The equation \(\sin z = 2\) may seem straightforward at first glance, but it opens the door to a rich exploration of complex analysis, transcendental functions, and the nature of solutions in the complex plane. Unlike real numbers, where the sine function is limited to the interval \([-1, 1]\), the extension to complex numbers allows the sine function to take on any complex value, including values outside this range. This tutorial aims to provide a thorough understanding of how to solve \(\sin z = 2\), including the derivation of solutions, the role of complex logarithms, the structure of solutions in the complex plane, and related concepts.
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Understanding the Sine Function in the Complex Plane
The Sine Function: Real vs. Complex
In real analysis, the sine function, \(\sin x\), is well-understood within the interval \([-1, 1]\). Its graph oscillates between these bounds, with zeros at integer multiples of \(\pi\), and is periodic with period \(2\pi\).
However, when extended to the complex plane, \(z = x + iy\), the sine function exhibits a much richer behavior. The complex sine function is defined via its exponential form:
\[ \sin z = \frac{e^{i z} - e^{-i z}}{2i} \]
This expression allows for solutions where \(\sin z\) exceeds the real bounds of \([-1, 1]\). Specifically, \(\sin z\) can take on any complex value, including real numbers greater than 1, such as 2.
Basic Properties
- Periodicity: \(\sin z\) is periodic with period \(2\pi\).
- Complex Values: For complex \(z\), \(\sin z\) can be unbounded.
- Zeroes: \(\sin z = 0\) at \(z = n\pi\), where \(n\) is an integer.
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Solving \(\sin z = 2\)
Recognizing the Nature of the Equation
Since \(\sin z = 2\) involves a complex variable, the solutions are complex numbers \(z\) satisfying the equation. Given the definition:
\[ \sin z = \frac{e^{i z} - e^{-i z}}{2i} \]
we aim to find all \(z\) such that:
\[ \frac{e^{i z} - e^{-i z}}{2i} = 2 \]
Multiplying both sides by \(2i\):
\[ e^{i z} - e^{-i z} = 4i \]
This equation involves exponential functions, which suggests solving using logarithmic methods and properties of complex analysis.
Deriving the Solutions
Step 1: Express \(e^{i z}\) in terms of a single variable.
Let:
\[ w = e^{i z} \]
Then:
\[ e^{-i z} = \frac{1}{w} \]
Our transformed equation becomes:
\[ w - \frac{1}{w} = 4i \]
Step 2: Multiply through by \(w\): It's also worth noting how this relates to size of business card.
\[ w^2 - 1 = 4i w \]
Rearranged as a quadratic in \(w\):
\[ w^2 - 4i w - 1 = 0 \]
Step 3: Solve the quadratic:
\[ w = \frac{4i \pm \sqrt{(4i)^2 - 4 \times 1 \times (-1)}}{2} \]
Calculate the discriminant:
\[ (4i)^2 - 4 \times 1 \times (-1) = 16 i^2 + 4 = 16(-1) + 4 = -16 + 4 = -12 \]
Thus:
\[ w = \frac{4i \pm \sqrt{-12}}{2} \] Some experts also draw comparisons with 3 4 solving complex 1 variable equations answer key.
Express \(\sqrt{-12}\):
\[ \sqrt{-12} = \sqrt{12} \times i = 2 \sqrt{3} \, i \]
Therefore:
\[ w = \frac{4i \pm 2 \sqrt{3} i}{2} = 2i \pm \sqrt{3} i \]
Factor out \(i\):
\[ w = i (2 \pm \sqrt{3}) \]
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Expressing \(z\) in terms of \(w\) This concept is also deeply connected to planning in business management function.
Recall that:
\[ w = e^{i z} \]
Taking the natural logarithm:
\[ i z = \ln w + 2 \pi i n, \quad n \in \mathbb{Z} \]
The addition of \(2 \pi i n\) accounts for the multi-valued nature of the complex logarithm.
Thus:
\[ z = -i \ln w + 2 \pi n \]
Substitute \(w\):
\[ z = -i \ln \left(i (2 \pm \sqrt{3})\right) + 2 \pi n \]
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Computing the Logarithm
The logarithm of a complex number \(w = r e^{i \theta}\) is:
\[ \ln w = \ln r + i \theta \]
where:
- \(r = |w|\), the modulus of \(w\),
- \(\theta = \arg(w)\), the argument of \(w\).
Calculate the modulus:
\[ |w| = |i (2 \pm \sqrt{3})| = |i| \times |2 \pm \sqrt{3}| = 1 \times |2 \pm \sqrt{3}| \]
Since \(2 + \sqrt{3} > 0\) and \(2 - \sqrt{3} > 0\), both are positive real numbers:
\[ |w| = 2 \pm \sqrt{3} \]
Calculate the arguments:
\[ \arg(w) = \arg(i (2 \pm \sqrt{3})) = \arg(i) + \arg(2 \pm \sqrt{3}) \]
- \(\arg(i) = \frac{\pi}{2}\),
- \(2 \pm \sqrt{3}\) are positive real numbers, so:
\[ \arg(2 \pm \sqrt{3}) = 0 \]
Therefore:
\[ \arg(w) = \frac{\pi}{2} + 0 = \frac{\pi}{2} \]
Now, the logarithm:
\[ \ln w = \ln |w| + i \arg(w) = \ln (2 \pm \sqrt{3}) + i \frac{\pi}{2} \]
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Final Expression for \(z\)
Recall:
\[ z = -i \left( \ln (2 \pm \sqrt{3}) + i \frac{\pi}{2} \right) + 2 \pi n \]
Distribute \(-i\):
\[ z = -i \ln (2 \pm \sqrt{3}) - i \times i \frac{\pi}{2} + 2 \pi n \]
Since \(i \times i = -1\):
\[ z = -i \ln (2 \pm \sqrt{3}) + \frac{\pi}{2} + 2 \pi n \]
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Summary of Solutions
The solutions to \(\sin z = 2\) are:
\[ \boxed{ z = \frac{\pi}{2} - i \ln (2 \pm \sqrt{3}) + 2 \pi n, \quad n \in \mathbb{Z} } \]
where the \(\pm\) accounts for the two roots derived from the quadratic solution.
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Geometric Interpretation and Properties of Solutions
Distribution in the Complex Plane
- The solutions are infinitely many, forming a discrete set in the complex plane.
- Each solution differs by a multiple of \(2\pi\) along the real axis, reflecting the periodicity of sine.
- The imaginary part involves the logarithm of real constants, resulting in fixed shifts vertically in the complex plane.
Nature of the Solutions
- These solutions are complex numbers with non-zero imaginary parts, indicating that solutions are not on the real axis.
- The imaginary parts involve \(\ln(2 \pm \sqrt{3})\), which are positive real numbers, ensuring the solutions are located away from the real axis.
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Related Concepts and Extensions
Sine Function Outside the Range \([-1, 1]\)
In real analysis, \(\sin x = 2\) has no solutions because \(\sin x\) is bounded. In complex analysis, the sine function is unbounded, allowing solutions for any complex value.
Multi-Valued Nature of Complex Logarithm
The logarithm function in complex analysis is multi-valued due to the periodic nature of the complex exponential:
\[ \ln w = \ln r + i (\theta + 2 \pi n) \]
Hence, solutions involve an infinite set parameterized by \(n \in \mathbb{Z}\).
General Solution for \(\sin z = c\)
For any complex constant
