Solving sin(𝑧)=2 (Trigonometric Equations with Complex Numbers)

solving_sin_z_equals_2.ipynb

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    "# Solving sin(z)=2 (Trigonometric Equations with Complex Numbers)\n",
    "\n",
    "## Step 1: Writing sin(z) in terms of z\n",
    "\n",
    "Start with Euler's formula:\n",
    "\n",
    "$$\n",
    "e^{i\\theta} = \\cos\\theta + i\\sin\\theta\n",
    "$$\n",
    "\n",
    "Put $\\theta = z$:\n",
    "\n",
    "$$\n",
    "e^{iz} = \\cos{z} + i\\sin{z}\n",
    "$$\n",
    "\n",
    "Put $\\theta = -z$:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "e^{-iz} &= \\cos(-z) + i\\sin(-z) \\\\\n",
    "&= \\cos{z} - i\\sin{z}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Subtract the 2 formulae above and make $\\sin{z}$ the subject:\n",
    "\n",
    "$$\n",
    "\\begin{array}{rrl}\n",
    "  &  e^{iz} \\!\\!\\!\\!\\! &= \\cos{z} + i\\sin{z} \\\\\n",
    "- & e^{-iz} \\!\\!\\!\\!\\! &= \\cos{z} - i\\sin{z} \\\\\n",
    "\\hline\n",
    "& e^{iz} - e^{-iz} \\!\\!\\!\\!\\! &= 2i\\sin{z} \\\\\n",
    "& \\implies \\sin{z} \\!\\!\\!\\!\\! &= \\displaystyle\\frac{e^{iz} - e^{-iz}}{2i}\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "## Step 2: Solving the equation\n",
    "\n",
    "Substitute the result from above:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\sin{z} &= 2 \\\\\n",
    "\\displaystyle\\frac{e^{iz} - e^{-iz}}{2i} &= 2 \\\\\n",
    "e^{iz} - e^{-iz} &= 4i \\\\\n",
    "(e^{iz})^2 - 1 &= 4i(e^{iz}) \\\\\n",
    "(e^{iz})^2 - 4i(e^{iz}) - 1 &= 0 \\\\\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Apply quadratic formula by substituting $u = e^{iz}$:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "e^{iz} &= \\displaystyle\\frac{-(-4i) \\pm \\sqrt{(-4i)^2 - 4(1)(-1)}}{2(1)} \\\\\n",
    "&= (2 \\pm \\sqrt{3})i \\\\\n",
    "iz &= \\ln((2 \\pm \\sqrt{3})i) \\\\\n",
    "&= \\ln(2 \\pm \\sqrt{3}) + \\ln(i) \\\\\n",
    "iz \\cdot (-i) &= \\left[ \\ln(2 \\pm \\sqrt{3}) + \\ln(i) \\right] \\cdot (-i) \\\\\n",
    "z &= -\\ln(2 \\pm \\sqrt{3})i - \\ln(i)i\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "## Step 3: Calculating ln(i)\n",
    "\n",
    "Calculate $\\ln(i)$ by writing $i$ in exponential form:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "i &= \\cos\\theta + i\\sin\\theta \\\\\n",
    "&= \\cos(\\pi/2 + 2k\\pi) + i\\sin(\\pi/2 + 2k\\pi) & \\forall k \\in \\mathbb{Z} \\\\\n",
    "&= e^{(\\pi/2 + 2k\\pi)i} \\\\\n",
    "\\ln(i) &= \\ln(e^{(\\pi/2 + 2k\\pi)i}) \\\\\n",
    "&= (\\pi/2 + 2k\\pi)i\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "## Step 4: Rewriting the solution\n",
    "\n",
    "Rewrite the solution:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "z &= -\\ln(2 \\pm \\sqrt{3})i - \\ln(i)i \\\\\n",
    "&= -\\ln(2 \\pm \\sqrt{3})i - [(\\pi/2 + 2k\\pi)i]i \\\\\n",
    "&= -\\ln(2 \\pm \\sqrt{3})i + (\\pi/2 + 2k\\pi) \\\\\n",
    "&= (\\pi/2 + 2k\\pi) - \\ln(2 \\pm \\sqrt{3})i\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "--------\n",
    "\n",
    "This gist was inspired by [Math for fun, sin(z)=2](https://youtu.be/3C_XD_cCeeI)."
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