\n", "
Tip: Trigonometric identities
\n", "\n", "Trigonometric identities are also much easier to be recovered this way. For example, let us think about angle addition $\\theta + \\varphi$.\n", "\n", "$$ e^{i(\\theta + \\varphi)} = e^{i \\theta} e^{i \\varphi} $$\n", "\n", "We Apply Euler's formula to each complex number.\n", "\n", "$$\\begin{align}\n", "\\cos (\\theta + \\varphi) + i \\sin(\\theta + \\varphi) \n", "& = (\\cos \\theta + i \\sin \\theta)(\\cos \\varphi + i \\sin \\varphi) \\\\\n", "& = \\cos \\theta \\cos \\varphi + i \\cos \\theta \\sin \\varphi + i \\sin \\theta \\cos \\varphi + i^2 \\sin \\theta \\sin \\varphi \\\\\n", "& = (\\cos \\theta \\cos \\varphi - \\sin \\theta \\sin \\varphi) + i(\\cos \\theta \\sin \\varphi + \\sin \\theta \\cos \\varphi) \\end{align} $$\n", "\n", "Equate real and imaginary parts on both sides:\n", "\n", "$$\\cos (\\theta + \\varphi) = \\cos \\theta \\cos \\varphi - \\sin \\theta \\sin \\varphi \\\\\n", "\\sin (\\theta + \\varphi ) = \\cos \\theta \\sin \\varphi + \\sin \\theta \\cos \\varphi $$\n", "\n", "The reader is encouraged to try to recover other trigonometric identities.\n", " \n", "
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