Question: Compute $(\cos 15^\circ + i \sin 15^\circ)^6$ and express the result in rectangular form.
["# Compute $(\cos 15^\circ + i \sin 15^\circ)^6$ and Express the Result in Rectangular Form", "When working with complex numbers in polar form, De Moivre’s Theorem becomes an essential tool. The problem of computing $(\cos 15^\circ + i \sin 15^\circ)^6$ is a perfect application of this powerful theorem. This article guides you through the computation step-by-step and reveals the rectangular form of the result.", "## Understanding the Expression", "The expression $(\cos 15^\circ + i \sin 15^\circ)^6$ is in polar form, where the complex number $\cos 15^\circ + i \sin 15^\circ$ has magnitude 1 and angle $15^\circ$. Using Euler’s formula, this can be written as:", "$$\nz = \ ext{cis}(15^\circ) = \cos 15^\circ + i \sin 15^\circ\n$$", "By De Moivre’s Theorem:", "$$\nz^n = \ ext{cis}(n \cdot \ heta) = \cos(n\ heta) + i \sin(n\ heta)\n$$", "Applying this to our case with $n = 6$ and $\ heta = 15^\circ$:", "$$\n(\cos 15^\circ + i \sin 15^\circ)^6 = \cos(90^\circ) + i \sin(90^\circ)\n$$", "## Evaluating the Result", "We now compute the cosine and sine of $90^\circ$:", "- $\cos 90^\circ = 0$\n- $\sin 90^\circ = 1$", "Therefore:", "$$\n(\cos 15^\circ + i \sin 15^\circ)^6 = 0 + i \cdot 1 = i\n$$", "## Expressing in Rectangular Form", "The rectangular form of a complex number is expressed as $a + bi$, where $a$ is the real part and $b$ is the imaginary part. In this case:", "$$\n\ ext{Rectangular form: } 0 + 1i = i\n$$", "## Conclusion", "Using De Moivre’s Theorem, we efficiently computed $(\cos 15^\circ + i \sin 15^\circ)^6$ and found that the result in rectangular form is simply $i$. This elegant solution highlights how powerful polar representations and trigonometric identities simplify complex exponentiation.", "---\nKeywords: $(\cos 15^\circ + i \sin 15^\circ)^6$, De Moivre’s Theorem, rectangular form, complex numbers, polar form, trigonometry, exponentiation, complex arithmetic", "---", "Ready to explore more applications of complex numbers? Check out our guides on De Moivre’s Theorem, complex number conversions, and polar coordinates!"]
