The value of (sin(π)/(6)+i cos(π)/(6))^7 is (a) (cos(π)/(6)+i sin(π)/(6)) (b) (cos(π)/(6)-i sin(

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The value of (sin(π)/(6)+i cos(π)/(6))^7 is (a)...

The value of $\left(\sin \frac{\pi}{6}+i \cos \frac{\pi}{6}\right)^{7}$ is (a) $\left(\cos \frac{\pi}{6}+\mathrm{i} \sin \frac{\pi}{6}\right)$ (b) $\left(\cos \frac{\pi}{6}-\mathrm{i} \sin \frac{\pi}{6}\right)$ (c) $\left(\sin \frac{\pi}{6}+i \cos \frac{\pi}{6}\right)$ (d) $\left(\sin \frac{\pi}{6}-i \cos \frac{\pi}{6}\right)$

The value of $\left(\sin \frac{\pi}{6}+i \cos \frac{\pi}{6}\right)^{7}$ is
(a) $\left(\cos \frac{\pi}{6}+\mathrm{i} \sin \frac{\pi}{6}\right)$
(b) $\left(\cos \frac{\pi}{6}-\mathrm{i} \sin \frac{\pi}{6}\right)$
(c) $\left(\sin \frac{\pi}{6}+i \cos \frac{\pi}{6}\right)$
(d) $\left(\sin \frac{\pi}{6}-i \cos \frac{\pi}{6}\right)$

Key Concepts

Complex Numbers in Polar Form

Any complex number can be represented in the form r(cos ? + i sin ?), where r is the modulus and ? is the argument. This representation simplifies operations like multiplication, division, and exponentiation, as it allows us to work directly with the magnitude and angle of the complex number.

Euler's Formula

Euler's formula, expressed as e^(i?) = cos ? + i sin ?, provides a powerful link between complex exponentials and trigonometric functions. This connection is especially useful when exponentiating complex numbers, as it transforms trigonometric expressions into exponential form for simpler manipulations.

De Moivre's Theorem

De Moivre's theorem states that for any complex number in polar form and any integer n, (cos ? + i sin ?)^n = cos(n?) + i sin(n?). This theorem is crucial in raising complex numbers to integer powers, as it directly relates the power to a multiplication of the angle in the trigonometric representation.

Trigonometric Identities

Trigonometric identities, such as relationships between the sine and cosine of complementary angles, help in rewriting and simplifying expressions. They provide alternative forms of trigonometric functions, facilitating conversion between different representations that can be more useful in a given context.

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