The values of ((cos30^∘+i sin30^∘)(cos45^∘+i sin45^∘))/((cos5^∘+i sin5^∘)(cos10^∘+i sin10^∘)(cos

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The values of ((cos30^∘+i sin30^∘)(cos45^∘+i...

The values of $\frac{\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)}{\left(\cos 5^{\circ}+i \sin 5^{\circ}\right)\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)}$ is (a) $\frac{\sqrt{2}}{2}(1+\mathrm{i})$ (b) $1+\mathrm{i}$ (c) $e^{\frac{i \pi}{3}}+e^{\frac{i n}{6}}$ (d) $(1+i) \frac{2}{5}$

The values of $\frac{\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)}{\left(\cos 5^{\circ}+i \sin 5^{\circ}\right)\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)}$ is
(a) $\frac{\sqrt{2}}{2}(1+\mathrm{i})$
(b) $1+\mathrm{i}$
(c) $e^{\frac{i \pi}{3}}+e^{\frac{i n}{6}}$
(d) $(1+i) \frac{2}{5}$

Key Concepts

Angle Addition and Subtraction

The principle of angle addition and subtraction in the context of complex numbers relates to how the arguments (angles) of complex numbers combine when they are multiplied or divided. This concept is essential for reducing complex expressions that involve multiple rotations or shifts in direction.

Multiplication and Division of Complex Numbers in Polar Form

When complex numbers are in polar form, their multiplication results in a new complex number whose magnitude is the product of the original magnitudes and whose argument is the sum of the original angles. Similarly, division of complex numbers is performed by dividing the magnitudes and subtracting the angles, making these operations more intuitive than in the rectangular form.

Euler's Formula

Euler's formula, stated as e^(i?) = cos? + i sin?, provides a powerful connection between exponential functions and trigonometry. This formula allows for a more straightforward representation and manipulation of complex numbers, especially when dealing with rotations and periodic functions.

Polar Representation of Complex Numbers

Expressing complex numbers in polar form involves representing a number by its magnitude and angle (or argument). This form simplifies operations like multiplication and division, because it converts these operations into additions and subtractions of angles and multiplications and divisions of magnitudes.

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