Convert to the polar form r e^i θ . For Problems 15 and 16, choose θin degrees, -180^∘ < θ≤180^∘

step 1 In this problem you have to convert to the polar form that is R E to the power iota theta Now in

Convert to the polar form r e^i θ . For Problems 15 and 16,...

Convert to the polar form $r e^{i \theta} .$ For Problems 15 and 16, choose $\theta$ in degrees, $-180^{\circ} < \theta \leq 180^{\circ} ;$ for Problems 17 and 18 choose $\theta$ in radians, $-\pi < \theta \leq \pi$ Compute the modulus and arguments for parts $A$ and $B$ exactly; compute the modulus and argument for part C to two decimal places. $(\mathrm{A})-1+i \sqrt{3}$ (B) $-3 i$ $(C)-7-4 i$

Convert to the polar form $r e^{i \theta} .$ For Problems 15 and 16, choose $\theta$ in degrees, $-180^{\circ} < \theta \leq 180^{\circ} ;$ for Problems 17 and 18 choose $\theta$ in radians, $-\pi < \theta \leq \pi$ Compute the modulus and arguments for parts $A$ and $B$ exactly; compute the modulus and argument for part C to two decimal places.
$(\mathrm{A})-1+i \sqrt{3}$
(B) $-3 i$
$(C)-7-4 i$

Key Concepts

Complex Numbers

A complex number is generally written in the form a + bi, where a and b are real numbers and i is the imaginary unit. This representation expresses a complex number in the Cartesian coordinate system and forms the basis for understanding its properties and operations.

Polar Representation

The polar form of a complex number expresses it as r e^(i?), where r is the modulus (or absolute value) and ? is the argument (or angle) the complex number makes with the positive real axis. This form is especially useful in multiplication, division, and finding powers and roots of complex numbers.

Modulus

The modulus of a complex number a + bi is calculated as r = ?(a² + b²). It represents the distance of the complex number from the origin in the complex plane, which is a key component in transitioning from Cartesian to polar representation.

Argument

The argument of a complex number is the angle ? formed by the line connecting the number to the origin and the positive real axis. It is typically computed using trigonometric functions such as arctan(b/a), taking into account the correct quadrant to ensure an accurate angle measurement.

Degree and Radian Measure

Angles in complex numbers can be expressed in either degrees or radians. Choosing the appropriate unit is important for consistency and clarity, as degrees and radians are different ways of measuring angles, with 360 degrees equal to 2? radians.

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