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# Polar Coordinates

Polar coordinates are a way to describe where a point is on a plane. Instead of using x and y, you use the angle theta and radius r, to describe the angle and distance of the point from the origin.

# Polar Coordinates - Complex Numbers

The complex number $$8+8i$$ can be expressed in polar form as $a\sqrt{2}(\cos b ^\circ+i\sin b ^\circ),$ where $$0 < b < 180$$. What is the value of $$a+b$$?

Let $$P$$ be the point in the complex plane that corresponds to complex number $$z$$ and let $$O$$ be the origin. If $$\vert z \vert = 33 \sqrt{2}$$ and the angle between the line segment $$OP$$ and the positive real axis is $$\frac{\pi}{4},$$ then $$z$$ can be expressed as $$a+bi,$$ where $$a$$ and $$b$$ are real numbers. What is the value of $$a+b?$$

Details and assumptions

$$i$$ is the imaginary number satisfying $$i^2 = -1$$.

Let $$P$$ be the point in the complex plane that corresponds to complex number $$z,$$ and let $$O$$ be the origin. If $$\vert z \vert = 12$$ and the angle between the line segment $$OP$$ and the positive real axis is $$\frac{\pi}{b},$$ then $$z$$ can be expressed as $$a+12i,$$ where $$a$$ and $$b$$ are positive integers. What is $$a+b?$$

Details and assumptions

$$i$$ is the imaginary number satisfying $$i^2 = -1$$.

Let $$O$$ be the origin on the complex plane, and let $$P$$ be the point representing the complex number $$z = 8 + 64i$$. If $$\theta$$ is the angle formed by the line segment $$OP$$ and the positive part of the real axis of the complex plane, what is the value of $$\tan \theta?$$

Details and assumptions

$$i$$ is the imaginary unit, where $$i^2=-1$$.

Let $$P$$ be the point in the complex plane that corresponds to complex number $$z$$ and let $$O$$ be the origin. If $$\vert z \vert = 25 \sqrt{2}$$ and the angle between the line segment $$OP$$ and the positive real axis is $$\frac{\pi}{4},$$ then $$z$$ can be expressed as $$a+bi,$$ where $$a$$ and $$b$$ are real numbers. What is the value of $$a+b?$$

Details and assumptions

$$i$$ is the imaginary number satisfying $$i^2 = -1$$.

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