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.

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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