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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.

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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 \(i^2 = -1\).

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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 \(i^2 = -1\).

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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 \(i^2=-1\).

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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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