Polar Coordinates
Let \[z = 3(\cos(15 ^\circ) + i\sin(15 ^\circ)),\] \[w = 5(\cos(54 ^\circ)+i\sin(54 ^\circ)).\] Then \(zw\) can be expressed as \(r(\cos \alpha ^ \circ + i\sin\beta ^ \circ )\), where \(r\) is a real number, \( 0 \leq \alpha \leq 90\) and \(0 \leq \beta \leq 90\). What is \(r+\alpha+\beta\)?
Let \(z_1\) and \(z_2\) be complex numbers such that \[z_1 = 15 \left( \cos \frac{5}{12}\pi + i \sin \frac{5}{12}\pi \right),\] \[z_2 = 2 \left( \cos \frac{1}{12}\pi + i \sin \frac{1}{12}\pi \right). \] The product \(z_1 z_2\) 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 \(i^2 = -1\).
Consider the complex numbers \[\begin{align} z_1 &= 5+5\sqrt{3}i, \\ z_2 &= 3\left(\cos \frac{\pi}{6}+i\sin \frac{\pi}{6}\right). \end{align} \] If the product \(z_1z_2\) can be expressed as \(a+bi\), where \(a\) and \(b\) are real numbers, what is \(a+b\)?
Let \(z_1, z_2\) and \(z_3\) be complex numbers such that \[\begin{align} z_1 &= 2 \left( \cos \frac{1}{12}\pi + i \sin \frac{1}{12}\pi \right), \\ z_2 &= 7 \sqrt{3} \left( \cos \frac{1}{12}\pi + i \sin \frac{1}{12}\pi \right), \\ z_3 &= 9 \left( \cos \frac{1}{6}\pi + i \sin \frac{1}{6}\pi \right). \end{align} \] If the product \(z_1 z_2 z_3\) can be expressed as \( a + bi, \) where \(a\) and \(b\) are real numbers, what is \( \frac{b^2}{a^2} \)?
Let \(z_1\) and \(z_2\) be complex numbers such that \[z_1 = 10 \left( \cos \frac{1}{24}\pi + i \sin \frac{1}{24}\pi \right),\] \[z_2 = 5 \sqrt{2} \left( \cos \frac{5}{24}\pi + i \sin \frac{5}{24}\pi \right). \] The product \(z_1 z_2\) 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 that satisfies \(i^2 = -1\).