De Moivre's Theorem
Let \(z\) be a complex number such that \[z = \sqrt{2}(\cos 7 ^{\circ} + i \sin 7 ^{\circ}).\] Then \( z^{8} \) can be expressed as \( r( \cos \alpha^{\circ} + i \sin \alpha^{\circ}) \), where \(r\) is a real number and \( 0 \leq \alpha \leq 90\). What is the value of \(r+\alpha ?\)
Details and assumptions
\(i\) is the imaginary unit, where \(i^2=-1\).
The two roots of \( f(x) = x^2 + 20x + 200 \) can be written in the form \( a \pm bi \). What is \( b-a, \) assuming \(b > 0 \)?
Details and assumptions
\( i \) is defined as the imaginary unit, such that \( i^2 = -1 \).
Let \(z=\cos \alpha^\circ+i\sin \alpha^\circ\) be a complex number such that \(0 < \alpha < 20\) and \(\displaystyle z^{15}=\frac{1+\sqrt{3}i}{2}.\) What is the value of \(\alpha\)?
Let \(a+bi\) and \(c+di\) be the two roots of the quadratic equation \[x^2+25=0,\] where \(b>d\). What is the value of \(ac-bd\)?
Details and assumptions
\(i\) is the imaginary number that satisfies \(i^2 = -1\).
Let \(z=\cos \alpha+i\sin \alpha \) be a complex number that satisfies \(12 ^\circ < \alpha < 36 ^\circ\) and \(z^{30}=1.\) If \(z\) can be expressed in polar form as \[z=\cos \alpha^\circ+i\sin \alpha^\circ,\] what is the value of \(\alpha\) (in degrees)?