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Algebra

# De Moivre's Theorem - Roots

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)?

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