Algebra

De Moivre's Theorem

De Moivre's Theorem - Roots

         

Let zz be a complex number such that z=2(cos7+isin7).z = \sqrt{2}(\cos 7 ^{\circ} + i \sin 7 ^{\circ}). Then z8 z^{8} can be expressed as r(cosα+isinα) r( \cos \alpha^{\circ} + i \sin \alpha^{\circ}) , where rr is a real number and 0α90 0 \leq \alpha \leq 90. What is the value of r+α?r+\alpha ?

Details and assumptions

ii is the imaginary unit, where i2=1i^2=-1.

The two roots of f(x)=x2+20x+200 f(x) = x^2 + 20x + 200 can be written in the form a±bi a \pm bi . What is ba, b-a, assuming b>0b > 0 ?

Details and assumptions

i i is defined as the imaginary unit, such that i2=1 i^2 = -1 .

Let z=cosα+isinαz=\cos \alpha^\circ+i\sin \alpha^\circ be a complex number such that 0<α<200 < \alpha < 20 and z15=1+3i2.\displaystyle z^{15}=\frac{1+\sqrt{3}i}{2}. What is the value of α\alpha?

Let a+bia+bi and c+dic+di be the two roots of the quadratic equation x2+25=0,x^2+25=0, where b>db>d. What is the value of acbdac-bd?

Details and assumptions

ii is the imaginary number that satisfies i2=1i^2 = -1.

Let z=cosα+isinαz=\cos \alpha+i\sin \alpha be a complex number that satisfies 12<α<3612 ^\circ < \alpha < 36 ^\circ and z30=1.z^{30}=1. If zz can be expressed in polar form as z=cosα+isinα,z=\cos \alpha^\circ+i\sin \alpha^\circ, what is the value of α\alpha (in degrees)?

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