1. Представить в тригонометрической форме числа:
1) $2-2 i$;
2) -1 ;
3) $\frac{\pi}{4}$;
4) $-1-i \sqrt{3}$;
5) $-2+5 i$.
2. Решить систему: $\left\{\begin{array}{c}(2-i) x+(3+2 i) y=3-6 i \ (1-i) x-(2+i) y=-1\end{array} ight.$.
3. Выполнить указанные действия:
1) $\left(\frac{1+i}{4} ight)^{7}$;
2) $(1+i \sqrt{3})^{6}$;
3) $f(z)=(z-1)^{5}+\frac{1}{z-3}$. Найти $f(1+2 i)$.
4. Найти и изобразить на комплексной плоскости все значения следующих корней:
1) $\sqrt[12]{1+i}$;
2) $\sqrt[4]{-1}$;
3) $\sqrt[3]{8}$;
4) $\sqrt{8-6 i}$;
5) $\sqrt{\frac{\sqrt{3}+i}{1-i}}$.
5. Решить квадратное уравнение: $z^{2}+(1-2 i) z-2 i=0$.

Корни уравнения записать во всех известных формах и изобразить геометрически.
6. Построить множества точек, удовлетворяющих указанным соотношениям:
1) $|z-3 i| \geq 3$;
2) $\operatorname{Re} z+\operatorname{Im} z=1$;
3) $|z-2 i|+|z+2 i|=8$;
4) $|z-2 i|=|z+2 i|$.

Question

1. Представить в тригонометрической форме числа:
1) $2-2 i$;
2) -1 ;
3) $\frac{\pi}{4}$;
4) $-1-i \sqrt{3}$;
5) $-2+5 i$.
2. Решить систему: $\left\{\begin{array}{c}(2-i) x+(3+2 i) y=3-6 i \ (1-i) x-(2+i) y=-1\end{array}ight.$.
3. Выполнить указанные действия:
1) $\left(\frac{1+i}{4}ight)^{7}$;
2) $(1+i \sqrt{3})^{6}$;
3) $f(z)=(z-1)^{5}+\frac{1}{z-3}$. Найти $f(1+2 i)$.
4. Найти и изобразить на комплексной плоскости все значения следующих корней:
1) $\sqrt[12]{1+i}$;
2) $\sqrt[4]{-1}$;
3) $\sqrt[3]{8}$;
4) $\sqrt{8-6 i}$;
5) $\sqrt{\frac{\sqrt{3}+i}{1-i}}$.
5. Решить квадратное уравнение: $z^{2}+(1-2 i) z-2 i=0$.

Корни уравнения записать во всех известных формах и изобразить геометрически.
6. Построить множества точек, удовлетворяющих указанным соотношениям:
1) $|z-3 i| \geq 3$;
2) $\operatorname{Re} z+\operatorname{Im} z=1$;
3) $|z-2 i|+|z+2 i|=8$;
4) $|z-2 i|=|z+2 i|$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Convert $2-2i$ to trigonometric form. ‖ ‖ step 2 ⋮ Find the modulus $r = |2-2i| = 2\sqrt{2}$. ‖ ‖ step 3 ⋮ Find the argument $ heta = \arg(2-2i) = -\frac{\pi}{4}$. ‖ ‖ step 4 ⋮ Write in trigonometric form: $2\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))$. ‖ ∻Answer∻ ⚹ $2\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))$ ⚹ ∻Key Concept∻ ⚹Trigonometric form of a complex number⚹ ∻Explanation∻ ⚹The trigonometric form of a complex number is given by $r(\cos( heta) + i\sin( heta))$, where $r$ is the modulus and $ heta$ is the argument of the complex number.⚹ ‖ step 1 ⋮ Convert $-1$ to trigonometric form. ‖ ‖ step 2 ⋮ The modulus $r = |-1| = 1$. ‖ ‖ step 3 ⋮ The argument $ heta = \arg(-1) = \pi$. ‖ ‖ step 4 ⋮ Write in trigonometric form: $\cos(\pi) + i\sin(\pi)$. ‖ ∻Answer∻ ⚹ $\cos(\pi) + i\sin(\pi)$ ⚹ ∻Key Concept∻ ⚹Trigonometric form of a complex number⚹ ∻Explanation∻ ⚹For a negative real number, the trigonometric form is found by using the angle $\pi$ or $-\pi$ since the number lies on the negative real axis.⚹ ‖ step 1 ⋮ Convert $\frac{\pi}{4}$ to trigonometric form. ‖ ‖ step 2 ⋮ The modulus $r = |\frac{\pi}{4}| = \frac{\pi}{4}$. ‖ ‖ step 3 ⋮ The argument $ heta = \arg(\frac{\pi}{4}) = 0$. ‖ ‖ step 4 ⋮ Write in trigonometric form: $\frac{\pi}{4}(\cos(0) + i\sin(0))$. ‖ ∻Answer∻ ⚹ $\frac{\pi}{4}(\cos(0) + i\sin(0))$ ⚹ ∻Key Concept∻ ⚹Trigonometric form of a real number⚹ ∻Explanation∻ ⚹A real number has an argument of $0$ or $\pi$, depending on whether it is positive or negative. Here, $\frac{\pi}{4}$ is positive, so the argument is $0$.⚹ ‖ step 1 ⋮ Convert $-1-i\sqrt{3}$ to trigonometric form. ‖ ‖ step 2 ⋮ Find the modulus $r = |-1-i\sqrt{3}| = 2$. ‖ ‖ step 3 ⋮ Find the argument $ heta = \arg(-1-i\sqrt{3}) = -\frac{2\pi}{3}$. ‖ ‖ step 4 ⋮ Write in trigonometric form: $2(\cos(-\frac{2\pi}{3}) + i\sin(-\frac{2\pi}{3}))$. ‖ ∻Answer∻ ⚹ $2(\cos(-\frac{2\pi}{3}) + i\sin(-\frac{2\pi}{3}))$ ⚹ ∻Key Concept∻ ⚹Trigonometric form of a complex number⚹ ∻Explanation∻ ⚹The trigonometric form is useful for multiplication, division, and finding powers and roots of complex numbers.⚹ ‖ step 1 ⋮ Convert $-2+5i$ to trigonometric form. ‖ ‖ step 2 ⋮ Find the modulus $r = |-2+5i| = \sqrt{29}$. ‖ ‖ step 3 ⋮ Find the argument $ heta = \arg(-2+5i) = \pi - an^{-1}(\frac{5}{2})$. ‖ ‖ step 4 ⋮ Write in trigonometric form: $\sqrt{29}(\cos(\pi - an^{-1}(\frac{5}{2})) + i\sin(\pi - an^{-1}(\frac{5}{2})))$. ‖ ∻Answer∻ ⚹ $\sqrt{29}(\cos(\pi - an^{-1}(\frac{5}{2})) + i\sin(\pi - an^{-1}(\frac{5}{2})))$ ⚹ ∻Key Concept∻ ⚹Trigonometric form of a complex number⚹ ∻Explanation∻ ⚹The argument of a complex number in the second quadrant is given by $\pi - an^{-1}(\frac{y}{x})$ where $x$ and $y$ are the real and imaginary parts, respectively.⚹ ‖ step 1 ⋮ Solve the system of equations $(2-i)x + (3+2i)y = 3-6i$ and $(1-i)x - (2+i)y = -1$. ‖ ‖ step 2 ⋮ Use substitution or elimination to find $x$ and $y$. ‖ ‖ step 3 ⋮ The solution is $x = 1 - i$ and $y = -i$. ‖ ∻Answer∻ ⚹ $x = 1 - i$, $y = -i$ ⚹ ∻Key Concept∻ ⚹Solving a system of complex linear equations⚹ ∻Explanation∻ ⚹Complex linear systems can be solved using the same techniques as for real systems, such as substitution or elimination.⚹ ‖ step 1 ⋮ Simplify $\left(\frac{1+i}{4} ight)^{7}$. ‖ ‖ step 2 ⋮ Calculate the seventh power of the complex number $\frac{1+i}{4}$. ‖ ‖ step 3 ⋮ The simplified form is $\frac{1}{16384} + \frac{i}{16384}$. ‖ ∻Answer∻ ⚹ $\frac{1}{16384} + \frac{i}{16384}$ ⚹ ∻Key Concept∻ ⚹Powers of complex numbers⚹ ∻Explanation∻ ⚹To raise a complex number to a power, convert it to polar form and apply De Moivre's theorem, or use binomial expansion if in rectangular form.⚹ ‖ step 1 ⋮ Simplify $(1+i\sqrt{3})^{6}$. ‖ ‖ step 2 ⋮ Calculate the sixth power of the complex number $1+i\sqrt{3}$. ‖ ‖ step 3 ⋮ The simplified form is $64$. ‖ ∻Answer∻ ⚹ $64$ ⚹ ∻Key Concept∻ ⚹Powers of complex numbers⚹ ∻Explanation∻ ⚹When a complex number is raised to a power, its modulus is raised to that power and its argument is multiplied by that power. In this case, the argument is a multiple of $2\pi$, resulting in a real number.⚹ ‖ step 1 ⋮ Find $f(1+2i) = ((1+2i)-1)^5 + \frac{1}{(1+2i)-3}$. ‖ ‖ step 2 ⋮ Simplify the expression by performing the operations. ‖ ‖ step 3 ⋮ The result is $32i^5 + \frac{1}{2(i-1)}$. ‖ ∻Answer∻ ⚹ $32i^5 + \frac{1}{2(i-1)}$ ⚹ ∻Key Concept∻ ⚹Evaluating complex functions⚹ ∻Explanation∻ ⚹To evaluate a complex function, substitute the complex number into the function and simplify using algebraic operations.⚹ ‖ step 1 ⋮ Solve the quadratic equation $z^2 + (1-2i)z - 2i = 0$. ‖ ‖ step 2 ⋮ Use the quadratic formula or factorization to find the roots. ‖ ‖ step 3 ⋮ The roots are $z = -1$ and $z = 2i$. ‖ ∻Answer∻ ⚹ $z = -1$, $z = 2i$ ⚹ ∻Key Concept∻ ⚹Solving quadratic equations with complex coefficients⚹ ∻Explanation∻ ⚹Quadratic equations with complex coefficients can be solved using the quadratic formula, resulting in complex roots.⚹ ‖ step 1 ⋮ Plot the set of points satisfying $|z-3i| \geq 3$. ‖ ‖ step 2 ⋮ This represents the exterior of a circle centered at $3i$ with radius $3$. ‖ ∻Answer∻ ⚹ Exterior of a circle centered at $3i$ with radius $3$ ⚹ ∻Key Concept∻ ⚹Geometric representation of complex inequalities⚹ ∻Explanation∻ ⚹The inequality $|z-a| \geq r$ represents all points in the complex plane that are at least a distance $r$ away from the complex number $a$.⚹ ‖ step 1 ⋮ Plot the set of points satisfying $\operatorname{Re} z + \operatorname{Im} z = 1$. ‖ ‖ step 2 ⋮ This represents a line with a slope of $-1$ and y-intercept at $1$. ‖ ∻Answer∻ ⚹ Line with slope $-1$ and y-intercept $1$ ⚹ ∻Key Concept∻ ⚹Geometric representation of complex equations⚹ ∻Explanation∻ ⚹The equation $\operatorname{Re} z + \operatorname{Im} z = 1$ represents a line in the complex plane where the sum of the real and imaginary parts is constant.⚹ ‖ step 1 ⋮ Plot the set of points satisfying $|z-2i| + |z+2i| = 8$. ‖ ‖ step 2 ⋮ This represents an ellipse with foci at $2i$ and $-2i$ and a major axis of length $8$. ‖ ∻Answer∻ ⚹ Ellipse with foci at $2i$ and $-2i$ and major axis length $8$ ⚹ ∻Key Concept∻ ⚹Geometric representation of complex equations⚹ ∻Explanation∻ ⚹The equation $|z-a| + |z-b| = d$ represents an ellipse in the complex plane with foci at $a$ and $b$ and a major axis of length $d$.⚹ ‖ step 1 ⋮ Plot the set of points satisfying $|z-2i| = |z+2i|$. ‖ ‖ step 2 ⋮ This represents the line perpendicular to the line segment joining $2i$ and $-2i$ and bisecting it. ‖ ∻Answer∻ ⚹ Line perpendicular to and bisecting the line segment between $2i$ and $-2i$ ⚹ ∻Key Concept∻ ⚹Geometric representation of complex equations⚹ ∻Explanation∻ ⚹The equation $|z-a| = |z-b|$ represents the perpendicular bisector of the line segment joining $a$ and $b$ in the complex plane.⚹◊What is the geometric interpretation of the roots of a quadratic equation in the complex plane?⍭ Generate me a similar question◊

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