Math Problems Vault

Problem 2408

Represent the number $1+i$ in trigonometric form.

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Problem 2409

Represent the number $1+i\sqrt{3}$ in trigonometric form.

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Problem 2410

Represent the number $\sqrt{3}-i$ in trigonometric form.

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Problem 2411

Represent the number $1-\cos \alpha +i\sin \alpha$ ; $0 < \alpha < 2\pi$ in trigonometric form.

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Problem 2412

Represent the number $1+\sin \alpha -i\cos \alpha$ in trigonometric form.

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Problem 2413

Represent the number $1+i\tg \alpha$ ; $-\frac{\pi }{2} < \alpha < \frac{\pi }{2}$ in trigonometric form.

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Problem 2414

Evaluate the following expression, where $\omega =\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}$ :

(a\omega +b\omega ^{2})(a\omega ^{2}+b\omega ) .

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Problem 2415

Evaluate the following expression, where $\omega =\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}$ :

(a+b+c)(a+b\omega +c\omega ^{2})(a+b\omega ^{2}+c\omega ) .

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Problem 2416

Evaluate the following expression, where $\omega =\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}$ :

(a+b\omega +c\omega ^{2})^{3}+(a+b\omega ^{2}+c\omega )^{3} .

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Problem 2417

Compute the square roots: $\sqrt{i}$ .

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Problem 2418

Compute the square roots: $\sqrt{3+4i}$ .

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Problem 2419

Compute the square roots: $\sqrt{-7+24i}$ .

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Problem 2420

Solve the quadratic equation $x^{2}+(5-2i)x+5(1-i)=0$ .

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Problem 2421

Solve the quadratic equation $x^{2}+(1-2i)x-2i=0$ .

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Problem 2422

Find all values of the following root: $\sqrt[3]{i}$ .

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Problem 2423

Find all values of the following root: $\sqrt[3]{-1+i}$ .

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Problem 2424

Find all values of the following root: $\sqrt[6]{-64}$ .

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Problem 2425

Find all values of the following root: $\sqrt[6]{64}$ .

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Problem 2426

Find the roots of the equation $\left(\dfrac {1+xi}{1-xi}\right)^{n}=\dfrac {1+i\tg \alpha }{1-i\tg \alpha }$ .

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Problem 2427

Find the roots of the equation $(x+i)^{n}+(x-i)^{n}=0$ .

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Problem 2428

Express the function $\cos 3\varphi$ in terms of $\cos \varphi$ and $\sin \varphi$ .

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Problem 2429

Express the function $\sin 3\varphi$ in terms of $\cos \varphi$ and $\sin \varphi$ .

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Problem 2430

Express the function $\cos 4\varphi$ in terms of $\cos \varphi$ and $\sin \varphi$ .

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Problem 2431

Express the function $\sin 5\varphi$ in terms of $\cos \varphi$ and $\sin \varphi$ .

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Problem 2432

Express the function $\cos^{3}\varphi$ in terms of trigonometric functions of multiple angles.

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Problem 2433

Express the function $\sin^{3}\varphi$ in terms of trigonometric functions of multiple angles.

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Problem 2434

Express the function $\cos^{4}\varphi$ in terms of trigonometric functions of multiple angles.

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Problem 2435

Express the function $\sin^{4}\varphi$ in terms of trigonometric functions of multiple angles.

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Problem 2436

Express the function $\cos^{5}\varphi$ in terms of trigonometric functions of multiple angles.

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Problem 2437

Express the function $\sin^{5}\varphi$ in terms of trigonometric functions of multiple angles.

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Problem 2438

Prove the identity:

\tg n\varphi =\dfrac {C_{n}^{1}\tg \varphi -C_{n}^{3}\tg ^{3}\varphi +C_{n}^{5}\tg ^{5}\varphi -\ldots }{1-C_{n}^{2}\tg ^{2}\varphi +C_{n}^{4}\tg ^{4}\varphi -\ldots },

where $C_{n}^{k}=\dfrac {n(n-1)\ldots (n-k+1)}{1 \cdot 2 \cdot 3 \ldots k}$ .

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Problem 2439

Prove the identity:

2^{2n}\cos ^{2n}\varphi =2\cos 2n\varphi +2C_{2n}^{1}\cos (2n-2)\varphi +\ldots +2C_{2n}^{n-1}\cos 2\varphi +C_{2n}^{n} .

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Problem 2440

Prove the identity:

\ctg x+\ctg \left(x+\frac{\pi }{n}\right)+\ctg \left(x+\frac{2\pi }{n}\right)+\ldots +\ctg \left(x+\frac{n-1}{n}\pi \right)=n\ctg nx .

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Problem 2441

Prove the identity:

\sin x\sin \left(x+\frac{\pi }{n}\right)\sin \left(x+\frac{2\pi }{n}\right)\ldots \sin \left(x+\frac{n-1}{n}\pi \right)=\dfrac {\sin nx}{2^{n-1}} .

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Problem 2442

Evaluate the sum $1+2\cos x+2\cos 2x+\ldots +2\cos nx$ .

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Problem 2443

Evaluate the sum $\sin x+\sin 3x+\sin 5x+\ldots +\sin (2n-1)x$ .

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Problem 2444

Evaluate the sum $\cos \varphi +a\cos 3\varphi +\ldots +a^{n}\cos (2n+1)\varphi$ .

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Problem 2445

Evaluate the sum $1+2a\cos \varphi +2a^{2}\cos 2\varphi +\ldots +2a^{n}\cos n\varphi$ .

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Problem 2446

Find the value of $\ln (-e)$ .

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Problem 2447

Find the value of $\ln (-2)$ .

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Problem 2448

Find the value of $\ln i$ .

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Problem 2449

Find the value of $\ln \dfrac {1+i}{\sqrt{2}}$ .

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Problem 2450

Find the value of $\ln (x+iy)$ .

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Problem 2451

Find the value of $e^{\pi i}$ .

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Problem 2452

Find the value of $i^{i}$ .

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Problem 2453

Find the value of $2^{i}$ .

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Problem 2454

Find the value of $\left(\dfrac {1+i}{\sqrt{2}}\right)^{-i}$ .

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Problem 2455

Find the value of $\tg \dfrac {\pi i}{2}$ .

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Problem 2456

Find the value of $\sin (x+iy)$ .

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Problem 2457

Find the value of $\cos (x+iy)$ .

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Problem 2458

Find the value of $\arctg xi$ .

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Problem 2459

Under what condition does the point $z=x+iy$ lie inside a circle of radius $R$ centered at the point $c=a+bi$ ?

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Problem 2460

A regular $n$ -gon is inscribed in a circle of radius $R$ centered at the point $c=a+bi$ , with one of its vertices at the point $z_{0}=a+(b+R)i$ . Where are the remaining vertices?

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Problem 2461

Two vertices of an equilateral triangle are at the points $z_{0}=1$ and $z_{1}=2+i$ . Find the third vertex.

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Problem 2462

Two adjacent vertices of a regular $n$ -gon are at the points $z_{0}$ and $z_{1}$ . Find the next vertex.

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Problem 2463

The vertices of a polygon are at the points $z_{k}=1+z+z^{2}+\ldots +z^{k-1}$ , $\left|z\right| < 1$ . Determine whether the point $z=0$ lies inside the polygon.

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Problem 2464

Three consecutive vertices of a parallelogram are at the points $z_{1}$ , $z_{2}$ , $z_{3}$ . Find the position of the fourth vertex.

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Problem 2465

The endpoints of a segment are at the points $z_{1}$ and $z_{2}$ . Find the midpoint.

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Problem 2466

Masses $m_{1}, m_{2}, \ldots , m_{n}$ are located at the points $z_{1}, z_{2}, \ldots , z_{n}$ . Find the position of the center of gravity.

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Problem 2467

The points $z_{1}, z_{2}, \ldots , z_{n}$ are the vertices of a convex $n$ -gon. Find the locus of points $z=\lambda_{1}z_{1}+\lambda_{2}z_{2}+\ldots +\lambda_{n}z_{n}$ , where $\lambda_{k} > 0$ , $\lambda_{1}+\lambda_{2}+\ldots +\lambda_{n}=1$ .

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Problem 2468

The consecutive vertices of a broken line lie at the points

z_{k}=a_{0}+a_{1}z+a_{2}z^{2}+\ldots +a_{k-1}z^{k-1}, \quad \left|z\right| \leqslant 1;

a_{0} > a_{1} > a_{2} > \cdots > a_{k-1} > 0 .

Prove that the broken line cannot close for any value of $k$ .

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Problem 2469

Prove that the equation

a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}=0, \qquad a_{0} > a_{1} > a_{2} > \cdots > a_{n} > 0

cannot have roots with modulus greater than unity.

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Problem 2470

Prove that the equalities $\lambda_{1}z_{1}+\lambda_{2}z_{2}+\lambda_{3}z_{3}=0$ , $\lambda_{1}+\lambda_{2}+\lambda_{3}=0$ , where $\lambda_{1}, \lambda_{2}, \lambda_{3}$ are real, are equivalent to the condition that the points $z_{1}, z_{2}, z_{3}$ lie on a single line.

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Problem 2471

Prove the identity:

\left|z_{1}+z_{2}\right|^{2}+\left|z_{1}-z_{2}\right|^{2}=2\left|z_{1}\right|^{2}+2\left|z_{2}\right|^{2}

and state its geometric meaning.

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Problem 2472

Prove the equivalence of the equalities

\left|z_{2}-z_{1}\right|^{2}=\left|z_{2}-z_{0}\right|^{2}+\left|z_{1}-z_{0}\right|^{2} \quad \text{and} \quad z_{2}-z_{0}=\lambda i(z_{1}-z_{0}),

where $\lambda$ is real.

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Problem 2473

Prove the identity:

2\left|z_{1}\right|+2\left|z_{2}\right|=\left|z_{1}+z_{2}-2\sqrt{z_{1}z_{2}}\right|+\left|z_{1}+z_{2}+2\sqrt{z_{1}z_{2}}\right| .

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Problem 2474

Prove the identity:

\left|x+y\right|+\left|x-y\right|=\left|x+\sqrt{x^{2}-y^{2}}\right|+\left|x-\sqrt{x^{2}-y^{2}}\right| .

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Problem 2475

Determine the type of the curve given by the equation $z=z_{0}+te^{\alpha i}$ , $z_{0}=x_{0}+iy_{0}$ , where $t$ is a real parameter.

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Problem 2476

Determine the type of the curve given by the equation $z=z_{0}+re^{ti}$ , where $t$ is a real parameter.

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Problem 2477

Determine the type of the curve given by the equation $z=z_{0}+ae^{ti}+be^{-ti}$ , where $t$ is a real parameter.

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Problem 2478

Determine the type of the curve given by the equation $z=ate^{ti}$ , where $t$ is a real parameter.

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Problem 2479

Determine the type of the curve given by the equation $z=ae^{\alpha t}$ , where $t$ is a real parameter, and $a$ and $\alpha$ are complex constants.

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Problem 2480

Trace how the argument of the quantity $x(x-1)$ changes as $x$ traces, counterclockwise, a closed curve enclosing the points $0$ and $1$ .

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Problem 2481

Study how the arguments of the quantities $\sqrt{x}$ , $\sqrt{x-1}$ , and $\sqrt{x(x-1)}$ change as $x$ traces the same contour as in problem 2480 (a closed curve traversed counterclockwise and enclosing the points $0$ and $1$ ).

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Problem 2482

How does the function $u=(z-a)^{\alpha }(z-b)^{\beta }$ change, for real $\alpha$ and $\beta$ , if $z$ traces a closed contour going counterclockwise around the points $a$ and $b$ ?

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Problem 2483

The same question for the function $u=(z-a)^{\alpha }\ln (z-a)$ and a contour going counterclockwise around the point $a$ .

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Remark.

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The question concerns how the function $u=(z-a)^{\alpha }\ln (z-a)$ changes, for real $\alpha$ , if $z$ traces a closed contour going counterclockwise around the point $a$ (by analogy with problem 2482).