Упр.625 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

Записать в тригонометрической форме комплексное число:

1. $$z=3$$;
2. $$z=-1$$;
3. $$z=3+3i$$;
4. $$z=-2+2\sqrt{3}i$$;
5. $$z=-1-\sqrt{3}i$$;
6. $$z=5-5i$$;
7. $$z=\sqrt{5}\left(\cos\frac{\pi}{3}-i\sin\frac{\pi}{3}\right)$$;
8. $$z=-\cos\frac{\pi}{7}-i\sin\frac{\pi}{7}$$.

1. $$z=3=3+0i$$
   $$|z|=\sqrt{3^2+0^2}=3,\qquad \varphi=0$$
   $$z=3(\cos 0+i\sin 0).$$
2. $$z=-1=-1+0i$$
   $$|z|=\sqrt{(-1)^2+0^2}=1,\qquad \varphi=\pi$$
   $$z=\cos \pi+i\sin \pi.$$
3. $$z=3+3i$$
   $$|z|=\sqrt{3^2+3^2}=3\sqrt{2},\qquad \tg\varphi=\frac{3}{3}=1$$
   $$\varphi=\frac{\pi}{4}$$
   $$z=3\sqrt{2}\left(\cos \frac{\pi}{4}+i\sin \frac{\pi}{4}\right).$$
4. $$z=-2+2\sqrt{3}\,i$$
   $$|z|=\sqrt{(-2)^2+(2\sqrt{3})^2}=4,\qquad \tg\varphi=\frac{2\sqrt{3}}{-2}=-\sqrt{3}$$
   Так как число лежит во II четверти, то
   $$\varphi=\frac{2\pi}{3}$$
   $$z=4\left(\cos \frac{2\pi}{3}+i\sin \frac{2\pi}{3}\right).$$
5. $$z=-1-\sqrt{3}\,i$$
   $$|z|=\sqrt{(-1)^2+(-\sqrt{3})^2}=2,\qquad \tg\varphi=\frac{-\sqrt{3}}{-1}=\sqrt{3}$$
   Так как число лежит в III четверти, то
   $$\varphi=\frac{4\pi}{3}$$
   $$z=2\left(\cos \frac{4\pi}{3}+i\sin \frac{4\pi}{3}\right).$$
6. $$z=5-5i$$
   $$|z|=\sqrt{5^2+(-5)^2}=5\sqrt{2},\qquad \tg\varphi=\frac{-5}{5}=-1$$
   $$\varphi=-\frac{\pi}{4}$$
   $$z=5\sqrt{2}\left(\cos \left(-\frac{\pi}{4}\right)+i\sin \left(-\frac{\pi}{4}\right)\right).$$
7. $$z=\sqrt{5}\left(\cos \frac{\pi}{3}-i\sin \frac{\pi}{3}\right)$$
   Так как $$-\sin \frac{\pi}{3}=\sin \left(-\frac{\pi}{3}\right),$$ то
   $$z=\sqrt{5}\left(\cos \left(-\frac{\pi}{3}\right)+i\sin \left(-\frac{\pi}{3}\right)\right).$$
8. $$z=-\cos \frac{\pi}{7}-i\sin \frac{\pi}{7}$$
   Используем формулы:
   $$-\cos \frac{\pi}{7}=\cos \left(\pi-\frac{\pi}{7}\right)=\cos \frac{6\pi}{7},$$
   $$-\sin \frac{\pi}{7}=\sin \left(-\frac{\pi}{7}\right).$$
   Тогда
   $$z=\cos \frac{6\pi}{7}+i\sin \left(-\frac{\pi}{7}\right).$$
   Удобнее записать в стандартной тригонометрической форме как
   $$z=\cos \frac{8\pi}{7}+i\sin \frac{8\pi}{7}.$$

Ответ

1) $$z=3(\cos 0+i\sin 0);$$
2) $$z=\cos \pi+i\sin \pi;$$
3) $$z=3\sqrt{2}\left(\cos \frac{\pi}{4}+i\sin \frac{\pi}{4}\right);$$
4) $$z=4\left(\cos \frac{2\pi}{3}+i\sin \frac{2\pi}{3}\right);$$
5) $$z=2\left(\cos \frac{4\pi}{3}+i\sin \frac{4\pi}{3}\right);$$
6) $$z=5\sqrt{2}\left(\cos \left(-\frac{\pi}{4}\right)+i\sin \left(-\frac{\pi}{4}\right)\right);$$
7) $$z=\sqrt{5}\left(\cos \left(-\frac{\pi}{3}\right)+i\sin \left(-\frac{\pi}{3}\right)\right);$$
8) $$z=\cos \frac{8\pi}{7}+i\sin \frac{8\pi}{7}.$$