Упр.14.12 ГДЗ Мерзляк 11 класс Углубленный уровень (Алгебра)

1) 4(cos(-п/4)+isin(п/4)); 4) -2(cos(2п/3)+isin(2п/3));
2) 3(cos(п/11)-isin(п/11)); 5) -cos(1/3)-isin(1/3).
3) 6(sin(2п/13)+icos(2п/13));

1. $$4\left(\cos\left(-\frac{\pi}{4}\right)+i\sin\left(\frac{\pi}{4}\right)\right)=4\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right),$$
   так как $$\cos(-\alpha)=\cos\alpha,$$ а $$\sin\left(\frac{\pi}{4}\right)=\sin\left(-\frac{\pi}{4}\right)$$ по модулю угла в записи тригонометрической формы.
2. $$3\left(\cos\frac{\pi}{11}-i\sin\frac{\pi}{11}\right)=3\left(\cos\left(-\frac{\pi}{11}\right)+i\sin\left(-\frac{\pi}{11}\right)\right).$$
3. $$6\left(\sin\frac{2\pi}{13}+i\cos\frac{2\pi}{13}\right)=6\left(\cos\left(\frac{\pi}{2}-\frac{2\pi}{13}\right)+i\sin\left(\frac{\pi}{2}-\frac{2\pi}{13}\right)\right)=6\left(\cos\frac{9\pi}{26}+i\sin\frac{9\pi}{26}\right).$$
4. $$-2\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)=2\left(\cos\frac{5\pi}{3}+i\sin\frac{5\pi}{3}\right).$$
5. $$-\cos\frac13-i\sin\frac13=\cos\left(\pi+\frac13\right)+i\sin\left(\pi+\frac13\right).$$

Ответ

1) $$4\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right);$$
2) $$3\left(\cos\left(-\frac{\pi}{11}\right)+i\sin\left(-\frac{\pi}{11}\right)\right);$$
3) $$6\left(\cos\frac{9\pi}{26}+i\sin\frac{9\pi}{26}\right);$$
4) $$2\left(\cos\frac{5\pi}{3}+i\sin\frac{5\pi}{3}\right);$$
5) $$\cos\left(\pi+\frac13\right)+i\sin\left(\pi+\frac13\right).$$