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Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.5

Solution

For z=r(cos⁡θ+isin⁡θ)z=r(\cos\theta+i\sin\theta):

x=rcos⁡θ,y=rsin⁡θ x=r\cos\theta,\quad y=r\sin\theta

(i) 4(cos⁡5π3+isin⁡5π3)4\left(\cos\dfrac{5\pi}{3}+i\sin\dfrac{5\pi}{3}\right)

cos⁡5π3=12,sin⁡5π3=−32\cos\frac{5\pi}{3}=\frac{1}{2},\quad \sin\frac{5\pi}{3}=-\frac{\sqrt{3}}{2} z=4(12−32i)=2−23 i\boxed{z=4\left(\frac{1}{2}-\frac{\sqrt{3}}{2}i\right)=2-2\sqrt{3}\,i}

(ii) 32(cos⁡7π6+isin⁡7π6)\dfrac{3}{2}\left(\cos\dfrac{7\pi}{6}+i\sin\dfrac{7\pi}{6}\right)

cos⁡7π6=−32,sin⁡7π6=−12\cos\frac{7\pi}{6}=-\frac{\sqrt{3}}{2},\quad \sin\frac{7\pi}{6}=-\frac{1}{2} z=32(−32−12i)=−334−34i\boxed{z=\frac{3}{2}\left(-\frac{\sqrt{3}}{2}-\frac{1}{2}i\right)=-\frac{3\sqrt{3}}{4}-\frac{3}{4}i}

(iii) ∣z∣=7, arg⁡(z)=23π12|z|=7,\ \arg(z)=\dfrac{23\pi}{12}

23π12=2π−π12\frac{23\pi}{12}=2\pi-\frac{\pi}{12} cos⁡23π12=cos⁡π12=6+24,sin⁡23π12=−sin⁡π12=−6−24\cos\frac{23\pi}{12}=\cos\frac{\pi}{12}=\frac{\sqrt{6}+\sqrt{2}}{4},\quad \sin\frac{23\pi}{12}=-\sin\frac{\pi}{12}=-\frac{\sqrt{6}-\sqrt{2}}{4} z=7(6+24−6−24i)=7(6+2)4−7(6−2)4i\boxed{z=7\left(\frac{\sqrt{6}+\sqrt{2}}{4}-\frac{\sqrt{6}-\sqrt{2}}{4}i\right)=\frac{7(\sqrt{6}+\sqrt{2})}{4}-\frac{7(\sqrt{6}-\sqrt{2})}{4}i}

(iv) ∣z∣=11, arg⁡(z)=−11π12|z|=11,\ \arg(z)=-\dfrac{11\pi}{12}

−11π12=−π+π12-\frac{11\pi}{12}=-\pi+\frac{\pi}{12} cos⁡(−11π12)=−cos⁡π12=−6+24\cos\left(-\frac{11\pi}{12}\right)=-\cos\frac{\pi}{12}=-\frac{\sqrt{6}+\sqrt{2}}{4} sin⁡(−11π12)=−sin⁡(11π12)=−sin⁡π12=−6−24\sin\left(-\frac{11\pi}{12}\right)=-\sin\left(\frac{11\pi}{12}\right)=-\sin\frac{\pi}{12}=-\frac{\sqrt{6}-\sqrt{2}}{4} z=11(−6+24−6−24i)=−11(6+2)4−11(6−2)4i\boxed{z=11\left(-\frac{\sqrt{6}+\sqrt{2}}{4}-\frac{\sqrt{6}-\sqrt{2}}{4}i\right)=-\frac{11(\sqrt{6}+\sqrt{2})}{4}-\frac{11(\sqrt{6}-\sqrt{2})}{4}i}

(v) ∣z∣=103, arg⁡(z)=−17π12|z|=\dfrac{10}{3},\ \arg(z)=-\dfrac{17\pi}{12}

−17π12=−π−5π12-\frac{17\pi}{12}=-\pi-\frac{5\pi}{12} cos⁡(−17π12)=cos⁡(π+5π12)=−cos⁡5π12=2−64\cos\left(-\frac{17\pi}{12}\right)=\cos\left(\pi+\frac{5\pi}{12}\right)=-\cos\frac{5\pi}{12}=\frac{\sqrt{2}-\sqrt{6}}{4} sin⁡(−17π12)=−sin⁡(π+5π12)=sin⁡5π12=6+24\sin\left(-\frac{17\pi}{12}\right)=-\sin\left(\pi+\frac{5\pi}{12}\right)=\sin\frac{5\pi}{12}=\frac{\sqrt{6}+\sqrt{2}}{4} z=103(2−64+6+24i)=5(2−6)6+5(6+2)6i\boxed{z=\frac{10}{3}\left(\frac{\sqrt{2}-\sqrt{6}}{4}+\frac{\sqrt{6}+\sqrt{2}}{4}i\right)=\frac{5(\sqrt{2}-\sqrt{6})}{6}+\frac{5(\sqrt{6}+\sqrt{2})}{6}i}

(vi) 2cos⁡(−33∘)+i 2sin⁡(−33∘)2\cos(-33^\circ)+i\,2\sin(-33^\circ)

cos⁡(−33∘)=cos⁡33∘,sin⁡(−33∘)=−sin⁡33∘\cos(-33^\circ)=\cos 33^\circ,\quad \sin(-33^\circ)=-\sin 33^\circ z=2cos⁡33∘−i 2sin⁡33∘\boxed{z=2\cos 33^\circ- i\,2\sin 33^\circ}
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