Chapter 8

Exer. 1-10: Find the absolute value. $$ |3-4 i| $$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (3+3 i)^{5} $$

Exer. 1-8: Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $$ \langle-2,5\rangle, \quad\langle 3,6\rangle $$

Solve \(\triangle A B C\). $$\alpha=41^{\circ}, \quad \gamma=77^{\circ}, \quad a=10.5$$

Exer. 1-10: Find the absolute value. $$ |5+8 i| $$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (1+i)^{12} $$

Exer. 1-8: Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $$ \langle 4,-7\rangle, \quad\langle-2,3\rangle $$

Solve \(\triangle A B C\). $$\beta=20^{\circ}, \quad \gamma=31^{\circ}, \quad b=210$$

Exer. 1-10: Find the absolute value. $$ |-6-7 i| $$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (1-i)^{10} $$

Exer. 1-8: Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $$ 4 \mathbf{i}-\mathbf{j}, \quad-3 \mathbf{i}+2 \mathbf{j} $$

Exer. 1-10: Find the absolute value. $$ |1-i| $$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (-1+i)^{8} $$

Exer. 1-8: Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $$ 8 \mathbf{i}-3 \mathbf{j}, \quad 2 \mathbf{i}-7 \mathbf{j} $$

Solve \(\triangle A B C\). $$\beta=50^{\circ} 50^{\prime}, \quad \gamma=70^{\circ} 30^{\prime}, \quad c=537$$

Exer. 1-8: Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $$ 9 \mathbf{i}, \quad 5 \mathbf{i}+4 \mathbf{j} $$

Solve \(\triangle A B C\). $$\alpha=42^{\circ} 10^{\prime}, \quad \gamma=61^{\circ} 20^{\prime}, \quad b=19.7$$

Exer. 1-10: Find the absolute value. $$ \left|i^{7}\right| $$

Exer. 1-10: Find the absolute value. $$ |8 i| $$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (1-\sqrt{3} i)^{5} $$

Exer. 1-8: Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $$ 6 \mathbf{j}, \quad-4 \mathbf{i} $$

Solve \(\triangle A B C\). $$\gamma=45^{\circ}, \quad b=10.0, \quad a=15.0$$

Solve \(\triangle A B C\). $$\alpha=103.45^{\circ}, \quad \gamma=27.19^{\circ}, \quad b=38.84$$

Exer. 1-10: Find the absolute value. $$ \left|i^{500}\right| $$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \left(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\right)^{15} $$

Exer. 1-8: Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $$ \langle 10,7\rangle, \quad\left\langle-2,-\frac{7}{5}\right\rangle $$

Solve \(\triangle A B C\). $$\beta=150^{\circ}, \quad a=150, \quad c=30$$

Solve \(\triangle A B C\). $$\gamma=81^{\circ}, \quad c=11, \quad b=12$$

Exer. 1-10: Find the absolute value. $$ |-15 i| $$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\right)^{25} $$

Exer. 1-8: Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $$ \langle-3,6\rangle, \quad\langle-1,2\rangle $$

Solve \(\triangle A B C\). $$\alpha=32.32^{\circ}, \quad c=574.3, \quad a=263.6$$

Exer. 1-10: Find the absolute value. $$ |0| $$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \left(-\frac{\sqrt{3}}{2}-\frac{1}{2} i\right)^{20} $$

Solve \(\triangle A B C\). $$\gamma=115^{\circ} 10^{\prime}, \quad a=1.10, \quad b=2.10$$

Solve \(\triangle A B C\). $$\gamma=53^{\circ} 20^{\prime}, \quad a=140, \quad c=115$$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \left(-\frac{\sqrt{3}}{2}-\frac{1}{2} i\right)^{50} $$

Exer. 9 -12: Show that the vectors are orthogonal. \(\langle 3,6\rangle\) \(\langle 4,-2\rangle\)

Solve \(\triangle A B C\). $$\alpha=27^{\circ} 30^{\prime}, \quad c=52.8, \quad a=28.1$$

Exer. 11-20: Represent the complex number geometrically. $$ 4+2 i $$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (\sqrt{3}+i)^{7} $$

Exer. 9 -12: Show that the vectors are orthogonal. $$ -4 \mathbf{j}, \quad-7 \mathbf{i} $$

Solve \(\triangle A B C\). $$a=2.0, \quad b=3.0, \quad c=4.0$$

Solve \(\triangle A B C\). $$\gamma=47.74^{\circ}, \quad a=131.08, \quad c=97.84$$

Exer. 11-20: Represent the complex number geometrically. $$ -5+3 i $$

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (-2-2 i)^{10} $$

Exer. 9 -12: Show that the vectors are orthogonal. $$ 8 \mathbf{i}-4 \mathbf{j}, \quad-6 \mathbf{i}-12 \mathbf{j} $$

Solve \(\triangle A B C\). $$a=10, \quad b=15, \quad c=12$$

Solve \(\triangle A B C\). $$\alpha=42.17^{\circ}, \quad a=5.01, \quad b=6.12$$

Exer. 11-20: Represent the complex number geometrically. $$ 3-5 i $$