Chapter 6

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$-2(1+\sqrt{3} i)$$

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ a=75.4, \quad b=52, \quad c=52 $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=55^{\circ}, \quad B=42^{\circ}, \quad c=\frac{3}{4}$$

In Exercises \(15-24,\) use the vectors \(u=\langle 3,3\rangle, \quad v=\langle- 4,2\rangle,\) and \(\mathbf{w}=\langle 3,-1\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$3 \mathbf{u} \cdot \mathbf{v}$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$\frac{5}{2}(\sqrt{3}-i)$$

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ a=1.42, \quad b=0.75, \quad c=1.25 $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$B=28^{\circ}, \quad C=104^{\circ}, \quad a=3 \frac{5}{8}$$

In Exercises \(15-24,\) use the vectors \(u=\langle 3,3\rangle, \quad v=\langle- 4,2\rangle,\) and \(\mathbf{w}=\langle 3,-1\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$(\mathbf{u} \cdot \mathbf{v}) \mathbf{v}$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$-5 i$$

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ A=120^{\circ}, \quad b=6, \quad c=7 $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=36^{\circ}, \quad a=8, \quad b=5$$

In Exercises \(15-24,\) use the vectors \(u=\langle 3,3\rangle, \quad v=\langle- 4,2\rangle,\) and \(\mathbf{w}=\langle 3,-1\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$(\mathbf{v} \cdot \mathbf{u}) \mathbf{w}$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$12i$$

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ A=48^{\circ}, \quad b=3, \quad c=14 $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=60^{\circ}, \quad a=9, \quad c=10$$

In Exercises \(15-24,\) use the vectors \(u=\langle 3,3\rangle, \quad v=\langle- 4,2\rangle,\) and \(\mathbf{w}=\langle 3,-1\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$(3 \mathbf{w} \cdot \mathbf{v}) \mathbf{u}$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$-7+4 i$$

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ B=10^{\circ} 35^{\prime}, \quad a=40, \quad c=30 $$

Finding the Component Form of a Vector In Exercises \(13-24,\) find the component form and magnitude of the vector v. \(\begin{array}{ll}{\text { Initial Point }} & {\text { Terminal Point }} \\\ {(-3,-5)} & {(5,1)}\end{array}\)

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$B=15^{\circ} 30^{\prime}, \quad a=4.5, \quad b=6.8$$

In Exercises \(15-24,\) use the vectors \(u=\langle 3,3\rangle, \quad v=\langle- 4,2\rangle,\) and \(\mathbf{w}=\langle 3,-1\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$(\mathbf{u} \cdot 2 \mathbf{v}) \mathbf{w}$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$3-i$$

Finding the Component Form of a Vector In Exercises \(13-24,\) find the component form and magnitude of the vector v. \(\begin{array}{ll}{\text { Initial Point }} & {\text { Terminal Point }} \\\ {(-2,7)} & {(5,-17)}\end{array}\)

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ B=75^{\circ} 20^{\prime}, \quad a=6.2, \quad c=9.5 $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$B=2^{\circ} 45^{\prime}, \quad b=6.2, \quad c=5.8$$

In Exercises \(15-24,\) use the vectors \(u=\langle 3,3\rangle, \quad v=\langle- 4,2\rangle,\) and \(\mathbf{w}=\langle 3,-1\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$\|\mathbf{w}\|-1$$

Finding the Component Form of a Vector In Exercises \(13-24,\) find the component form and magnitude of the vector v. \(\begin{array}{ll}{\text { Initial Point }} & {\text { Terminal Point }} \\\ {(1,3)} & {(-8,-9)}\end{array}\)

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ B=125^{\circ} 40^{\prime}, \quad a=37, \quad c=37 $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=145^{\circ}, \quad a=14, \quad b=4$$

In Exercises \(15-24,\) use the vectors \(u=\langle 3,3\rangle, \quad v=\langle- 4,2\rangle,\) and \(\mathbf{w}=\langle 3,-1\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$2-\|\mathbf{u}\|$$

Finding the Component Form of a Vector In Exercises \(13-24,\) find the component form and magnitude of the vector v. \(\begin{array}{ll}{\text { Initial Point }} & {\text { Terminal Point }} \\\ {(1,11)} & {(9,3)}\end{array}\)

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ C=15^{\circ} 15^{\prime}, \quad a=7.45, \quad b=2.15 $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=100^{\circ}, \quad a=125, \quad c=10$$

In Exercises \(15-24,\) use the vectors \(u=\langle 3,3\rangle, \quad v=\langle- 4,2\rangle,\) and \(\mathbf{w}=\langle 3,-1\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$(\mathbf{u} \cdot \mathbf{v})-(\mathbf{u} \cdot \mathbf{w})$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$2 \sqrt{2}-i$$

Finding the Component Form of a Vector In Exercises \(13-24,\) find the component form and magnitude of the vector v. \(\begin{array}{ll}{\text { Initial Point }} & {\text { Terminal Point }} \\\ {(-1,5)} & {(15,12)}\end{array}\)

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ C=43^{\circ}, \quad a=\frac{4}{9}, \quad b=\frac{7}{9} $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=110^{\circ} 15^{\prime}, \quad a=48, \quad b=16$$

In Exercises \(15-24,\) use the vectors \(u=\langle 3,3\rangle, \quad v=\langle- 4,2\rangle,\) and \(\mathbf{w}=\langle 3,-1\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$(\mathbf{v} \cdot \mathbf{u})-(\mathbf{w} \cdot \mathbf{v})$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$-3-i$$

Finding the Component Form of a Vector In Exercises \(13-24,\) find the component form and magnitude of the vector v. \(\begin{array}{ll}{\text { Initial Point }} & {\text { Terminal Point }} \\\ {(-3,11)} & {(9,40)}\end{array}\)

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ C=101^{\circ}, \quad a=\frac{3}{8}, \quad b=\frac{3}{4} $$

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$C=95.20^{\circ}, \quad a=35, \quad c=50$$

Finding the Magnitude of a Vector In Exercises \(25-30\) , use the dot product to find the magnitude of u. $$\mathbf{u}=\langle- 8,15\rangle$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$5+2 i$$

Using the Law of Sines. Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=110^{\circ}, \quad a=125, \quad b=100$$

Finding the Magnitude of a Vector In Exercises \(25-30\) , use the dot product to find the magnitude of u. $$\mathbf{u}=\langle 4,-6\rangle$$

Trigonometric Form of a Complex Number \(\mathrm{In}\) Exercises \(11-30\) , represent the complex number graphically. Then write the trigonometric form of the number. $$8+3 i$$

Using the Law of Sines. Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=110^{\circ}, \quad a=125, \quad b=200$$

Finding the Magnitude of a Vector In Exercises \(25-30\) , use the dot product to find the magnitude of u. $$\mathbf{u}=20 \mathbf{i}+25 \mathbf{j}$$