Chapter 11

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. $$r=4 \cos 2 \theta$$

Solve triangle. There may be two, one, or no such triangle. $$B=74.3^{\circ}, a=859 \text { meters }, b=783 \text { meters }$$

Write each complex number in rectangular form. Give exact values for the real and imaginary parts. Do not use a calculator. $$\sqrt{2} \text { cis } \pi$$

Answer each of the following. Explain why a positive real number must have a positive real \(n\) th root.

Refer to the guidelines to solve oblique triangles to decide on the procedure to use to solve each triangle. Then solve the triangle. \(a=7.031, b=9.947, A=41^{\circ} 12^{\prime}\)

Graph each pair of parametric equations for \(0 \leq t \leq 2 \pi\) in the window \([0,6.6]\) by \([0,4.1] .\) Identify the letter of the alphabet that is being graphed. $$\begin{aligned} &x_{1}=2+0.8 \cos 0.85 t, \quad y_{1}=2+\sin 0.85 t\\\ &x_{2}=1.2+\frac{t}{1.3 \pi}, \quad y_{2}=2 \end{aligned}$$

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. $$r=3 \cos 5 \theta$$

Solve triangle. There may be two, one, or no such triangle. $$C=82.2^{\circ}, a=10.9 \text { kilometers, } c=7.62 \text { kilometers }$$

Write each complex number in rectangular form. Give exact values for the real and imaginary parts. Do not use a calculator. $$\sqrt{3} \operatorname{cis} \frac{3 \pi}{2}$$

True or false: (a) Every real number must have two real square roots. (b) Some real numbers have three real cube roots.

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. $$r^{2}=4 \cos 2 \theta$$

Solve triangle. There may be two, one, or no such triangle. $$A=142.13^{\circ}, b=5.432 \text { feet, } a=7.297 \text { feet }$$

Write each complex number in the trigonometric form \(r(\cos \theta+i \sin \theta)\), where \(r\) is exact and \(0^{\circ} \leq \theta<360^{\circ}\) $$3-3 i$$

Find all complex solutions for each equation. Leave your answers in trigonometric form. $$x^{4}+1=0$$

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. $$\langle- 5,8\rangle$$

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. $$r^{2}=4 \sin 2 \theta$$

Solve triangle. There may be two, one, or no such triangle. $$B=113.72^{\circ}, a=189.6 \text { yards, } b=243.8 \text { yards }$$

Write each complex number in the trigonometric form \(r(\cos \theta+i \sin \theta)\), where \(r\) is exact and \(0^{\circ} \leq \theta<360^{\circ}\) $$-2+2 i \sqrt{3}$$

Find all complex solutions for each equation. Leave your answers in trigonometric form. $$x^{4}+16=0$$

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. $$\langle 6,-3\rangle$$

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. $$r=4-4 \cos \theta$$

Solve triangle. There may be two, one, or no such triangle. $$A=42.5^{\circ}, a=15.6 \text { feet, } b=8.14 \text { feet }$$

Write each complex number in the trigonometric form \(r(\cos \theta+i \sin \theta),\) where \(r\) is exact and \(0^{\circ} \leq \theta<360^{\circ}\) $$1+i \sqrt{3}$$

Find all complex solutions for each equation. Leave your answers in trigonometric form. $$x^{5}-i=0$$

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. $$\langle 2,0\rangle$$

Refer to the guidelines to solve oblique triangles to decide on the procedure to use to solve each triangle. Then solve the triangle. \(a=2634, c=2200, C=73^{\circ} 30^{\prime}\)

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. $$r=6-3 \cos \theta$$

Solve triangle. There may be two, one, or no such triangle. $$C=52.3^{\circ}, a=32.5 \text { yards, } c=59.8 \text { yards }$$

Write each complex number in the trigonometric form \(r(\cos \theta+i \sin \theta),\) where \(r\) is exact and \(0^{\circ} \leq \theta<360^{\circ}\) $$-3-3 i \sqrt{3}$$

Find all complex solutions for each equation. Leave your answers in trigonometric form. $$x^{4}-i=0$$

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. $$\langle 0,-4\rangle$$

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. \(r=2 \sin \theta \tan \theta\) (This is a cissoid.)

Solve triangle. There may be two, one, or no such triangle. $$B=72.2^{\circ}, b=78.3 \text { meters, } c=145 \text { meters }$$

Write each complex number in the trigonometric form \(r(\cos \theta+i \sin \theta),\) where \(r\) is exact and \(0^{\circ} \leq \theta<360^{\circ}\) $$-2 i$$

Find all complex solutions for each equation. Leave your answers in trigonometric form. $$x^{3}+1=0$$

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. Direction angle \(45^{\circ},\) magnitude 8

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. \(r=\frac{\cos 2 \theta}{\cos \theta}\) (This is a cissoid with a loop.)

Solve triangle. There may be two, one, or no such triangle. $$C=68.5^{\circ}, c=258 \text { centimeters, } b=386 \text { centimeters }$$

Find all complex solutions for each equation. Leave your answers in trigonometric form. $$x^{3}+i=0$$

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. Direction angle \(210^{\circ},\) magnitude 3

CONCEPT CHECK related to the geometric property that the sum of the lengths of any two sides of a triangle must be greater than the remaining side?

The screen shown to the right is an example of a Lissajous figure. Lissajous figures occur in electronics and may be used to find the frequency of an unknown voltage. Graph each Lissajous figure for \(0 \leq t \leq 6.5\) in the window \([-6.6,6.6]\) by \([-4.1,4.1]\). $$x=2 \cos t, y=3 \sin 2 t$$

Answer each question.How do you graph \((r, \theta)\) by hand if \(r<0 ?\).

Solve triangle. There may be two, one, or no such triangle. $$A=38^{\circ} 40^{\prime}, a=9.72 \text { kilometers, } b=11.8 \text { kilometers }$$

Write each complex number in trigonometric form, where \(r\) is exact and \(0 \leq \theta<2 \pi\) $$4 \sqrt{3}+4 i$$

Find all complex solutions for each equation. Leave your answers in trigonometric form. $$x^{3}-8=0$$

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. Direction angle \(115^{\circ},\) magnitude 0.6

The screen shown to the right is an example of a Lissajous figure. Lissajous figures occur in electronics and may be used to find the frequency of an unknown voltage. Graph each Lissajous figure for \(0 \leq t \leq 6.5\) in the window \([-6.6,6.6]\) by \([-4.1,4.1]\). (GRAPH CANNOT COPY) $$x=2 \sin 2 t, y=3 \cos 3 t$$

Answer each question.For \(r>0,\) the points \((r, \theta)\) and \(\left(-r, \theta+180^{\circ}\right)\) have the same graph. Why this is so?

Solve triangle. There may be two, one, or no such triangle. $$C=29^{\circ} 50^{\prime}, a=8.61 \text { meters, } c=5.21 \text { meters }$$