Chapter 11

Fill in the blank to correctly complete each sentence. For the plane curve defined by $$ x=t^{2}+1, y=2 t+3, \quad \text { for } t \text { in }[-4,4] $$, the ordered pair that corresponds to \(t=-3\) is__________.

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left[3\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\right]^{3}$$

Consider case and determine whether the law of sines should be used to solve the triangle. Two angles and the side included between them are known.

Fill in the blank to correctly complete each sentence. For the plane curve defined by \(x=-3 t+6, y=t^{2}-3, \quad\) for \(t\) in \([-5,5]\),the ordered pair that corresponds to \(t=4\) is __________.

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left[2\left(\cos 135^{\circ}+i \sin 135^{\circ}\right)\right]^{4}$$

Consider case and determine whether the law of sines should be used to solve the triangle. Two angles and a side opposite them are known.

Fill in the blank to correctly complete each sentence. For the plane curve defined by $$ x=\cos t, y=2 \sin t, \quad \text { for } t \text { in }[0,2 \pi] $$,the ordered pair that corresponds to \(t=\frac{\pi}{3}\) is __________.

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)^{8}$$

Fill in the blank to correctly complete each sentence. For the plane curve defined by $$ x=-\sin t, y=-2 \cos t, \quad \text { for } t \text { in }[0,2 \pi] $$,the ordered pair that corresponds to \(t=\frac{\pi}{6}\) is__________.

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)^{5}$$

Match the ordered pair from Column II with the pair of parametric equations in Column I on whose graph the point lies. In each case, consider the given value of \(t\). I $$x=3 t+6, y=-2 t+4 ; \quad t=2$$ II A. \((5,25)\) B. \((7,2)\) C. \((12,0)\) D. \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)

For each point given in polar coordinates, state the quadrant in which the point lies if it is graphed in a rectangular coordinate system. (a) \(\left(5,135^{\circ}\right)\) (b) \(\left(2,60^{\circ}\right)\) (c) \(\left(6,-30^{\circ}\right)\) (d) \(\left(4.6,213^{\circ}\right)\)

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left[2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\right]^{3}$$

Which one of the following sets of data does not determine a unique triangle? A. \(A=40^{\circ}, B=60^{\circ}, C=80^{\circ}\) B. \(a=5, b=12, c=13\) C. \(a=3, b=7, c=50^{\circ}\) D. \(a=2, b=2, c=2\)

Match the ordered pair from Column II with the pair of parametric equations in Column I on whose graph the point lies. In each case, consider the given value of \(t\). I $$x=\cos t, y=\sin t ; \quad t=\frac{\pi}{4}$$ II A. \((5,25)\) B. \((7,2)\) C. \((12,0)\) D. \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)

For each point given in polar coordinates, state the axis on which the point lies if it is graphed in a rectangular coordinate system. Also, state whether it is on the positive portion or the negative portion of the axis. (For example, \(\left(5,0^{\circ}\right)\) lies on the positive \(x\) -axis.) (a) \(\left(7,360^{\circ}\right)\) (b) \(\left(4,180^{\circ}\right)\) (c) \(\left(2,-90^{\circ}\right)\) (d) \(\left(8,450^{\circ}\right)\)

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left[3\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)\right]^{4}$$

Which one of the following sets of data determines a unique triangle? A. \(A=50^{\circ}, B=50^{\circ}, C=80^{\circ}\) B. \(a=3, b=5, c=20\) C. \(A=40^{\circ}, B=20^{\circ}, C=30^{\circ}\) D. \(a=7, b=24, c=25\)

Assume triangle \(A B C\) has standard labeling and complete the following. (a) Determine whether SAA, ASA, SSA. SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. \(a, c,\) and \(A\)

Match the ordered pair from Column II with the pair of parametric equations in Column I on whose graph the point lies. In each case, consider the given value of \(t\). I $$x=t, y=t^{2} ; \quad t=5$$ II A. \((5,25)\) B. \((7,2)\) C. \((12,0)\) D. \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)

Plot each point, given its polar coordinates. Give two other pairs of polar coordinates for each point. Do not use a calculator. $$\left(1,45^{\circ}\right)$$

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$6-5 i$$

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left[3 \text { cis } 100^{\circ}\right]^{3}$$

Assume triangle \(A B C\) has standard labeling and complete the following. (a) Determine whether SAA, ASA, SSA. SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. \(a, B,\) and \(C\)

Match the ordered pair from Column II with the pair of parametric equations in Column I on whose graph the point lies. In each case, consider the given value of \(t\). I $$x=t^{2}+3, y=t^{2}-2 ; \quad t=2$$ II A. \((5,25)\) B. \((7,2)\) C. \((12,0)\) D. \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)

Plot each point, given its polar coordinates. Give two other pairs of polar coordinates for each point. Do not use a calculator. $$\left(3,120^{\circ}\right)$$

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$-3+2 i$$

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left[3 \text { cis } 40^{\circ}\right]^{3}$$

Given the following angles and sides, decide whether solving triangle \(A B C\) results in the ambiguous case. Do not use a calculator. A. \(C\) and \(c\)

Assume triangle \(A B C\) has standard labeling and complete the following. (a) Determine whether SAA, ASA, SSA. SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. \(b, c,\) and \(A\)

Find a rectangular equation for each curve and describe the curve. $$x=3 \sin t, y=3 \cos t ; \text { for } t \text { in }[-\pi, \pi]$$

Plot each point, given its polar coordinates. Give two other pairs of polar coordinates for each point. Do not use a calculator. $$\left(-2,135^{\circ}\right)$$

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$2-2 i \sqrt{3}$$

Find each power. Write the answer in rectangular form. $$ (\sqrt{3}+i)^{3} $$

Find a rectangular equation for each curve and describe the curve. $$x=2 \sin t, y=2 \cos t ; \text { for } t \text { in }[0,2 \pi]$$

Plot each point, given its polar coordinates. Give two other pairs of polar coordinates for each point. Do not use a calculator. $$\left(-4,27^{\circ}\right)$$

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$\sqrt{2}+i \sqrt{2}$$

Find each power. Write the answer in rectangular form. Do not use a calculator. $$(2 \sqrt{2}-2 i \sqrt{2})^{6}$$

Find a rectangular equation for each curve and describe the curve. $$x=2 \cos ^{2} t, y=2 \sin ^{2} t ; \text { for } t \text { in }\left[0, \frac{\pi}{2}\right]$$

Plot each point, given its polar coordinates. Give two other pairs of polar coordinates for each point. Do not use a calculator. $$\left(5,-60^{\circ}\right)$$

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$-4 i$$

Find each power. Write the answer in rectangular form. Do not use a calculator. $$(1+i \sqrt{3})^{4}$$

Find a rectangular equation for each curve and describe the curve. $$x=\sqrt{5} \sin t, y=\sqrt{3} \cos t ; \text { for } t \text { in }[0,2 \pi]$$

Plot each point, given its polar coordinates. Give two other pairs of polar coordinates for each point. Do not use a calculator. $$\left(2,-45^{\circ}\right)$$

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$3 i$$

Find each power. Write the answer in rectangular form. Do not use a calculator. $$(2-2 i \sqrt{3})^{4}$$

Find a rectangular equation for each curve and describe the curve. $$x=3 \tan t, y=2 \sec t ; \text { for } t \text { in }\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$

Plot each point, given its polar coordinates. Give two other pairs of polar coordinates for each point. Do not use a calculator. $$\left(-3,-210^{\circ}\right)$$

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$-8$$

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\right)^{4}$$