Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus Enhanced with Graphing Utilities
In Problems 19-32, plot each point given in polar coordinates. (5, 5 3 )In Problems 19-32, plot each point given in polar coordinates. (2, 135 )In Problems 19-32, plot each point given in polar coordinates. (3, 120 )In Problems 19-32, plot each point given in polar coordinates. (4, 2 3 )In Problems 19-32, plot each point given in polar coordinates. (2, 5 4 )In Problems 19-32, plot each point given in polar coordinates. (1, 3 )In Problems 19-32, plot each point given in polar coordinates. (3, 3 4 )In Problems 19-32, plot each point given in polar coordinates. (2,)In Problems 19-32, plot each point given in polar coordinates. (3, 2 )In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates ( r, ) of the point for which: a. r0,20 b. r0,02 c. r0,24 ( 5, 2 3 )In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates ( r, ) of the point for which: a. r0,20 b. r0,02 c. r0,24 ( 4, 3 4 )In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates ( r, ) of the point for which: a. r0,20 b. r0,02 c. r0,24 ( 2,3 )In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates ( r, ) of the point for which: a. r0,20 b. r0,02 c. r0,24 ( 3,4 )In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates ( r, ) of the point for which: a. r0,20 b. r0,02 c. r0,24 ( 1, 2 )In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates ( r, ) of the point for which: a. r0,20 b. r0,02 c. r0,24 ( 2, )In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates ( r, ) of the point for which: a. r0,20 b. r0,02 c. r0,24 ( 3, 4 )In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates ( r, ) of the point for which: a. r0,20 b. r0,02 c. r0,24 ( 2, 2 3 )In Problems 41-56, polar coordinates of a point are given. Find the rectangular coordinates of each point. ( 3, 2 )In Problems 41-56, polar coordinates of a point are given. Find the rectangular coordinates of each point. ( 4, 3 2 )In Problems 41-56, polar coordinates of a point are given. Find the rectangular coordinates of each point. ( 2,0 )42AYU43AYU44AYU45AYU46AYU47AYU48AYU49AYUIn Problems 41-56, polar coordinates of a point are given. Find the rectangular coordinates of each point. ( 3, 90 )In Problems 41-56, polar coordinates of a point are given. Find the rectangular coordinates of each point. ( 7.5, 110 )In Problems 41-56, polar coordinates of a point are given. Find the rectangular coordinates of each point. ( 3.1, 182 )In Problems 41-56, polar coordinates of a point are given. Find the rectangular coordinates of each point. ( 6.3,3.8 )In Problems 41-56, polar coordinates of a point are given. Find the rectangular coordinates of each point. ( 8.1,5.2 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 3,0 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 0,2 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 1,0 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 0,2 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 1,1 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 3,3 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 3 ,1 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 2,2 3 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 1.3,2.1 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 0.8,2.1 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 8.3,4.2 )In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point. ( 2.3,0.2 )In Problems 69-76, the letters xandy represent rectangular coordinates. Write each equation using polar coordinates ( r, ) . 2 x 2 +2 y 2 =3In Problems 69-76, the letters xandy represent rectangular coordinates. Write each equation using polar coordinates ( r, ) . x 2 + y 2 =xIn Problems 69-76, the letters xandy represent rectangular coordinates. Write each equation using polar coordinates ( r, ) . x 2 =4yIn Problems 69-76, the letters xandy represent rectangular coordinates. Write each equation using polar coordinates ( r, ) . y 2 =2xIn Problems 69-76, the letters xandy represent rectangular coordinates. Write each equation using polar coordinates ( r, ) . 2xy=1In Problems 69-76, the letters xandy represent rectangular coordinates. Write each equation using polar coordinates ( r, ) . 4 x 2 y=1In Problems 69-76, the letters xandy represent rectangular coordinates. Write each equation using polar coordinates ( r, ) . x=4In Problems 69-76, the letters xandy represent rectangular coordinates. Write each equation using polar coordinates ( r, ) . y=3In Problems 69-76, the letters rand represent polar coordinates. Write each equation using rectangular coordinates ( x,y ) . r=cosIn Problems 69-76, the letters rand represent polar coordinates. Write each equation using rectangular coordinates ( x,y ) . r=sin+1In Problems 69-76, the letters rand represent polar coordinates. Write each equation using rectangular coordinates ( x,y ) . r 2 =cosIn Problems 69-76, the letters rand represent polar coordinates. Write each equation using rectangular coordinates ( x,y ) . r=sincosIn Problems 69-76, the letters rand represent polar coordinates. Write each equation using rectangular coordinates ( x,y ) . r=2In Problems 69-76, the letters rand represent polar coordinates. Write each equation using rectangular coordinates ( x,y ) . r=4In Problems 69-76, the letters rand represent polar coordinates. Write each equation using rectangular coordinates ( x,y ) . r= 4 1cosIn Problems 69-76, the letters rand represent polar coordinates. Write each equation using rectangular coordinates ( x,y ) . r= 3 3cosChicago ln Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of inlines they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 inlines west of State Street and 36 inlines north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive x-axis . a. Write the location of Wrigley Field using rectangular coordinates. b. Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express in degrees. c. U.S. Cellular Field, home of the White Sox, is located at 35th and Princeton, which is 3 inlines west of State Street and 35 inlines south of Madison. Write the location of U.S. Cellular Field using rectangular coordinates. d. Write the location of U.S. Cellular Field using polar coordinates. Use the east direction for the polar axis. Express in degrees.84AYUIn converting from polar coordinates to rectangular coordinates, what formulas will you use?Explain how to convert from rectangular coordinates to polar coordinates.Is the street system in your town based on a rectangular coordinate system, a polar coordinate system, or some other system? Explain.1AYUThe difference formula for cosine is cos( AB )= . (p.490)3AYUIs the sine function even, odd, or neither? (p. 404)sin 5 4 = . (pp. 385-387)cos 2 3 = . (pp. 385-387)An equation whose variables are polar coordinates is called a(n) _____ ________.True or False The tests for symmetry in polar coordinates are always conclusive.9AYU10AYUTrue or False A cardioid passes through the pole.Rose curves are characterized by equations of the form r=acos( n ) or r=asin( n ),a0 . If n0 is even, the rose has _____ petals; if n1 is odd, the rose has _______ petals.In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r=4In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r=2In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. = 3In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. = 4In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. rsin=4In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. rcos=4In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. rcos=2In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. rsin=2In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r=2cosIn Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r=2sinIn Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r=4sinIn Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r=4cosIn Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. rsec=4In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. rcsc=8In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. rcsc=2In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. rsec=429AYUIn problems 31-38, match each of the graphs ( A ) through ( H ) to one of the following polar equations. = 4In problems 31-38, match each of the graphs ( A ) through ( H ) to one of the following polar equations. = 3 4In problems 31-38, match each of the graphs ( A ) through ( H ) to one of the following polar equations. rcos=2In problems 31-38, match each of the graphs ( A ) through ( H ) to one of the following polar equations. r=1+cosIn problems 31-38, match each of the graphs ( A ) through ( H ) to one of the following polar equations. r=2sinIn problems 31-38, match each of the graphs ( A ) through ( H ) to one of the following polar equations. = 3 4In problems 31-38, match each of the graphs ( A ) through ( H ) to one of the following polar equations. rsin=2In Problems 39-62, identify and graph each polar equation. r=2+2cosIn Problems 39-62, identify and graph each polar equation. r=1+sinIn Problems 39-62, identify and graph each polar equation. r=33sinIn Problems 39-62, identify and graph each polar equation. r=22cosIn Problems 39-62, identify and graph each polar equation. r=2+sinIn Problems 39-62, identify and graph each polar equation. r=2cosIn Problems 39-62, identify and graph each polar equation. r=42cosIn Problems 39-62, identify and graph each polar equation. r=4+2sinIn Problems 39-62, identify and graph each polar equation. r=1+2sinIn Problems 39-62, identify and graph each polar equation. r=12sinIn Problems 39-62, identify and graph each polar equation. r=23cosIn Problems 39-62, identify and graph each polar equation. r=2+4cosIn Problems 39-62, identify and graph each polar equation. r=3cos( 2 )In Problems 39-62, identify and graph each polar equation. r=2sin( 3 )In Problems 39-62, identify and graph each polar equation. r=4sin( 5 )In Problems 39-62, identify and graph each polar equation. r=3cos( 4 )In Problems 39-62, identify and graph each polar equation. r 2 =9cos( 2 )In Problems 39-62, identify and graph each polar equation. r 2 =sin( 2 )In Problems 39-62, identify and graph each polar equation. r= 2In Problems 39-62, identify and graph each polar equation. r= 3In Problems 39-62, identify and graph each polar equation. r=1cosIn Problems 39-62, identify and graph each polar equation. r=3+cosIn Problems 39-62, identify and graph each polar equation. r=13cosIn Problems 39-62, identify and graph each polar equation. r=4cos( 3 )In Problems 63-68, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. r=8cos;r=2secIn Problems 63-68, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. r=8sin;r=4cscIn Problems 63-68, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. r=sin;r=1+cosIn Problems 63-68, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. r=3;r=2+2cosIn Problems 63-68, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. r=1+sin;r=1+cosIn Problems 63-68, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. r=1+cos;r=3cosIn problems 69-72, the polar equation for each graph is either r=a+bcos or r=a+bsin,a0 . Select the correct equation and find the values of aandb .In problems 69-72, the polar equation for each graph is either r=a+bcos or r=a+bsin,a0 . Select the correct equation and find the values of aandb .In problems 69-72, the polar equation for each graph is either r=a+bcos or r=a+bsin,a0 . Select the correct equation and find the values of aandb .In problems 69-72, the polar equation for each graph is either r=a+bcos or r=a+bsin,a0 . Select the correct equation and find the values of aandb .In Problems 73-82, graph each polar equation. r= 2 1cos (parabola)72AYU73AYUIn Problems 73-82, graph each polar equation. r= 1 1cos (parabola)In Problems 73-82, graph each polar equation. r=,0 (spiral of Archimedes)In Problems 73-82, graph each polar equation. r= 3 (reciprocal spiral)In Problems 73-82, graph each polar equation. r=csc2,0 (conchoid)In Problems 73-82, graph each polar equation. r=sintan (cissoid)In Problems 73-82, graph each polar equation. r=tan, 2 2 (kappa curve)In Problems 73-82, graph each polar equation. r=cos 2Show that the graph of the equation rsin=a is a horizontal line a units above the pole if a0and| a | units below the pole if a0 .Show that the graph of the equation rcos=a is a vertical line a units to the right of the pole if a0and| a | units to the left of the pole if a0 .83AYU84AYU85AYU86AYUExplain why the following test for symmetry is valid: Replace r by r and by in a polar equation. If an equivalent equation results, the graph is symmetric with respect to the line = 2 (y-axis) . (a) Show that the test on page 591 fails for r 2 =cos , yet this new test works. (b) Show that the test on page 591 works for r 2 =sin , yet this new test fails.Explain why the vertical-line test used to identify functions in rectangular coordinates does not work for equations expressed in polar coordinates.The tests for symmetry given on page 591 are sufficient, but not necessary. Explain what this means.Explain why the vertical-line test used to identify functions in rectangular coordinates does not work for equations expressed in polar coordinates.The conjugate of 43i is _______. (p. A59)The sum formula for the sine function sin(A+B)= .(p. 493)The sum formula for the cosine function is cos(A+B)= .(p. 490)sin 120 = ; cos 240 = . (pp. 385-387)In the complex plane, the x-axis is referred to as the _____ axis, and the y-axis is called the _____ axis.When a complex number z is written in the polar form z=r(cos+isin) , the nonnegative number r is the _____ or _____ of z , and the angle , 02 , is the _____ of z .Let z 1 =r 1 (cos 1 +isin 1 ) and z 2 =r 2 (cos 2 +isin 2 ) be two complex numbers. Then z 1 z 2 =[ cos( )+isin( ) ] .If z=r( cos+isin ) is a complex number, then z n =[ cos( )+isin( ) ] .Every nonzero complex number will have exactly _____ distinct complex cube roots.True or False The polar form of a nonzero complex number is unique.In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. 1+i12AYU13AYU14AYUIn Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. 3i16AYU17AYUIn Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. 9 3 +9iIn Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. 34iIn Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. 2+ 3 iIn Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. 2+3iIn Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. 5 iIn Problems 25-34, write each complex number in rectangular form. 2( cos 120 +isin 120 )In Problems 25-34, write each complex number in rectangular form. 3( cos 210 +isin 210 )In Problems 25-34, write each complex number in rectangular form. 4( cos 7 4 +isin 7 4 )In Problems 25-34, write each complex number in rectangular form. 2( cos 5 6 +isin 5 6 )In Problems 25-34, write each complex number in rectangular form. 3( cos 3 2 +isin 3 2 )In Problems 25-34, write each complex number in rectangular form. 4( cos 2 +isin 2 )29AYU30AYUIn Problems 25-34, write each complex number in rectangular form. 2( cos 18 +isin 18 )In Problems 25-34, write each complex number in rectangular form. 3( cos 10 +isin 10 )33AYUIn Problems 35-42, find zw and z w . Leave your answers in polar form. z=cos 120 +isin 120 w=cos 100 +isin 10035AYU36AYUIn Problems 35-42, find zw and z w . Leave your answers in polar form. z=2( cos 8 +isin 8 ) w=2( cos 10 +isin 10 )In Problems 35-42, find zw and z w . Leave your answers in polar form. z=4( cos 3 8 +isin 3 8 ) w=2( cos 9 16 +isin 9 16 )In Problems 35-42, find zw and z w . Leave your answers in polar form. z=2+2i w= 3 iIn Problems 35-42, find zw and z w . Leave your answers in polar form. z=1i w=1 3iIn Problems 43-54, write each expression in the standard form . [ 4( cos 40 +isin 40 ) ] 3In Problems 43-54, write each expression in the standard form a+bi . [ 3( cos 80 +isin 80 ) ] 3In Problems 43-54, write each expression in the standard form a+bi . [ 2( cos 10 +isin 10 ) ] 5In Problems 43-54, write each expression in the standard form a+bi . [ 2 ( cos 5 16 +isin 5 16 ) ] 4In Problems 43-54, write each expression in the standard form a+bi . [ 3 ( cos 10 +isin 10 ) ] 6In Problems 43-54, write each expression in the standard form a+bi . [ 1 2 ( cos 72 +isin 72 ) ] 5In Problems 43-54, write each expression in the standard form a+bi . [ 5 ( cos 3 16 +isin 3 16 ) ] 4In Problems 43-54, write each expression in the standard form a+bi . [ 3 ( cos 5 18 +isin 5 18 ) ] 6In Problems 43-54, write each expression in the standard form a+bi . ( 1i ) 5In Problems 43-54, write each expression in the standard form a+bi . ( 3 i ) 6In Problems 43-54, write each expression in the standard form a+bi . ( 2 i ) 6In Problems 43-54, write each expression in the standard form a+bi . ( 1 5 i ) 8In Problems 55-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex cube roots of 88iIn Problems 55-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex cube roots of 3 iIn Problems 55-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex cube roots of 44 3 iIn Problems 55-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex cube roots of 88iIn Problems 55-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex cube roots of 16iIn Problems 55-60, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex cube roots of 859AYU60AYUFind the four complex fourth roots of unity (1) and plot them.Find the six complex sixth roots of unity (1) and plot them.Show that each complex n th root of a nonzero complex number w has the same magnitude.Use the result of Problem 65 to draw the conclusion that each complex n th root lies on a circle with center at the origin. What is the radius of this circle?Refer to Problem 66. Show that the complex n th roots of a nonzero complex number w are equally spaced on the circle.Prove formula (6).Mandelbrot Sets (a) Consider the expression a n = ( a n1 ) 2 +z , where z is some complex number (called the seed) and a 0 =z . Compute a 1 ( = a 0 2 +z ) , a 2 ( = a 1 2 +z ) , a 3 ( = a 2 2 +z ) , a 4 , a 5 and a 6 for the following seeds: z 1 =0.10.4i , z 2 =0.5+0.8i , z 3 =0.9+0.7i , z 4 =1.1+0.1i , z 5 =01.3i , and z 6 =1+1i . (b) The dark portion of the graph represents the set of all values z=x+yi that are in the Mandelbrot set. Determine which complex numbers in part (a) are in this set by plotting them on the graph. Do the complex numbers that are not in the Mandelbrot set have any common characteristics regarding the values of found in part (a)? (c) Compute | z |= x 2 + y 2 for each of the complex numbers in part (a). Now compute | a 6 | for each of the complex numbers in part (a). For which complex numbers is | a 6 || z | and | z |2 ? Conclude that the criterion for a complex number to be in the Mandelbrot set is that | a n || z | and | z |2 .1AYU2AYU3AYU4AYU5AYU6AYU7AYU8AYU9AYU10AYU11AYU12AYU13AYU14AYU15AYU16AYU17AYU18AYU19AYU20AYU21AYU22AYU23AYU24AYU25AYU26AYU27AYU28AYU29AYU30AYU31AYU32AYU33AYU34AYU35AYU36AYU37AYU38AYU39AYU40AYU41AYU42AYU43AYU44AYU45AYU46AYU47AYU48AYU49AYU50AYU51AYU52AYU53AYU54AYU55AYU56AYU57AYU58AYU59AYU60AYU61AYU62AYU63AYU64AYU65AYU66AYU67AYU68AYU69AYU70AYU71AYU72AYU73AYU74AYU75AYU76AYU77AYU