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NAME DATE PERIOD9-1EnrichmentDistance on the Earth’s SurfaceAs you learned in Lesson 9-1, lines of longitude onEarth’s surface intersect at the North and South Poles.A line of longitude that passes completely around Earthis called a great circle. All great circles have thesame circumference, found by calculating thecircumference of a circle with Earth’s radius, 3963.2miles. (Since Earth is slightly flattened at the poles, itis not precisely spherical. The difference is so small,however, that for most purposes it can be ignored.)1. Find the circumference of a great circle.On a great circle, position is measured in degrees north or south ofthe equator. Pittsburgh’s position of 40 26’ N means that radii fromEarth’s center to Pittsburgh and to the point of intersection of theequator and Pittsburgh’s longitude line form an angle of 40 26’.(See the figure above.)2. Find the length of one degree of arc on a longitude line.3. Charleston, South Carolina (32 46’ N), and Guayaquil, Ecuador(2 9’ S), both lie on Pittsburgh’s longitude line. Find the distancefrom Pittsburgh to each of the other cities.Because circles of latitude are drawn parallel to theequator, their radii and circumferences grow steadilyshorter as they approach the poles. The length of onedegree of arc on a circle of latitude depends on how farnorth or south of the equator the circle is located. Thefigure at the right shows a circle of latitude of radius rlocated degrees north of the equator. Because theradii of the equator and the circle of latitude arerparallel, m NEO . Therefore, cos , whichRgives r R cos , where R represents the radius ofEarth.4. Find the radius and circumference of a circle of latitude located70 north of the equator.5. Find the length of one degree of arc on the circle described inExercise 4.6. Bangor, Maine, and Salem, Oregon, are both located at latitude44 50’ N. Their respective longitudes are 68 46’ and 123 2’west of Greenwich. Find the distance from Bangor to Salem. Glencoe/McGraw-Hill369Advanced Mathematical Concepts

NAME DATE PERIOD9-2Study GuideGraphs of Polar EquationsA polar graph is the set of all points whose coordinates (r, )satisfy a given polar equation. The position and shape of polargraphs can be altered by multiplying the function by anumber or by adding to the function. You can also alter thegraph by multiplying by a number or by adding to it.Example 1 Graph the polar equation r 2 cos 2 .Make a table of values. Graph the ordered pairsand connect them with a smooth curve. 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 Example 22 cos 2 210 1 2 101210 1 2 101(r, )(2, 0 )(1, 30 )(0, 45 )( 1, 60 )( 2, 90 )( 1, 120 )(0, 135 )(1, 150 )(2, 180 )(1, 210 )(0, 225 )( 1, 240 )( 2, 270 )( 1, 300 )(0, 315 )(1, 330 )This type of curve is called a rose.Notice that the farthest points are2 units from the pole and the rosehas 4 petals.Graph the system of polar equations. Solve thesystem using algebra and trigonometry, andcompare the solutions to those on your graph.r 2 2 cos r 2 2 cos To solve the system of equations,substitute 2 2 cos for r in the secondequation.2 2 cos 2 2 cos cos 03 or 22Substituting each angle into either of theoriginal equations gives r 2. The solutions3 and 2, .of the system are therefore 2, 2 2 Tracing on the curves shows that these solutionscorrespond with two of the intersection points onthe graph. The curves also intersect at the pole. Glencoe/McGraw-Hill370Advanced Mathematical Concepts

NAME DATE PERIOD9-2PracticeGraphs of Polar EquationsGraph each polar equation. Identify the type of curve eachrepresents.1. r 1 cos 2. r 3 sin 3 3. r 1 2 cos 4. r 2 2 sin 5. r 0.5 6. r2 16 cos 2 Graph each system of polar equations. Solve the systemusing algebra and trigonometry. Assume 0 2 .7. r 1 2 sin 8. r 1 cos r 2 sin r 3 cos 9. Design Mikaela is designing a border for her stationery.Suppose she uses a rose curve. Determine an equation fordesigning a rose that has 8 petals with each petal 4 units long. Glencoe/McGraw-Hill371Advanced Mathematical Concepts

NAME DATE PERIOD9-2EnrichmentSymmetry in Graphs of Polar EquationsIt is sometimes helpful to analyze polar equations for certainproperties that predict symmetry in the graph of the equation. Thefollowing rules guarantee the existence of symmetry in the graph.However, the graphs of some polar equations exhibit symmetry eventhough the rules do not predict it.1. If replacing by – yields the same equation, then the graph ofthe equation is symmetric with respect to the line containing thepolar axis (the x-axis in the rectangular coordinate system).2. If replacing by yields the same equation, then the graphof the equation is symmetric with respect to the line 2(the y-axis in the rectangular coordinate system).3. If replacing r by – r yields the same equation, then the graphof the equation is symmetric with respect to the pole.ExampleIdentify the symmetry of and graph r 3 3 sin .Since sin ( ) sin , by rule 2 the graph is symmetric with respect to the line . Therefore, it is only necessary2to plot points in the first and fourth quadrants. 3 3 sin 20– 30.4– 61.503.0 64.5 35.6 26.0– (r, ) 0, – 0.4, – 1.5, – 2 3 6(3.0, 0) 4.5, 5.6, 6.0, 6 3 2The points in the second and third quadrants are found by usingsymmetry.Identify the symmetry of and graph each polar equation on polar grid paper.2. r2 4 sin 2 1. r 2 3 cos Glencoe/McGraw-Hill372Advanced Mathematical Concepts

NAME DATE PERIOD9-3Study GuidePolar and Rectangular CoordinatesUse the conversion formulas in the following examples toconvert coordinates and equations from one coordinate systemto the other.Example 1Find the rectangular coordinates of each point.3 a. P 3, 4 b. Q(20, 60 )3 3 , r 3 and .For P 3, 4 4Use the conversion formulasx r cos and y r sin .x r cos 3 3 cos 4 2 3 2 3 2 or 2y r sin 3 3 sin 4 2 3 2 3 2 or 2The rectangular coordinatesFor Q(20, 60 ), r 20 and 60 .x r cos 20cos ( 60 ) 20(0.5) 10y r sin 20 sin( 60 ) 3 20 2 10 3 The rectangular coordinates of Q are(10, 10 3 ), or approximately(10, 17.32)3 2 , 3 2 , or of P are 22 ( 2.12, 2.12) to the nearesthundredth.Example 2Find the polar coordinates of R(5, 9).For R(5, 9), x 5 and y 9.r x 2 y2 ( 9 ) 252 1 0 6 or about 10.30y Arctan x x 0 9 Arctan 5 1.06To obtain an angle between 0 and 2 you can add2 to the -value. This results in 5.22.The polar coordinates of R are approximately(10.30, 5.22).Example 3Write the polar equation r 5 cos inrectangular form.r 5 cos r2 5r cos x2 y2 5x Glencoe/McGraw-HillMultiply each side by r.r 2 x 2 y 2 and r cos x373Advanced Mathematical Concepts

NAME DATE PERIOD9-3PracticePolar and Rectangular CoordinatesFind the rectangular coordinates of each point with thegiven polar coordinates.1. (6, 120 )2. ( 4, 45 ) 3. 4, 6 13 4. 0, 3 Find the polar coordinates of each point with the givenrectangular coordinates. Use 0 2 and r 0.5. (2, 2)6. (2, 3) )7. ( 3, 38. ( 5, 8)Write each polar equation in rectangular form.9. r 410. r cos 5Write each rectangular equation in polar form.12. y 311. x2 y2 913. Surveying A surveyor records the polar coordinates of the locationof a landmark as (40, 62 ). What are the rectangular coordinates? Glencoe/McGraw-Hill374Advanced Mathematical Concepts

NAME DATE PERIOD9-3EnrichmentPolar RosesThe polar equation r a sin n graphs as a rose.When n 1, the rose is a circle — a flower with one leaf.Sketch the graphs of these roses.1. r 2 sin 2 2. r –2 sin 3 3. r –2 sin 4 4. r 2 sin 5 5. The graph of the equation r a sin n is a rose. Use yourresults from Exercises 1–4 to complete these conjectures.a. The distance across a petal is ? units.b. If n is an odd integer, the number of leaves is ? .c. If n is an even integer, the number of leaves is ? .6. Write r 2 sin 2 in rectangular form.7. The total area A of the three leaves in the three-leaved roser a sin 3 is given by A area is A 1 a2 .21 4a2 . For a four-leaved rose, thea. Find the area of a four-leaved rose with a 6.b. Write the equation of a three-leaved rose with area 36 . Glencoe/McGraw-Hill375Advanced Mathematical Concepts

NAME DATE PERIOD9-4Study GuidePolar Form of a Linear EquationExample 1Write the equation x 3y 6 in polar form.The standard form of the equation is x 3y 6 0. To find the values of p and , write theequation in normal form. To convert to normal 2 B2 .form, find the value of A A 2 B2 1 2 32 or 1 0 Since C is negative, use 1 0 .The normal form of the equation is 1 03 1 0y 3 1 0 0. 1 x 3 y 6 0 or x 10105 1 0 1 0 1 0Using the normal form x cos y sin p 0,3 1 0 . Since cos and sin we can see that p 6 or 5 1 0 are both positive, the normal lies in Quadrant I.sin tan cos tan 3 1.25 3 1 0 1 3 1 0Use the Arctangent function.Substitute the values for p and into the polar form.p r cos( )3 1 0 r 5Example 2cos( 1.25)Polar form of x 3y 6Write 3 r cos( 30 ) in rectangular form.3 r cos( 30 )3 r(cos cos 30 sin sin 30 )Difference identity for cosine cos 1 sin 33 r 22 , sin 30 1 3cos 30 22 r cos 1 r sin 33 22Distributive property x 1 y33 22r cos x, r sin y x y6 3 x y 60 3Multiply each side by 2.Subtract 6 from each side.The rectangular form of 3 rcos( 30 ) is 3 x y 6 0. Glencoe/McGraw-Hill376Advanced Mathematical Concepts

NAME DATE PERIOD9-4PracticePolar Form of a Linear EquationWrite each equation in polar form. Round to the nearestdegree.1. 3x 2y 162. 3x 4y 153. 3x 4y 124. y 2x 1Write each equation in rectangular form.5 6. 2 r cos ( 90 )5. 4 r cos 6 7. 1 r cos 4 Graph each polar equation.9. 3 r cos ( 60 )8. 3 r cos ( 240 ) 10. 1 r cos 3 11. Landscaping A landscaper is designing a garden with hedgesthrough which a straight path will lead from the exterior of thegarden to the interior. If the polar coordinates of the endpoints ofthe path are (20, 90 ) and (10, 150 ), where r is measured in feet,what is the equation for the path? Glencoe/McGraw-Hill377Advanced Mathematical Concepts

NAME DATE PERIOD9-4EnrichmentDistance Using Polar CoordinatesSuppose you were given the polar coordinates of twopoints P1(r1, 1) and P2(r2, 2) and were asked to findthe distance d between the points. One way would beto convert to rectangular coordinates (x1, y1) and(x2, y2), and apply the distance formula2 d (x x ) (y y ) 2.2121A more straightforward method makes use of the Lawof Cosines.1. In the above figure, the distance d between P1 and P2 is the lengthof one side of OP1 P2. Find the lengths of the other two sides.2. Determine the measure of P1OP2.3. Write an expression for d 2 using the Law of Cosines.4. Write a formula for the distance d between the pointsP1 (r1, 1 ) and P2 (r2, 2 ).5. Find the distance between the points (3, 45 ) and (5, 25 ). Roundyour answer to three decimal places. and 4, . Round6. Find the distance between the points 2, 2 8your answer to three decimal places.7. The distance from the point (5, 80 ) to the point (r, 20 ) is 2 1 .Find r. Glencoe/McGraw-Hill378Advanced Mathematical Concepts

NAME DATE PERIOD9-5Study GuideSimplifying Complex NumbersAdd and subtract complex numbers by performing the chosenoperation on both the real and imaginary parts. Find theproduct of two or more complex numbers by using the sameprocedures used to multiply binomials. To simplify thequotient of two complex numbers, multiply the numerator anddenominator by the complex conjugate of the denominator.To find the value of i n, let R be theremainder when n is divided by 4.if R 0in 1if R 1in iif R 2i n 1if R 3i n iExample 1Powers of ii1 ii 3 i 2 i ii5 i4 i ii 7 i 4 i 3 ii 2 1i 4 (i 2)2 1i 6 i 4 i 2 1i 8 (i 2)4 1Simplify each power of i.b. i 11a. i30Method 1Method 2Method 1Method 230 (i4)7 i2 11 4 3 R 1 i 11 (i4) 3 i130 4 7 R 2in72If R 2, i 1. (1) iIf R 1, in i. (1) 3 i1i30 1 1i 11 i iExample 2Simplify each expression.a. (3 2i) (5 3i)(3 2i) (5 3i) (3 5) (2i 3i) 8 iExample 3b. (8 4i) (9 7i)(8 4i) (9 7i) 8 4i 9 7i 1 3iSimplify (4 2i)(5 3i).(4 2i)(5 3i) 5(4 2i) 3i(4 2i) 20 10i 12i 6i2 20 10i 12i 6( 1) 14 22iExample 4Distributive propertyDistributive propertyi2 1Simplify (4 5i) (2 i).4 5i(4 5i) (2 i) 2 i4 5i 2 i 2 i2 i2i is the conjugate of 2 i.8 10i 4i 5i 4 i228 14i 5( 1) 4 ( 1)i2 35 154 iWrite the answer in the form a bi.13 14i 5 Glencoe/McGraw-Hill379Advanced Mathematical Concepts

NAME DATE PERIOD9-5PracticeSimplifying Complex NumbersSimplify.1. i382. i 173. (3 2i) (4 5i)4. ( 6 2i) ( 8 3i)5. (8 i) (4 i)6. (1 i)(3 2i)7. (2 3i)(5 i)8. (4 5i)(4 5i)9. (3 4i)211. (2 i) (2 i)10. (4 3i) (1 2i)8 7i12. 1 2i13. Physics A fence post wrapped in two wires has two forcesacting on it. Once force exerts 5.3 newtons due north and4.1 newtons due east. The second force exerts 6.2 newtons duenorth and 2.8 newtons due east. Find the resultant force on thefence post. Write your answer as a complex number. (Hint: Avector with a horizontal component of magnitude a and a verticalcomponent of magnitude b can be represented by the complexnumber a bi.) Glencoe/McGraw-Hill380Advanced Mathematical Concepts

NAME DATE PERIOD9-5EnrichmentCycle QuadruplesFour nonnegative integers are arranged in cyclic orderto make a “cyclic quadruple.” In the example, this quadruple is 23, 8, 14, and 32.The next cyclic quadruple is formed from the absolutevalues of the four differences of adjacent integers: 23 – 8 15 8 – 14 6 14 – 32 18 32 – 23 9By continuing in this manner, you will eventually getfour equal integers. In the example, the equal integersappear in three steps.Solve each problem.1. Start with the quadruple 25, 17, 55, 47. In how manysteps do the equal integers appear?2. Some interesting things happen when one or more of theoriginal numbers is 0. Draw a diagram showing abeginning quadruple of three zeros and one nonnegativeinteger. Predict how many steps it will take to reach4 equal integers. Also, predict what that integer willbe. Complete the diagram to check your predictions.3. Start with four integers, two of them zero. If the zeros areopposite one another, how many steps does it take for the zeros todisappear?4. Start with two equal integers and two zeros. The zeros are nextto one another. How many steps does it take for the zeros todisappear?5. Start with two nonequal integers and two zeros. The zeros arenext to one another. How many steps does it take for the zeros todisappear?6. Start with three equal integers and one zero. How many stepsdoes it take for the zero to disappear?7. Describe the remaining cases with one zero and tell how manysteps it takes for the zero to disappear. Glencoe/McGraw-Hill381Advanced Mathematical Concepts

NAME DATE PERIOD9-6Study GuideThe Complex Plane and Polar Form ofComplex NumbersIn the complex plane, the real axis is horizontal and theimaginary axis is vertical. The absolute value of a complexnumber is its distance from zero in the complex plane.The polar form of the complex number a bi is r(cos i sin ), which is often abbreviated as r cis . In polar form,r represents the absolute value, or modulus, of the complexnumber. The angle is called the amplitude or argument ofthe complex number.Example 1Graph each number in the complex plane andfind its absolute value.a. z 4 3i 2 32 z 4 5iOb. z 2iz 0 2i 2 ( 2 ) 2 z 0 2(4, 3)iROR(0, –2 )Example 2Express the complex number 2 3i in polar form.First plot the number in the complex plane.i(2, 3)Then find the modulus.r 2 2 32 or 1 3 ORNow find the amplitude. Notice that is inQuandrant I. Arctan 32 Arctan ab if a 0 0.98Therefore, 2 3i 1 3 (cos 0.98 i sin 0.98) or 1 3 cis 0.98. Glencoe/McGraw-Hill382Advanced Mathematical Concepts

NAME DATE PERIOD9-6PracticeThe Complex Plane and Polar Form of ComplexNumbersGraph each number in the complex plan and find its absolute value.1. z 3i2. z 5 i3. z 4 4iExpress each complex number in polar form.4. 3 4i5. 4 3i6. 1 i7. 1 iGraph each complex number. Then express it in rectangularform.3 3 5 5 4 4 i sin i sin i sin 9.

ACT Practice questions that appear in the Student Edition on page 613. Page A2 is an answer sheet for the SAT and ACT Practice master. These improve students'familiarity with the answer formats they may encounter in test taking. The answers for the lesson-by-lesson masters are provided as reduced pages with answers appearing in red.