ࡱ>  R{r 0mbjbjWW 055]&&ii///0000103l =A=A=AA,,,^``````$!n-/,ii=AAe~~~i=Ap/A^~^~~ (~A<80z ~J0~V~/~,~,,,f,,,,,,,,,,,,& /: PreCalculus: Trig Functions DayTopicsHWGrade1Right triangle trigonometryHW Trig Functions 1 2Circle functionsHW Trig Functions 2 3Graphs of Sine and Cosine IHW Trig Functions 3 4Graphs of Sine and Cosine IIHW Trig Functions 4 5Practice HW Trig Functions 5 6Inverse trig functionsHW Trig Functions 6 7Simple trig equationsHW Trig Functions 7 8Trig modelsHW Trig Functions Review 9Review10**TEST** PreCalculus HW Trig Functions - 1 1. A TV camera is located 4000 feet from the base of a rocket launch pad. A rocket rises vertically from the pad. Let h represent the rockets height in feet; let ( represent the angle of elevation of the camera as it tracks the rocket, and let s represent the distance between the camera and the rocket. a. Find h as a function of (. b. Find s as a function of h.  2. A lighthouse is on a small island 3 km away from a straight length of shoreline on the mainland. Measured from the point on the shore closest to the lighthouse, how far along the shore is the beam of light when the beam makes an angle of (? 3. On a certain day in Honolulu, the sun passes directly overhead. When the angle of elevation from the ground to the sun is (, a 150 foot tall apartment building casts a shadow of length x. Write an equation to give x as a function of (. 4. An aircraft in level flight overflies a radar station at an altitude of 10 km. a. What is the angle of elevation from the radar to the aircraft as a function of the aircrafts distance from the point on its path directly above the radar? b. What is the straight-line distance from the radar station to the aircraft as a function of the aircrafts distance from the point on its path directly above the radar? 5. The Dorkosian Hilton Hotel is 800 meters high and has a glass elevator on the outside of the building with spectacular views of the city of Thermos. Unfortunately, you are staying at the Dorkosian Motel 6 located 200 m away from the Hilton and all you get to do is watch the elevator from your window 150 meters above the ground. Find an equation relating the angle of elevation from your window to the elevator as a function of h, the elevators height above the ground.  6. A horizontally revolving spotlight with a narrowly focused beam shines on a long straight wall 20 feet away. The intensity of the light where it hits the wall is inversely proportional to the distance from the source (i.e.,  EMBED Equation.DSMT4  for some constant k). Find an equation for I as a function of (.  7. Two sides and the included angle of a triangle are known values a, b and C respectively (see the diagram). a. Find the height h in terms of known values (a, b and/or C). b. Find the area of the triangle in terms of known values. (The answer should look familiar.) PreCalculus HW Trig Functions - 2 You should be able to do this assignment without using your calculator. 1. Evaluate the following. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  2. Find the exact value of each of the following without using your calculator and learn them well enough to be able to do them quickly and accurately in your head (seriously). a.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  3. Find the reference angle for each of the following. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  4. Evaluate the following: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d. tan2  EQ \f(5p,6)  e. sin  EMBED Equation.DSMT4  f. 4sin2  EQ \f(5p,4)  g. tan  EQ \f(1,2) ( 5. For the function  EMBED Equation.DSMT4 , evaluate a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  0p/2p3p/22psincos 6. a. Fill in the table at right with the key values of the sine and cosine functions. b. Use the table to sketch one period each of the graphs of y = sin x and y = cos x. PreCalculus HW Trig Functions - 3 1. a. Is the sine function even or odd? sin((x) = ? b. Is the cosine function even or odd? cos((x) = ? 2. Sketch at least one full period of each function without using your calculator. Show appropriate scales on the axes. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  3. Write the equation of the given graph.   a. b. c. 4. Normal electrical outlets in the US provide alternating current. The voltage varies sinusoidally with a maximum of about 115 volts and a minimum of 115 volts (measured from a 0 level called ground). The period of the current is 1/60 of a second. If the voltage is represented by V = Asin Bt, find the values of A and B. 5. While Norman Nerdling is doing math homework, his blood pressure can be (approximately) modeled by BP = 30sin(150(t) + 98 where t is in minutes. a. What is Normans highest (systolic) blood pressure? His lowest (diastolic)? b. How does the + 98 affect the graph of Normans pulse rate? c. What is the period of Normans blood pressure in minutes? d. What is Normans pulse rate (number of beats per minute)? 6. The minute hand of Big Ben (famous clock on a famous tower in a famous city) is 14 feet long. Let the x-axis go through the center of Big Ben. a. Write an equation to represent the height of the tip of the minute hand above the x-axis as a function of time in minutes. Let t = 0 represent on the hour. b. The center of the clock is about 190 feet above ground level. Write an equation for the height of the tip of the minute hand above the ground as a function of time in minutes. 7. Erin has tied one end of a rope to a wall and is shaking the other end up and down. Her brother Darrin snapped the picture shown at right. If the highest point of the rope is 5 feet off the ground, the lowest point is 2 feet off the ground and Erins hand is 12 feet from the wall, write an equation for the shape of the rope. Let the origin be at ground level beneath Erins hand.  SHAPE \* MERGEFORMAT  PreCalculus HW Trig Functions - 4 1. Sketch at least one full period of each function without using the graphing or table functions of your calculator. Show appropriate scales on the axes. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  2. Write the equation of the given graph.  a. b.  c. 4. A weight hangs from a spring in the Physics lab next to a vertical meter stick. Phyllis pulled the weight down a certain amount and let go. Then she started a stopwatch and observed the weights position along the meter stick at various times. Her data is recorded in the table below. Time (s)0.00.20.40.60.81.01.21.41.61.82.0Position (cm)14.720.025.327.525.320.014.712.514.720.025.3 Write an equation for the weights position, y, as a function of time, t. 5. Billy the Barnacle clings to the outer edge of a paddle on the paddlewheel of a steamboat. As the paddlewheel turns, Billys height above the water is given by  EMBED Equation.DSMT4  where h is in feet and t is in seconds. a. What is Billys greatest height above the water? b. How deep below the waters surface does Billy go? c. How long does it take the Billy to go around once on the paddlewheel? d. Sketch the graph of Billys height above the water for one complete revolution of the paddlewheel without using your calculator. e. During one trip around on the paddlewheel, how many seconds does Billy spend under water? (Use your calculator for this part.) PreCalculus Trig Functions - 5 1. Evaluate the following without using your calculator: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e. ln (cos2() f.  EMBED Equation.DSMT4  g. log 50 log 5 h. ln (sin 2() 2. For the function  EMBED Equation.DSMT4 , a > 0, a. Find the range of the function. b. Find the period of the function. c. Find the average value of y over one period. d. Find a value of x where the function takes on its maximum value. 3. Sunspot activity is cyclical. Suppose the number of sunspots per year is modeled by  EMBED Equation.DSMT4  where t is in years. According to this model, a. What is the maximum number of sunspots you would expect in one year? b. What is the minimum number of sunspots you would expect in one year? c. How many years is one complete sunspot cycle? d. In this model, the year that corresponds to t = 0 was a year of (multiple choice) (1) minimum sunspot activity (2) average sunspot activity (3) maximum sunspot activity  4. The graph shows the time of sunrise in Moosebutt, ON during a year. Write an equation for the time of sunrise, S, as a function of time t in months.  5. Little Willy Fugnutter is on an inner tube in the wave pool at MatheMagic Fun Park. The graph shows Willys height in feet above the bottom of the pool as a function of time in seconds as the waves go by. Write an equation for the function shown in the graph. (This assignment is continued on the next page.) 6. Fritz Fugnutter is riding the Ferris wheel at MatheMagic Fun Park. Meanwhile, his brother Willy is exploring the tree house right next to the Ferris wheel. Willy observes that as the Ferris wheel turns, his brother is sometimes higher than he is and sometimes lower. Fritzs height relative to Willy is given by  EMBED Equation.DSMT4  where y is Fritzs height above Willy in feet and t is time in seconds.  a. Sketch a graph of Fritzs height above Willy as a function of time. Do not use your calculator. b. How long does it take the Ferris wheel to go around once? c. If Fritzs maximum height above the ground is 30 feet, how high is the tree house Willy is in? PreCalculus HW Trig Functions - 6 1. a. Sketch a graph of  EMBED Equation.DSMT4 . Clearly label important features. (Use your calculator.) b. Identify an appropriate restricted domain for  EMBED Equation.DSMT4 . c. Find the domain and range of  EMBED Equation.DSMT4  (a.k.a.  EMBED Equation.DSMT4 ) 2. Evaluate the following without using your calculator. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  h.  EMBED Equation.DSMT4   READ: (You should already know this from A2Trig but read it anyway.) To solve an equation of the form  EMBED Equation.DSMT4 , where trg may be any trig function, on the domain [0, 2(), 1. Determine what quadrants x may be in. Draw a diagram. 2. Find the reference angle:  EMBED Equation.DSMT4 . (Note absolute value signs.) 3. Use the reference angle and your diagram to find the solutions.  Ex: Solve  EMBED Equation.DSMT4  for x( [0, 2() 1. cos < 0 in QII and QIII 2.  EMBED Equation.DSMT4  (second diagram) 3. QII:  EMBED Equation.DSMT4  QIII:  EMBED Equation.DSMT4  Solutions:  EMBED Equation.DSMT4  3. Without using your calculator, find all solutions in the interval [0, 2() for each of the following. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  4. Using your calculator, find all the angles in the interval [0, 2() for which a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  PreCalculus HW Trig Functions - 7 Without using your calculator, find the general solution to each equation. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3. 3tan2 x = 1 4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  Using your calculator to do the inverse functions, solve the following algebraically. 8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  PreCalculus HW Trig Functions - 8 1. The city of Thermos is located in the northern hemisphere of the planet Dorkos. The mean daily high temperature in Thermos can be modeled by T = 12 + 18 sin 0.0280294t where T is temperature and t is time in (Dorkosian) days after New Year's Day. a. What is the mean high temperature of the warmest day of the year in Thermos? b. What is the mean high temperature of the coldest day of the year in Thermos? c. How many Dorkosian days make a Dorkosian year? d. How many days after the beginning of the year is the warmest day of the year in Thermos? e. When during the year would the mean high temperature be 0? f. On Dorkos, New Year's Day is most likely celebrated on (multiple choice) (1) the winter solstice (2) the vernal equinox (3) the summer solstice (4) the autumnal equinox g. How often do Dorkosians have leap years? 2. The Dorkosians use a base 16 number system. As a result, they have divided the Dorkosian day into 16 hours. In Thermos on New Year's Day, the mean high temperature is 12 and the mean low temperature, occurring 8 hours later, is 2. a. Write a model for the temperature T in Thermos on New Year's Day as a function of time t measured in hours after the hottest part of the day. (Hint: you should figure out amplitude, angular frequency, vertical shift, and which trig function is most appropriate.) b. When will the temperature be 10? c. It turns out that the hottest part of the day is 2 hours after noon. Rewrite the model from part a so that t is measured in hours after noon. d. How many fingers do Dorkosians have? 3. The longest day of the year in Seattle has approximately 16 hours of daylight; the shortest day has approximately 8.5 hours of daylight. a. Write a trigonometric model for the number of hours of daylight in Seattle as a function of t where t represents the number of days since the Vernal Equinox (March 21). b. Given that March 21 is the 80th day of the calendar year, rewrite the model so that t represents the day of the year (t = 1 corresponds to Jan 1). c. Approximately how many hours of daylight are there in Seattle on Halloween? d. What is the date of the longest day of the year in Seattle (according to the model)? e. Given that Norwood-Norfolk is a little (about 3) south of Seattle in latitude, how would a model for number of daylight hours here be different from the model for Seattle? PreCalculus Review: Trigonometric Functions 1. Give the exact value of  EMBED Equation.DSMT4 . 2. Find, in terms of (, all the values of ( in the interval [0, 2() that satisfy  EMBED Equation.DSMT4  3. If A is the radian measure a first quadrant angle and cos A = k, find, in terms of A, the measures of all the angles in the interval [0, 2() that satisfy the equation cos x = -k  4. Little Willy has lost control of his remote controlled airplane and is watching it fly away from him. The plane is flying at a constant altitude a above the ground and Willys eyes are at a height h off the ground; x, d and ( are variables as represented in the diagram. a. Find d as a function of x. b. Find x as a function of (. c. Find d as a funciton of (. d. Find ( as a function of x. 5. A weight on a spring bounces up and down. Its height is given by  EMBED Equation.DSMT4 , where a > 0, ho > a, y is the weights height above the floor in centimeters and t is time in seconds. a. What is the amplitude of the weights motion? b. What is the vertical shift of the function? c. What was the height of the weight at time t = 0? d. How many seconds does it take the weight to bounce up and down once? e. How many bounces will the weight make in one minute? f. What is the weights greatest height above the floor? g. Why do we need ho > a? h. What is the phase shift of the weights motion (amount and direction)? 6. Find the general solution for each of the following equations: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  7. Write an equation for the graph below.  8. The center of the clock in the clock tower of the School of Advanced Mathematics is 24 meters above the ground. The hour hand of the clock is 3 meters in length (measured from the center of the clock to the tip of the hand). An acrophobic bug clings to the tip of the hour hand. Sketch a graph of the bugs bug's height above the ground as a function of time in hours since noon and write an equation for the function. 9. At 6:00 PM exactly, the hapless bug from problem #8 lets go of the clock hand and falls to the sidewalk below. There he is promptly run over by a passing bicyclist and gets caught in the tread of the bike's front tire. Although he is blissfully unaware of this, the bug's height above the ground is now given by  EMBED Equation.DSMT4 where h is in meters and t is time in seconds since the bug got run over. a. What is the bug's greatest height above the ground while caught in the tire? b. How long does it take the bug to make one complete revolution? c. How fast is the bicyclist traveling in km/hr? Answers to Selected Problems  HW - 1 1a. h = 4000tan ( 2. x = 3tan( 3.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4   4a.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  7a. h = asinC b.  EMBED Equation.DSMT4  HW - 2 1a. e b. 0 c. 0 d.  EMBED Equation.DSMT4  2a.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  , 1 b.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  3a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  4a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c. 2 d.  EQ \f(1,3)  e.  EMBED Equation.DSMT4  f. 2 g. undef. 5a. (2 b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  HW - 3 1a. Odd; (sin x b. Even; cos x 2a. b. 3a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5a. 128, 68 b. Vertical translation up 98 c. 1/75 s d. 75 6a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  HW 4 1a. b. 2a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5a. 25 ft b. 3 ft c. 6 s d. e. 1.27 s HW 5 1a. (1 b. (1 c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e. 0 f. 1 g. 1 h. undef. 2a. [d ( a, d + a] b. 6 c. d d. 1/4 3a. 110 b. 10 c. 11ish h. (2) 4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6a. b. 12 s c. 14 ft HW 6 1b.  EMBED Equation.DSMT4  b. Domain: all real x; range:  EMBED Equation.DSMT4  2a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  h.  EMBED Equation.DSMT4  3a.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  4a. 2.498, 3.785 b. 0.622, 5.662 c. 1.521, 4.662 HW 7 1.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  HW 8 1a. Max value of sin ( is 1 so warmest day of year will be 12 + 18(1) = 30(. b. Min value of sin ( is -1 so warmest day of year will be 12 + 18(-1) = -6(. c. Presumably, one year is one full period. So one year =  EMBED Equation.DSMT4  days. d. One way: Warmest day will be when sin 0.02803t = 1 ( t = 56. Easier way: Warmest day occurs one fourth of the way through the year: 224/4 = 56. e. 12 + 18sin 0.02803t = 0 ( t = 138 and t = 198 f. On New Years, the temperature is near average and going up. That would describe spring. (2) g. A Dorkosian year is about 224.1641 days long. If they celebrate New Years every 224 days, then they are gaining 0.1641 day per year. At that rate, they will gain a day every 6.094 years. So they should probably have a leap year every 6 years. Of course, since thats a little too often, they will have to skip leap years periodically (about every 390 years). 2a. Vertical shift =  EMBED Equation.DSMT4 . Amplitude = 12 7 = 5. Period = 16 ( ( = EMBED Equation.DSMT4 . We want t = 0 to correspond to the hottest part of the day, so use cosine (starts at its max). So  EMBED Equation.DSMT4 . b.  EMBED Equation.DSMT4  ( t = 2.36 and t = 13.64 c. We need to shift our graph two hours to the right. So  EMBED Equation.DSMT4 . 3a. Vertical shift =  EMBED Equation.DSMT4 . Amplitude = 16 12.25 = 3.75. Period ( 365 ( ( = EMBED Equation.DSMT4 . If t = 0 is the vernal equinox, then we want to use sine. So  EMBED Equation.DSMT4 . b. Need to move the graph 80 days to the right so  EMBED Equation.DSMT4  c. On October 31, t = 365 31 30 = 304. H(304) = 9.79 hours. d. H is a max when t = 171 = 31 + 28 + 31 + 30 + 31 + 20 is about June 20. e. Slightly smaller amplitude. Review 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  3. ( ( A 4a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  5a. a b. ho c. ho 0.5a d. 1.5 e. 40 f. ho + a g. Otherwise, the weight would hit the floor h. right 0.25 6a.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  (  EMBED Equation.DSMT4  7. y = 3 + 5sin EMBED Equation.DSMT4 or y = 3 + 5cos 8. a = 3, d = 24, period = 12 ( b =  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  9a. Vertical shift = amplitude =  EMBED Equation.DSMT4 . Max height = d + a =  EMBED Equation.DSMT4  b. Period =  EMBED Equation.DSMT4  seconds. c. Circumference = 2(r =  EMBED Equation.DSMT4  meters. So bike travels  EMBED Equation.DSMT4  m/s ( 3.333 m/s = 12 km/hr 48 y 2( ( 2 4 3 -3 8 y x 2 h -8 Drawings not to scale! Big Fritz Little Willy 6 60 120 HYPERLINK "http://images.google.com/imgres?imgurl=http://www.raindrop.org/rugrat/fun/ctreehse.gif&imgrefurl=http://www.raindrop.org/rugrat/color8.shtml&h=891&w=700&sz=21&tbnid=IzMS39mZDy4J:&tbnh=144&tbnw=113&start=15&prev=/images%3Fq%3Dtree%2Bhouse%26hl%3Den%26lr%3D%26rls%3DGGLD,GGLD:2004-40,GGLD:en%26sa%3DN" INCLUDEPICTURE "http://images.google.com/images?q=tbn:IzMS39mZDy4J:www.raindrop.org/rugrat/fun/ctreehse.gif" \* MERGEFORMATINET  Month 16 (11, 4) (3, 24) . . 8 y x 2 ( ( y x 2 ( 2 3( 10 5 2 ( y x 6 x Sunrise Time (AM) 9 8 7 6 5 4 3 J F M A M J J A S O N D  420 . 300 12 240 y Time (seconds)  4 12 24 36 48 60 -4 9 y 1 -1 x 3( y 3 x 80 60 40 20 2 6 ( y x C h b a ( Looking down. s ( Looking down. x t 24 12 21 24 27  ( 2( ( y x y s ( x ( 10 R A ( 200 h 150 150 ( x h s ( 4000 R C L x 6 4 2 -2 -4 -6 180 12 16 12 8 -8 9 y x 3 6 24 16 4 8 Height (feet) 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 20 -20 24 y x 6 3 -3 -6  EMBED Equation.DSMT4  y x 30 360 ( x d a h    EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  2 1 25PQil|}   ) , B C [ ^ s t u v : ; 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