This implies 9x =9 Giving x = 1 as required. (d) The number of fish in the pond will not decrease below p. Write down the value of p. . Mathematics applications and interpretation (a) 2890 () (b) 83501 7. We set up a one-to-one correspondence between a set of numbers and a set of cows. y = 0. On this view, mathematics is just like the glass bead game. 15 In questions 32-33, find the area of the shaded sector. We can sketch the graph to explore its features. (b) Use the model to calculate the boiling point of water at a height of 1.37 km above sea level. statistic, using the values above as actual observed data. Also, since the squares perimeter is 4x, and the total length of wire is 4m,wehavedx + 7 (a) Estimate, to 1 significant figure, the coordinates where the local maximum value of the function occurs. Piecewise linear models For more information about piecewise models, refer to Chapter 9. (@) (-1+15 (b) (-1 1i* ( (=v3+it (@ (=3 5. Since the vertex must be halfway along the length of the bridge, its x-coordinate must be 251.5 (251.5, 118) (503,0) Then, we can use the general quadratic equation y = ax? Erfahren Sie, wie wir und unser Anzeigenpartner Google Daten sammeln und verwenden. (a) (%) il ( 347 5a+2 . The methods mathematics teaching you have experienced reflects a Platonist or constructivist view of mathematics? Symmetry in the underlying algebra led mathematical physicists to propose the existence of new fundamental particles, which were subsequently discovered. In Euler form, it is V = 10e". To obtain a more realistic model we need to take account of the fact that as the population increases, quite a few factors will begin to affect the growth rate. (@) Using the TVM solver, we make the following entries and solve for the future value, FV, PMT: END The number of payments, N, is 6, because interest is compounded once per year for six years. 28. In that case, it can also be considered a direct variation model, which we will study later in this chapter. Webinfinite geometric series calculator symbolab lingering covid cough and sore throat lingering covid cough and sore throat should not use the model to predict the length wide. (a) d =433 () () 304 1L (a) d= 27 (i) 0.330% () 307g @ (i) An underestimate. * has an amplitude, which is the measure of its potential difference or current * has a specific starting point, called the phase, typically measured in degrees relative to its basic waveform which is generally set at the origin. WebNew math course starting in September 2019 for IB Class of 2021 Course description from IB Approximate current equivalent Recommended prior math background Mathematics: 13" 2 27. Ris the radius of the cylinder and is on the parabola = 12 = 2ph and volume of cylinder = wr2h = - 2ph - h = 2mph? It is assumed that the prey do not die from natural causes or that the predators do not come into existence except through the provision of food. 1 a(positive arigle) v Figure 5.9 Coterminal angles Figure 5.9 shows a positive angle a and a negative angle 3 that are coterminal in quadrant III. Suppose $2500 is invested in an account paying compound interest at a rate of 3% per year. (c) The temperature of the sun is 9941F. What shape will this distribution take? 0 fix)=2/x % (g) (h) fit) = 3t> 2sint h(0) = 3sinf + 4cosb (i) fix) = Vx2x5) N (k)i h(f) = e*=! Method to estimate the integral ppmdl? (a) 1+2i (b) x=3+i () x=5%2i (b) 2 +3i () x=4+3i (f) x=4+4i (c) 35 2i 7. (d) The acceleration due to gravity of the model rocket is @ = 9.3 ms~2, This is less that the gravitational constant of g = 9.81 ms 2. Substitute y = vx and dy = vdx + xdv into the given equation (%2 y? 409 simple graphs 454, 457-8, 460 definition 451,452 simple interest 64-5 simple random sampling 367-8 sine function 13945, 563-5 sine graph 145-7,330-2, 563 sine model 332-5 sine rule 110-13 singular matrix 217 sinusoidal curves 192 sinusoidal functions 193 sites (Voronoi diagrams) 153-62 skewed distributions 377, 386, 389, 397 slope fields 860-3,865-8, 871 social facts 904 solid of revolution 706-14 solution curves 836,837,839, 8604 spanning trees 484-8 minimum 489-95, 506-7 Spearmansr, 795-8, 800-2 speed 313-14, 526, 530, 589-90, 715 falling objects 8557, 861-2 see also velocity spheres 115, 116,706-7 spread 376, 390402, 647-9 square matrix 203, 204-5,216 diagonalisation 243-6 eigenvectors/eigenvalues 239-43, 244, 245-6 stability of coupled systems 875-8 standard deviation 395-402, 635, 7456, 760 normal distribution 644-6, 649 and outliers 397 pooled data 762-3 random variables 635,645, 6478, 701-2 transformed data 398-9, 647-8 unbiased estimators 736-8 uniform distribution 647-9 standard error of the mean 745-6,759, 760 standard error of the proportion 744-5,759 standard position 130 standardised normal distribution 645, 745 standardised scores 645 state (experiment outcome) 247-50 statistical inference 734 statistical tests and analyses 752-80 chi squared test of goodness of fit 765-9 chi squared test of independence 769-74 comparing outcomes from different groups 769-74 hypothesis testing see hypothesis testing multinomial outcomes 765-9, 769-74 single sample 754-5 ttests 752-6, 7614 two samples 7614 statistically significant result 754 statistics see descriptive statistics; inferential statistics steady state distributions 653, 654-5 step size 863 straight line motion 283-8, 561, 590, 714-24 stratified random sampling 368, 370 strength (scatter diagrams) 784,785 stretch 46-8,253-5 strongly connected graphs 478-81 Student t-test see t-tests subgraphs 453,455, 484-6 submatrix 204 substitution, integration by 670-3, 685-6 subtraction complex numbers 177 matrices 205-6 vectors 276 subtrees 483 sum to infinity 84-7 supply and demand 590-1, 698700 surface areas 115-16, 119 symbols 896,897, 907-8 symmetric distributions 389, 397 symmetric matrix 205 symmetry 894,912-13,914 systematic sampling 3689 systems of equations 215-19, 229-38, 597 T tdistribution 752-3 Ltests 752-6 single sample 754-5 two samples 7614 tangent function 139-45, 564-5 tangent graph 147-8 tangent identity 1434 tangents 532-5, 550, 595-601, 863 temperature models 324-5,353, 843,851 terminal side 130, 141 terminal velocity 719 three-dimensional measurements 115-19, 2867 tide model 148-9 Time-Value-Money (TVM) solver 72,73,75, 84,90-5 trails 4634, 467-9 transformation matrices 252-67 1009 Index composition of 259-62 dilation 253-5, 261,264 fractals 262-5 reflection 252-3,254-5, 258 rotation 256-8, 259, 260, 264 scaling 254,261 shearing 256 translation 256, 259, 260, 264 transformations 45-51 reflections 46,47, 48-9 sequences of 48-9,259-62 stretch/shrink 46-8 translations 45-6,47-9 see also transformation matrices transformed data 3989, 647-9 transition matrices 247-50 probability distributions 653-8 strongly connected graphs 479-81 translations 45-6,47-9 phase shift 331,334 transformation matrix 256, 259, 260, 264 transpose (ofa matrix) 222,245 trapezoidal rule 686-8 travelling salesman problem 501-8 tree diagrams 247, 421-3,434-5, 653 conditional probabilities 430,432-3 trees (graphs) 482-8 minimum spanning trees 48995, 506-7 spanning trees 484-8 trials see experiments (trials) triangle inequality 503 triangles area 109-10,112-13,220 cosinerule 113-14 finding unknowns 1056, 111-14 right-angled 105-6,109-10, 113,117 similar triangles 118 sine rule 110-13 transformation 254-5 triangular matrix 205,243 trigonometric equations 148-52 trigonometric functions 138-52 definition 140 derivatives 563-5 graphical analysis 145-52 inverse 105, 194,274,292 of real numbers 141-3 reciprocal 105 inunit circle 139-41 trigonometric identities 1434 trigonometric models 148-51,330-8, 352 trigonometric ratios 105-6, 138-9 trigonometry 105-14,120-2 area of a triangle 109-10,112-13 modelling with 106-9 right-angled triangles 105-6, 117 trigonometric ratios 105-6 1010 trivial solution 218 two-point form of a line 221-2 two-tailed test 752-3,756 two-way table 769-70 2-sample t-test 762-3 typeland Il errors 758-61 u unbiased estimators 736-8 undirected graphs 4514, 466 ungrouped data 373, 390 uniform distribution 647-9, 738 uniformly accelerated motion 719-21 union (of events) 418 union (of graphs) 454 unitcircle 132-4,139-41,145-6 unit vectors 2767 unknowns and cubes problem 899 unpaired data 761, 762-3 unpaired t-tests 761,762-3 unpooled data 762 upper bounds 506-8, 581-2, 636 v validity (data) 734-6 variability, measures of 390-402 variables 16-18, 299-301, 3634 associated 783-5,787-8 graphical representation 372-84 optimisation problems 604-11 relationship between see bivariate analysis substituting 619-22 see also random variables variables-separable differential equations 83844, 848 variance 3967, 634-6,806 binomial distribution 63941 combined distribution 64952 F-test 762,763 normal distribution 644, 649 Poisson distribution 641,650 population variance 736-7, 739, 746, 752-3,760 sample means distribution 738-9 transformed data 6479 unbiased estimators 736-8 uniform distribution 647-9 variation direct variation model 344-5,347-9, 352 explanatory/response variables 805-10 inverse variation model 344, 346-9, 352 predicting 805-10 vectors 204, 274-96 addition 276-9 angle between 289, 292-3 equation of aline 280-3 finding minimum distances 285-7 modelfing with 479-81 moving objects 283-8 properties 275-6 representation 274-9 scalar multiplication 276 scalar product 288-90,291,292-3 unit vectors 2767 vector product 290-1 velocity 560-1, 588-90, 715, 716-24 averagefinstantaneous 5304, 583, 589 velocity function 716-22 velocity-time graph 717-18 Venn diagrams 417-20,429, 431 vertex (of an angle) 130 vertex (of a parabola) 305 vertical line test 17 vertices (graph theory) 451-7 adjacency matrices 457-9,460-1, 464-6, 478-9 Eulers formula 475-7 incidence matrices 459-60 shortest path 495-510 trees 4834 walks/paths/trails 463-73 vertices (Voronoi diagrams) 153 volume 115-16,119, 705-14 volume models 309, 311-12 direct variation 345 optimisation problems 6024 related rates 619-20 Voronoi diagrams 15368, 170-2 adding new sites 154-9 creating new 15962 largest empty circle (LEC) 1634 nearest-neighbour interpolation 162-3 w walks 463-6,467-8 and adjacency matrices 464-6 washers 708-10 wave height 330 wavelength 331 weighted graphs 479, 488-95,496-510 work done 289-90 wrapping function 141-3, 145 Y Yates' continuity correction 771 z zero matrix 204 zeros 178-9,305 complex numbers 175,176, 178-9 zoom fit function 150, 151,308, 312. Some symbols such as %, sin~! Total cost will include the cost of producing x units plus fixed costs which are costs involved in running the operation, regardless of the number of units produced. 4. (a) 3+ 3i (d) 3-3i (b) =3 +3i (e) 5+ 5/3i (c) 33i (f) 5/3 +5i () 5+5/3i (h) 5v3 5i 2. In one experiement, a patient is given a drug and the patients blood is tested at regular intervals to determine the concentration in millimoles per litre (mmol L!). To the nearest metre, what is the width of the stream? 106=2%-1 x=H 3 1 20 2 2 o 2 4 o 4 8 23 bound=(-199, 1E99} 108=2%-1 Noting the number of the square and the exponent n =10 s So, we need to find a solution to y = 2! derivative changes sign atx = c, then (c, fio) is apoint of inflection. Hence, if the ordered pair (a, b) is a point on the graph of y = f{x), then the reversed ordered pair (b, a) must be on the graph ofy = f~!(x). (a) 1000 = 10 (b) 64 =43 () 100 = 1000 d9'=3 () 2/2=8 ) 100=1 M) 6= @ =1 I\ W (3) =8 el ey D= L @omi=w0 _ @e2=-s 2. ty! Information boxes As you read this textbook, you will encounter numerous boxes of different colours containing a wide range of helpful information. Basic definitions relate to the use of a right-angled triangle, but any triangle will suffice. (a) j=6cm, k=10cm, [ = 120 (b) j=6VZ cm, k=12cm,[ = 135 T Solution (a) P=62+1022-6-10cos120 =36+ 100 120 - (%) =196 =1=y/196 = l4cm (b) 2=1(6y2)>+ 122 2. 30 11 3 Exercise 1.3 () x=logy(5~y) (e) x= llugz% 4. (a) the largest empty circle (LEC) must be centred on a (b) All vertices in the diagram are adjacent to cell K and 4@ 2+i vertex (15, 8) is the vertex farthest from site K, therefore it must be the centre of the largest empty circle. The concepts of a set and membership of a set are primitive. 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State the null and alternative hypotheses for each given claim about the true value of the population mean. For example, the number mapped to the point (1, 0) because 7 is half of the circumference circle. (c) Discuss the long-term trend of the deer population. The variance is unknown. The patient is tested just before being given the drug, then twice in 30 minutes, then every six hours. 0.999999 really does look very different to 1 but if the proof works then we are forced to believe that they are the same. Clearly it sets up a one-to-one correspondence between the set of natural numbers and the set of even numbers (check this yourself). In 2016, the population growth rate in India was 1.2% and the population was 1.324 billion. 36. amatrix product Here are some examples of matrix multiplication. We can also think of the average value of a function in the same manner. These seem to be rather modest beginnings on which to build the complexities of modern mathematics. (b) Find f' and state its range. V-D=p, 2. Find the tenth term. For example, the approximate distance from the Earth to the sun is 149600000 km. We will consider impedance in AC circuits here. (b) What are its coordinates when it has covered three-quarters of the total distance? (27 =1 W R 5t=2 2y 2 @ 2f=2 Express each equation in the form x = (a) y=2* (b) y = 10* () y=e* (= e G) y=2%2 () y=3 W) y=e (Fy=css W y= %eu (@ y=2% 3. Last updated: After 6 years, Amount = 1500 X 1.045 = $1953.39 (b) After n years, an expression for the amount in the account is: Amount = 1500 X 1.045" We need to solve the inequality 1500 X 1.045" > 10 000 This can be solved using a graph or table on a GDC, or algebraically using logarithms. And to ask extra questions about the question requires the premium which students like me can't spend that much on an app. As most functions that we encounter in applications are composite functions, it can be argued that the chain rule is the most important, and most widely used, rule of differentiation. 7. Here are some examples of estimation and approximation. We will consider composite and inverse functions, and investigate how the graphs of functions can be transformed by means of translations, stretches and reflections. =9 Solution (a) Since the diameter of the Riesenrad is 200 feet, the wave height of the model must be 200, so the amplitude a = 100 (b) Since the minimum value of the function is 12 feet, the principal axis must be located at 12 + 100 = 112. n 10 logm( 1072) N45 (b) We need to solve the equatio The intersection is at (316, 145), so the sound intensity is 316 W m 2. (a) Find a linear function to model the cost of model A. 14. (a) 9= @ 300(0.01 105 q)iing (b) g 104 no (9) qity= 1041 e~0003), (i) Afier one year the amount of pollutant wil be approximately 9278 g i 4 000031)) (d") lim g(1041 = e700%) = 10 104 (@ 5 = 300(~10%); g0 = 9278 dt () qit) = 9278(e~0%9%), affer one year there will be 670 g approximately () 2.6 years 4 s gZ &2 g 3 3 v kv, where k = V367 and is a constant dt of proportionality that depends on several factors like 25, eigenvectors H3) temperatutre and friction. Therefore, angle AOB = % = % radians. @) (-1,3,2) L (a) () ) =26 PG 1 ++ 195 16r) Styg5t, (=7,3,-2) i s inconsistent 5. 1. His data are Temp | Solubility C)_| (gper100mD) 0.0 66.7 4.0 71.0 10 76.3 15 80.6 21 85.7 29 929 (d) Use your least-squares regression line to predict the solubility of NaNO; at 25C. %=1 X To find the inverse of a function f; use the following steps: Determine if the function is one-to-one; if not, restrict the domain so that it is. e Solution Concave P Find the values of x where the first derivative is equal to zero or undefined. each element of set X All the various ways of representing a mathematical function illustrate that its defining characteristic is that it is a rule by which each number in the domain determines a unique number in the range. 3.5% -3 K2187 mass of the tub, so the amount of ice cream will likely be less than 307 g 12. Find a formula for the total amount of silver that will be consumed within years from 2017 and estimate when the known world reserves of 533 thousand metric tons will be exhausted. To do this, we can use a GDC to graph the model and look for a maximum value. Table 19.13 826 3. The sequence G,is defined by G, = G, = 1and G, = 2G,, + G5, n>2 A second sequence b, is defined by b, = G G, n=1 (a) Use a spreadsheet to find the first 30 terms of both G, and b, (b) Hence, write down the limit of b, as n tends to infinity, 1}9:27 b, Arithmetic sequences Here are three sequences and the most likely recursive formula for each one. Therefore, we choose the cosine function, and we keep both a and b positive since no reflection across the x or y-axis is needed. The power of symbolic representation is that it allows us to build abstract worlds - virtual realities where the what if conditions are true. With radian measure, calculating the area of a sector is a bit . T=2p (b) 832 PHP (c) 800 THB 4. beaba g tongson (b) A tissue sample from a body discovered on the border between Austria and Itvaly has an average acfnflty 0f 6.92Bq. 2. (a) Write down A(0). _eeeeeeeeeee Solution (a) MCis the rate of change of cost, thus, cost itself is an antiderivative of MC. 1 (g) Eigenvalues: 2, 5, eigenvectors (1), 2 o e +5Ceot y=Ce 3 - - -z o e , z ZEr g A E Ee Ty Ex :::: : LA '( L s ' a Trajectories approach equilibrium and then move away. The truths of the natural sciences are always subject to revision, but mathematical truths are eternal. (31 9.91.812 10. Experiment with different values of to see how the graph changes. Domain and range of a function In Example 2.1 we found the function In general, when writing a function, you should Cld) = 71/4900 + d? The data in the table shows a sample of the data collected. (a) . This chapter revises and extends topics that may have been introduced in earlier years. IB Applications and Interpretations SL and HL Resources. = 2u,,3+ 1 andu; =0 (h) a,=a,;+2anda; =1 Find a recursive definition for each sequence. (c) 1:Only the 8 is significant. IB Mathematics: Applications and Interpretation SL in 70 Pages - May 11 2020 This revision guide will be a valuable resource and reference for students, assisting them to understand and learn the theory of IB Mathematics: Applications and Interpretation Standard Level. We know that neither of the assumptions in (1) is true. (a) ) 2X 5 4 g2 (b) %+%+%+2x 5 47+ sint+ cE+ & (e) 2sinf+ %cusz +e 3 (@) () 3sin62tan6 +c 21+ 11 20 (b) T 8/ (5)675+ 2) + (d) 3x* dx? 3.32 2,34.32 ] 5. 3 Modelling real-life phenomena 1 2x _ = The width and length of the paper get reduced by twice the length of the When we expand the right hand side of the corners we cut out. )dx + 3x2v(vdx + xdv) = 0 = xX1 + 2v)dx + 3x3vdv =0 Divide by x2 simplify and integrate dx 2 Gy (1 + 29 dx +3wdv=0= [T = vdy =3[0 Inx| = %m(l +202) + C = 41n|x| = 31n(1 + 202 + 4C, Now let 4C; = InC and simplify Inx* = In(C(1 + 2v? on the same set of axes. Here is an example to illustrate how we might have to correct our intuition by careful mathematical reasoning. The diagram shows the major highways connecting some European cities: Vienna (V), Munich (M), Frankfurt (F), Stuttgart (S), Zurich (Z), Milan (L), and Paris (P). The second type of soil is sandy and the cost of building the pipeline is $2 million per km. The origin is an asymptotically stable focus, also known as a spiral sink. We will use seconds as the units for all times. Example 4.21 A bird lands on the zip line in Example 4.19, two-thirds of the way along the line from point M(0, 0, 0) to T(30, 60, 20). min ! This also leads to two fundamental trigonometric identities. () =025~ 5sq (b) g(t) = 75 + ce*"*, with initial conditions q(t) =75 + (g0 75)e ) 75 Exercise 20.3 1. (c) Suppose sound b has an intensity x W m 2. Get the best Homework answers from top Homework helpers in the field. + -3+ i (d) 1 18(3x2 + 5) =4 vx + 3) 36 T 5 (2x + 3)8 g -FFIS L @ S, +c 18x (h) %ln[costz ~D+3)+ (@) tan(s +e ) Lsinim +3) + R Leen e W %zu itc () Inlln2z] + (m)%ln) fe 4. Hersh sees a political and pedagogical dimension to his thinking about mathematics. I understand every breakdown. Costs are decreasing for 0 < x < 120 and increasing for 120 < x < 1000, so there must be a local minimum at x = 120. () 6. Wells) value, = A1) = [ o 1000 0d =500(1 e"!) (a) Show that the 110th term is 8926. Saddle point. (Assume no further money is invested or withdrawn from the account.) (a) 9x 8 (b) 27x 26 (c) 81x 80 d) 3% 3"+1 (e) 10 9. 2k 3] = g = 1 )T 5= X0 =AYy =02 13 Since the domain of gis x ] oc, 0], then the range of g~! T M 17 Geometry and trigonometry 1 Solution If the mound is considered to be at M(0, 0, 0), then the starting point up the tree is T(30, 60, 20). (a) 5630 (d) 3.14 (g) 0.0201 Exercise 1.2 5 (g1 1201 3.14 (d) 83 (d) 0.00 0.02% 0.05% 2 (b) 3100 () 0.000207 (h) 0.0200 (d) 0.56% (d) 100% 5 (h) 2 (c) 4760000 (f) 100 4.3 8. In the trigonometry part of this chapter, we will consider angles of rotation with the unit circle to enable us to generalise the three trigonometric ratios of sine, cosine, and tangent to apply to any angle. from the graph of f. Given the reversing effect that a pair of inverse functions have on each other, if fla) = b then f~! (b) How much interest has been earned on the investment in 30 years? However, our data indicates that the height of the rocket at time t = 0 is zero, so there must be error present in the data collection or in the model. (a) 2.22 (b) 28.272 23 272 CF (B (c) 42.2? ) (a) Geometric (b) r= 7. (i) For what intervals is fincreasing or decreasing? 6x dx Three functions of importance to an economist are: the cost function: C(x) = total cost of producing x units of a product during some time period the revenue function: R(x) = total revenue from selling x units of the product during the time period 673 the profit function: P(x) = total profit obtained by selling x units of the product during the time period If all units produced are sold, then P(x) = R(x) Clx) profit = revenue cost The total cost C(x) of producing x units is made up of two parts: overhead or fixed costs, a, and variable cost or manufacturing cost, M(x) C(x) = a + M(x) Ifa firm can sell all the items it produces forp units of money (, , or $, ) each, its total revenue R(x) will be R(x) = px Also, economists call P'(x), R'(x), and C'(x) the marginal profit, marginal revenue, and marginal cost; and they interpret these quantities as the additional profit, revenue, or cost that result from producing and selling an additional unit of the product when production/sales levels are at x units. (b) Find (i) h(29) (i) h(53) (c) State the range of h. 3. The confidence interval here is 0.38 < p < 0.62 If the sample size is increased further to 100, the 90% confidence interval should be narrower still. A company produces artificial leather wallets with maximum capacity of 800 units per day. If we write down the values of b and the length of the period, we get Table 9.12. This agrees with the graph. To pay off the loan, Phil makes annual deposits of $P at the end of every year in a savings account, paying an annual interest rate of 2%. WebHandwritten notes on the recently changed syllabus for IBDP Math, for first exams in 2021 - this is the Applications and Interpretation HL course (A&I HL / AIHL).It is 112 pages long - 19 on Algebra, 16 on Functions, 26 on Trig/Geometry, 28 on Stats/Probability, 23 on Calculus. Your goal should be understanding, not just remembering. We therefore need to revise some of the derivative formulae. Solution (a) u;=3andr = 4,s0u;, = 3(4)!"' The number of cells in the culture at 12:00 on Monday is 47. 2. There are photocopiable resource for general and credit students. _ Solution (a) A graph of fproduced on a GDC reveals that it is not monotonic over its domain ] o0 , co[. In 99% of these cases the test does its job and records a negative result. Regardless of the size of the circle (i.e. (c) Find the dimensions of the open box with the largest volume that can be created using this method. Unstable. One revolution takes 20 minutes. That makes sense because the centre of the wheel is 30 cm above the ground. The wheel rotates at a constant rate in an anticlockwise direction. cosla + ) 11 (a) (,-m(a B sin(x + ) coslat BJ) (b) M(B)M(a) = (cosn cosB sinasinf sina cos cosa sinf cosasing + sinacosB cosa cosp sinasin (c) By comparison we recognise the formulas for cos(a + B)and sin(a + ). Period, we can also be considered a direct variation model, which were subsequently discovered below Write... 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