# COLLEGE MATHEMATICS PDF Author: ESSIE REAVER Language: English, Portuguese, French Country: Maldives Genre: Politics & Laws Pages: 136 Published (Last): 01.01.2016 ISBN: 443-7-63319-180-3 ePub File Size: 26.68 MB PDF File Size: 9.65 MB Distribution: Free* [*Register to download] Downloads: 25868 Uploaded by: MARITA mathematics student who has yet to realize that formulas are about structure and through the second year of college regardless of major. In addition, there are. tion of America (MAA) its report on "A General Curriculum in Mathematics for evaluation of this General College Curriculum in Mathematics (CMC) which. In the Third Edition of College Mathematics, I have maintained the point-of-view of the ﬁrst two editions. Students who are engaged in learning mathematics in.

Differential Analysis. Differential Calculus. Differential Equations. Differential Geometry. Differential Topology. Discrete Mathematics. Elliptic Curves. Fourier Analysis. Functional Analysis. Fractional Calculus. Geometric Algebra. Geometric Topology. Groups Theory. Graph Theory. Harmonic Analysis. Higher Algebra. History of Mathematics. Homological Algebra. Integral Calculus. Lie Algebra. Linear Algebra. Mathematical Analysis.

Mathematical Series. Modern Geometry. Multivairable Calculus. Number Theory. Numerical Analysis. Are there any which are not diagonals? Plane trigonometry, considered in the next several chapters, is restricted to triangles lying in planes. Spherical trigonometry deals with certain triangles which lie on spheres. The early applications of the trigonometric functions were to surveying, navigation, and engineering.

These functions also play an important role in the study of all sorts of vibratory phenomena—sound, light, electricity, etc. As a consequence, a considerable portion of the subject matter is concerned properly with a study of the properties of and relations among the trigonometric functions.

The point O is called the vertex and the half lines are called the sides of the angle. Then O is again the vertex, OX is called the initial side, and OP is called the terminal side of the angle. An angle, so generated, is called positive if the direction of rotation indicated by a curved arrow is counterclockwise and negative if the direction of rotation is clockwise. The angle is positive in Figs.

Throughout the remainder of this book, degree measure and radian measure will be used. The reader should make certain that he or she knows how to use a calculator in both of these modes of angle measure. How far does the tip of the hand move during 20 min? Express the measure of the central angle y in radians and in degrees. What radius should be used if the track is to change direction by 25— in a distance of ft? Find their difference in latitude. The linear velocity is The end of a in.

Through what angle does the pendulum swing? Through what angle has it turned in 1 min? The central angle of one measures 20— with radius ft and the central angle of the other measures 25— with radius ft. Find the total length of the two arcs. How fast rpm does the wheel turn on the axle when the automobile maintains a speed of 45 mph? With respect to a rectangular coordinate system, an angle is said to be in standard position when its vertex is at the origin and its initial side coincides with the positive x axis.

See Figs. There are an unlimited number of angles coterminal with a given angle. The angles 0— ; 90— ; — ; — , and all angles conterminal with them are called quadrantal angles. The values of the functions of a given angle y are, however, independent of the choice of the point P on its terminal side.

Since r is always positive, the signs of the functions in the various quadrants depend upon the signs of x and y. To determine these signs one may visualize the angle in standard position or use some device as shown in Fig. When, however, the value of one function of an angle is given, the angle is not uniquely determined. In general, two possible positions of the terminal side are found—for example, the terminal sides of 30— and — in Fig. The exceptions to this rule occur when the angle is quadrantal.

For a quadrantal angle, the terminal side coincides with one of the axes. For example, the terminal side of the angle 0— coincides with the positive x axis and the ordinate of P is 0. The trigonometric functions of the quadrantal angles are given in Table Table Recall that r is always positive. Thus, y is a third quadrant angle. Thus, y is a second quadrant angle. Then y may be positive or negative and y is a second or third quadrant angle.

Thus, y may be a second or fourth quadrant angle. To draw Fig. Since cos y is positive, y is in quadrant I or IV. Thus, y is in quadrant II. In dealing with any right triangle, it will be convenient see Fig. With respect to angle A, a will be called the opposite side and b will be called the adjacent side; with respect to angle B, a will be called the adjacent side and b the opposite side.

Side c will always be called the hypotenuse. Thus, any function of an acute angle is equal to the corresponding cofunction of the complementary angle. Problems The results in Table For this purpose, Table Note that 1: In any right triangle ABC: Side a 1.

Find the value of the trigonometric functions of 45—. In any equilateral triangle ABD see Fig. The bisector of any angle, as B, is the perpendicular bisector of the opposite side. Let the sides of the equilateral triangle be of length 2 units.

Find the measure of the angle of elevation of the sun. How far from the ground is the top of the ladder and how long is the ladder if it makes an angle of 70— with the ground? The top of ladder is 32 m above the ground. The ladder is 35 m long. How far is the boat from the lighthouse? In the right triangle ACD of Fig. Let A and B be the centers of two consecutive holes on the circle of radius r and center O. Consider a right triangle having as acute angle the given angle.

How high above his starting point is he? If the broken part makes an angle of 50— with the ground and if the top of the tree is now 20 ft from its base, how tall was the tree? Find the shortest distance from one road to a gas station on the other road m from the junction. From the roof of the shorter building, 40 ft in height, the angle of elevation to the edge of the roof of the taller building is 40—.

How high is the taller building? If the ladder is 50 ft long, how wide is the street? Express each of the following in terms of functions of a positive acute angle in two ways: Then any function of f is numerically equal to the same function of y. The algebraic sign in each case is that of the function in the quadrant in which the terminal side of f lies. The related angle is 45— ; — is in quadrant III. There will be two angles see Chapters 24 and 25 , one in the third quadrant and one in the fourth quadrant.

The required solutions have measures 36— ; — ; — ; 60— ; —: Let y be any given angle in standard position. See the Figs. Draw MP perpendicular to OX; draw also the tangents to the circle at A and B meeting the terminal side of y or its extension through O in the points Q and R, respectively.

OQ and OR are to be considered positive when measured on the terminal side of the angle and negative when measured on the terminal side extended. Using Figs. In Table Notice, too, the relationship between the graphs for tan x and cot x. The smallest range of values of x which corresponds to a complete cycle of values of the function is called the period of the function.

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It is evident from the graphs of the trigonometric functions that the sine, cosine, secant, and cosecant are of period 2p while the tangent and cotangent are of period p. More complicated forms of wave motions are obtained by combining two or more sine curves. The method of adding corresponding ordinates is illustrated in the following example. Note the position of the y axis. For proofs of the quotient and Pythagorean relations, see Problems The reciprocal relations were treated in Chapter See also Problems It is frequently desirable to transform or reduce a given expression involving trigonometric functions to a simpler form.

Each form is equally useful. In general, one begins with the more complicated side. Success in verifying identities requires a Complete familiarity with the fundamental relations b Complete familiarity with the processes of factoring, adding fractions, etc.

Quad I: Quad II: Quad IV: Then see Problem Such a triangle contains either three acute angles or two acute angles and one obtuse angle.

## Mathematics

The convention of denoting the measures of the angles by A, B, C and the lengths of the corresponding opposite sides by a, b, c will be used here. The four cases of oblique triangles are Case I. Given one side and two angles Case II. Given two sides and the angle opposite one of them Case III. Given two sides and the included angle Case IV. CASE 1. Given one side and two angles Suppose a, B, and C are given. Given two sides and the angle opposite one of them Suppose b, c, and B are given. If sin C One solution if the side opposite the given angle is equal to or greater than the other given side. No solution, one solution right triangle , or two solutions if the side opposite the given angle is less than the other given side. When the given angle is obtuse, there will be c No solution when the side opposite the given angle is less than or equal to the other given side.

This, the so-called ambiguous case, is solved by the law of sines and may be checked by the projection formulas. In any triangle ABC, the square of any side is equal to the sum of the squares of the other two sides diminished by twice the product of these sides and the cosine of their included angle; i. Suppose a, b, and C are given. With a, b, and c given, solve the law of cosines for each of the angles. Let ABC be any oblique triangle.

In the right triangle BCD of Fig. Thus, in Fig. Find the length of AB. See Fig From the top of the tower the angle of depression of a point on the opposite shore is 28— and from the base of the tower the angle of depression of the same point is 18— Find the width of the river and the height of the cliff.

The river is ft wide and the cliff is ft high. Let b, c, and B be the given parts. With A as center and radius equal to b the side opposite the given angle describe an arc.

Figures a — e illustrate the special cases which may occur when the given angle B is acute, while Figs. When b [CHAP. One triangle isosceles is determined. The given angle is obtuse. When b c; only one triangle is formed as in Fig For B: For a: For C: For b: Derive the law of cosines.

The remaining equations may be obtained by cyclic changes of the letters. In the triangle ABC see Fig. Find AC and BC. One is mi N 42— 40 0 E and the other is mi N 45— 10 0 W of a shore station. Can the two ships communicate directly? No; they are mi apart. A ship leaves the dock at 9 A. At what time will it be 8 mi from the lighthouse?

Two forces of lb and lb acting on an object have a resultant of magnitude lb. Find the angle between the directions in which the given forces act. At a point m down the hill the angle between the surface of the hill and the line of sight to the top of the tower is 12— 30 0. Find the inclination of the hill to a horizontal plane.

We call the function g the inverse of f and we call f the inverse of g. This relationship is written as follows: Thus, functions that are not one-to-one do not have inverse functions. One particularly important class of inverse functions is the class of inverse trigonometric functions. But when x is given, the equation may have no solution or many solutions.

For example: Similarly, the graphs of the remaining inverse trigonometric relations are those of the corresponding trigonometric functions except that the roles of x and y are interchanged. It is at times necessary to consider the inverse trigonometric relations as single-valued i. To do this, we agree to select one out of the many angles corresponding to the given value of x.

This selected value is called the principal value of arcsin x. When only the principal value is called for, we shall write Arcsin x, Arccos x, etc. The portions of the graphs on which the principal values of each of the inverse trigonometric relations lie are shown in Figs. Note that Arcsin x, Arccos x, etc. They are called the inverse trigonometric functions.

Thus, the portions of the graphs shown in a heavier line are the graphs of these functions. Let y be an inverse trigonometric relation of x.

Since the value of a trigonometric function of y is known, there are determined in general two positions for the terminal side of the angle y see Fig.

Let y1 and y2 respectively be angles determined by the two positions of the terminal side. One of the values y1 or y2 may always be taken as the principal value of the inverse trigonometric relation with the domains properly restricted.

From Figs. If a given equation has one solution, it has in general an unlimited number of solutions. In this chapter we shall list only the particular solutions for which 0 x The equation may be factorable.

C Both members of the equation are squared. Note that Solved Problems Solve each of the trigonometric equations First Solution.

The solutions are 45— , 71— , — , — As in Problem Since we require x such that 0 x All of these values are solutions. To avoid the substitution for sin 3x, we use one of the procedures below. Each of the values obtained is a solution. Dividing 1 0 by 2 0 , Thus, x must be positive. Thus, x must be negative. If x is negative, a and b terminate in quadrant IV; thus, x must be positive and b acute. Arcsin 0: No solution in given interval Notice the cyclic nature of the powers of i.

Complex numbers may be thought of as including all real numbers and all pure imaginary numbers. To add complex numbers, add the real parts and add the pure imaginary parts. To subtract two complex numbers, subtract the real parts and subtract the pure imaginary parts. To divide two complex numbers, multiply both numerator and denominator of the fraction by the conjugate of the denominator.

For this reason, the x axis is called the axis of reals. The y axis is called the axis of imaginaries. The plane on which the complex numbers are represented is called the complex plane. In addition to representing a complex number by a point P in the complex plane, the number may be represented by the directed line segment or vector OP. The vector OP is sometimes! The modulus of the product of two complex numbers is the product of their moduli, and the amplitude of the product is the sum of their amplitudes.

The modulus of the quotient of two complex numbers is the modulus of the dividend divided by the modulus of the divisor, and the amplitude of the quotient is the amplitude of the dividend minus the amplitude of the divisor. For a proof of these theorems, see Problem We state, without proof, the theorem: The procedure for determining these roots is given in Example 9.

See also Problem For a and b , draw as in Figs. The theorem may be established for n a positive integer by mathematical induction. Chapter 35 The Conic Sections I. The axis intersects the parabola in the point V, the midpoint of FD, called the vertex.

The line segment joining any two distinct points of the parabola is called a chord. The distance between the directrix and vertex and the distance between the vertex and focus are the same as given in the section above. The line FF 0 joining the foci intersects the ellipse in the points V and V0 , called the vertices.

The segment! V 0 V intercepted on the line FF 0 by the ellipse is called its major axis; the segment B 0 B intercepted on the! A line segment whose extremities are any two points on the ellipse is called a chord. A chord which passes through a focus is called a focal chord. The lengths of the major and minor axes, the distance between the foci, the distance from the center to a directrix, and the eccentricity are as given in the section above.

Since a2 is under the term in y, the major axis is parallel to the y axis. Since the directrices are perpendicular to the major 0 17 25 33 axis, their equations are d: The line FF 0 joining the foci intersects the hyperbola in the points V and V 0 , called the vertices. The segment V 0 V intercepted on the line FF 0 by the hyperbola is called its transverse axis. The line l! The ratio e is called the eccentricity of the hyperbola. The lengths of the transverse and conjugate axes, the distance between the foci, the distance from the center to a directrix, the slope of the asymptotes, and the eccentricity are as given in the sections above.

The transverse axis is parallel to the x axis the positive term contains x. Hence, they may be obtained most readily by the simple trick of changing the right member of the equation of the hyperbola from 1 to 0.

Their foci lie on a circle whose center is the common center of the hyperbolas. The cable of a suspension bridge has supporting towers which are 50 ft high and ft apart and is in the shape of a parabola. Take the origin of coordinates at the lowest point of the cable and the positive y axis directed upward along the axis of symmetry of the parabola. Sketch the curve. The center is at the origin and the major axis is along the x axis a2 under x2.

The center is at the origin and the major axis is along the y axis a2 under y2.

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Sketch each locus. The center is at the origin and the transverse axis is along the x axis a2 under x2. The center is at the origin and the transverse axis is along the y axis. The curves are shown in Fig. Sketch each curve.

The difference in equations is due to their positions with respect to the coordinate axes. In order to make a detailed study of the loci represented by The operations by which 1 is eventually replaced by 2 are two transformations. The general effect of these transformations may be interpreted as follows: Recall that the transformation which moves the coordinate axes to a new position while keeping them always parallel to their original position is called a!

When the values of x and y from The locus, an ellipse, together with the original and new system of coordinates, is shown in Fig. Under a rotation of the coordinate axes with equations of transformation Under this transformation, the general equation When Under a suitable translation the semireduced form Note that the new origin was chosen on the line l.

We have, using Equations Using the transformation Since the coordinates are unchanged, the origin is called an invariant point of the transformation. Sketch the locus, showing each set of coordinate axes. In order to distinguish between the three coordinate systems, we shall use the term unprimed for the original system, primed for the system after the rotation, and double-primed for the system after the translation. We begin with the original unprimed axes in the usual position.

Applying the transformation Thus, prove Consider the three mutually perpendicular planes of Fig. These three planes the xy plane, the xz plane, the yz plane are called the coordinate planes; their three lines of intersection are called the coordinate axes the x axis, the y axis, the z axis ; and their common point O is called the origin.

Positive direction is indicated on each axis by an arrow-tip. The coordinate system of Fig. When the x and y axes are interchanged, the system becomes right-handed. The coordinate planes divide the space into eight regions, called octants. Then distances on parallels to the x and z axes will be drawn to full scale while distances 7 of full scale. Se Fig. In the plane a directed line l [positive direction upward in Figs.

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However, in our study of the line in the plane we have favored the angle a over the angle b, calling it the angle of inclination of the line and its tangent the slope of the line. Indeed, it will be the direction cosines which will be generalized in our study of the straight line in space. If, as in Fig. The sum of the squares of the direction cosines of any line is equal to 1; i. Instead of the direction cosines of a line, it is frequently more convenient to use any triple of numbers, preferably small integers when possible, which are proportional to the direction cosines. Any such triple is called a set of direction numbers of the line. Thus, two undirected lines are parallel if and only if their direction cosines are the same or differ only in sign.

In terms of direction numbers, two lines are parallel if and only if corresponding direction numbers are proportional.

If l1: This procedure will be called the direction number device. Note, however, that it is a mechanical procedure for obtaining one solution of two homogeneous equations in three unknowns and thus has other applications. Find also the midpoint of the segment. The three lines joining the midpoints of the opposite edges of a tetrahedron pass through a point P which bisects each of them.

Let the tetrahedron, shown in Fig. Find the direction cosines of l when directed upward. Since the two sets are proportional, the lines are parallel; since the lines have a point in common they are coincident and the points are collinear.

Consider the triangle OP1 P2 , in Fig. The line segment OP1 is of length 1 why? Find the acute angle between them. The required angle is 40 Since the two sets are proportional, the two lines are parallel.

Since [see Equation Let P have coordinates 0,b; c and express the condition see Problem P 0,3,2 Let P divide AB in the ratio 1: Prove that each of the four line segments is divided in the ratio 1: G, the point P of Problem Solve the linear equation for one of the two unknowns your choice and substitute in the quadratic equation. Since this results in a quadratic equation in one unknown, the system can always be solved. Substitute in the quadratic equation: The locus of the linear equation is the straight line and the locus of the quadratic equation is the ellipse in Fig.

In general, solving a system of two quadratic equations in two unknowns involves solving an equation of the fourth degree in one of the unknowns. Since the solution of the general equation of the fourth degree in one unknown is beyond the scope of this book, only those systems which require the solution of a quadratic equation in one unknown will be treated here. Eliminate one of the unknowns by the method of addition for simultaneous equations in Chapter 5.

By convention, we read the two upper signs and the two lower signs in the latter form. A homogeneous expression equated to zero is called a homogeneous equation. A homogeneous quadratic equation in two unknowns can always be solved for one of the unknowns in terms of the other. Solve the systems see Example 1: Solve, as in Example 3, the system consisting of this homogeneous equation and either of the given equations. Frequently, a careful study of a given system will reveal some special device for solving it. Substitute in 1: The straight line is tangent to the parabola see Fig. Multiply 2 by 5: Alternate Solutions. Here we give an alternate solution. Substitute in 2: Substitute 2: Find the radius of each circle. It is found that by adding three more to the group, the cost per person would be reduced by 50 cents.

For how many was the party originally planned? It will be understood throughout this chapter that b is positive and different from 1. Since 0: Thus in, f and g are inverse functions.

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Such logarithms are called common logarithms. Such equations are solved by means of logarithms. Take logarithms of both sides: Evaluate log 82, rounded to six decimal places. On screen: Then x Solved Problems Express the logarithms of the given expressions in terms of the logarithms of the individual letters involved.

Since log 2: Using a calculator, press 1. The result on the screen is Taking logarithms, x log 1: Taking logarithms, x log We shall assume that the points corresponding to intermediate values of x lie on a smooth curve joining the points given in the table. See Table The general properties are a The curve passes through the point 1, 0. Find and use patterns of reasoning or structure, make and test conjectures, try multiple representations e.

Make abstractions and generalizations and verify that solutions are correct, approximate, or reasonable. Use mathematical models to guide their understanding of the world around us. Be at the mathematical analysis pre-calculus level or above. Mathematical analysis that includes the mathematical development of the trigonometric, logarithmic, and exponential functions can be approved for UC honors credit. Honors-level courses in mathematics can be designed as differentiation within heterogeneous classrooms, as long as the depth of instruction and assessment parallel the rigor of AP Advanced Placement and IB International Baccalaureate course expectations.

Calculus, with four years of college-preparatory mathematics as prerequisite, qualifies as an honors-level course if it is substantially equivalent to an AP Calculus course. Statistics, with a three-year mathematics prerequisite, may be approved for honors credit if it is substantially equivalent to an AP Statistics course. A proclivity to put time and thought into using mathematics to grasp and solve unfamiliar problems.Solve, as in Example 3, the system consisting of this homogeneous equation and either of the given equations.

Marti Garlett, Dean of the Teachers College at Western Governors University, for her professional support as I struggled to meet deadlines while beginning a new position at the University. Given the need to reach more women, minorities, and other underserved student populations, it is not surprising that professional mathematics groups have made recommendations to change mathematics education. The American Experience Table of Mortality begins with persons all of age 10 years and indicates the number of the group who die each year thereafter.

Transpose one of the radicals: Note that the unknown appears in the same polynomials in both the expressions free of radicals and under the radical. Each form is equally useful.

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