As in Prob. Find the width of the river. Find the height of the hill if the height of the flagpole is Manual: Let the line of the pole meet the horizontal through A in C. Then Find the equal sides and the altitude.
What is the height of the center of the wheel above the base of the incline when the wheel has rolled 5 ft up the incline? A window in the house is A fire escape slide attaches to the bottom of the window and to the top of the wall opposite the window. How long a slide must be purchased? If the angle of depression from the window to the car is What is the angle of elevation of the sun? What is the angle the diagonal makes with the longer side?
How far must the train go up the track for it to gain 30 m in altitude? The bearing is then read from the north or south line toward the east or west. The angle used in expressing a bearing is usually stated in degrees and minutes. For example, see Fig. For exam- ple, see Fig. A vector quantity may be represented by a directed line segment arrow called a vector.
The direction of the vector is that of the given quantity and the length of the vector is proportional to the mag- nitude of the quantity. Its velocity is represented by the vector AB in Fig. Thus, vector AB is three times as long as vector CD. Both vectors are drawn to the same scale. A vector has no fixed po- sition in a plane and may be moved about in the plane provided that its magnitude and direction are not changed. Place the tail ends of both vectors at any point O in their plane and complete the parallelogram having these vectors as adjacent sides.
The directed diagonal issuing from O is the resultant or vector sum of the two given vectors. Thus, in Fig. Choose one of the vectors and label its tail end O.
Place the tail end of the other vec- tor at the arrow end of the first. The resultant is then the line segment closing the triangle and directed from O. Thus, in Figs. Figure 5. It is often very useful to resolve a vector into two components along a pair of perpendicular lines. Note that F is the vector sum or resultant of Fh and Fv. The heading is measured clockwise from the north and expressed in degrees and minutes.
The airspeed determined from a reading of the airspeed indicator is the speed of the airplane in still air. The course or track of an airplane is the direction in which it moves relative to the ground. The course is measured clockwise from the north.
The groundspeed is the speed of the airplane relative to the ground. The drift angle or wind-correction angle is the difference positive between the heading and the course. The groundspeed vector is the resultant of the airspeed vector and the wind vector. In constructing the figure, put in the airspeed vector at O, then follow through note the directions of the arrows with the wind vector, and close the triangle.
Note further that the groundspeed vector does not follow through from the wind vector. The forces Fa and Fd are the component vectors for the weight W. The minimum force needed to keep an object from sliding down an inclined plane ignoring friction has the same magnitude but is in the opposite direction from Fd.
What is the minimum force ignoring friction needed to keep the barrel from rolling down the incline and what is the force the barrel exerts against the surface of the inclined plane? How far north and how far east does it travel?
Suppose the boat S leaves A. In the right triangle ABC, see Fig. What is the bearing of a of B from A and b of A from B? Find the distance between the wrecked ship and the tower.
One hour later the boat is at A, If the course is continued, how close will the ship approach the light? A body at O is being acted upon by two forces, one of lb due north and the other of lb due east. Find the magnitude and direction of the resultant. Find the resultant speed and direction of the bullet. Find the horizontal and vertical components Fh and Fv of the pull F. Then Fh is the force tending to move the sled along the ground, and Fv is the force tending to lift the sled.
Neglect friction. Resolve the weight W of the block into components F1 and F2, respectively parallel and perpendicular to the ramp. F1 is the force tending to move the block down the ramp and F2 is the force of the block on the ramp.
Construction: Put in the airspeed vector from O, follow through with the wind vector, and close the triangle. Construction: The groundspeed vector is along ON. Lay off the wind vector from O, follow through with the airspeed vector units from the head of the wind vector to a point on ON , and close the triangle. Construction: Lay off the groundspeed vector from O, put in the wind vector at O so that it does not follow through to the groundspeed vector, and close the triangle.
How far south and how far east of the starting point is it? Find the magnitude and direction of the resultant force. Find the magnitude and direction of the actual velocity. Find the distance of P from C and the required course to reach C.
What is the horizontal pull on the top of the pole? Find the pull on the string. What is the length of the shortest inclined plane he can use if his pulling strength is lb? Find a the force of the shell against the runway and b the force required to drag the shell. Find the magnitude of the resultant force and the measure of the angle the resultant force makes with the lb force.
Find the drift angle, the groundspeed, and the course of the plane. Drift angle 8. If one of the forces is 45 N, what is the other force and the angle it makes with the resultant? What angle does the inclined surface make with the horizontal if a force of 10 kg is exerted on the inclined surface? For a proof of these relations, see Prob. The values of the six trigonometric functions of the reference angle for u, R, agree with the function values for u except possibly in sign. When the signs of the functions of R are determined by the quadrant of angle u, as in Sec.
Thus, our tables can be used to find the value of a trigonometric function of any angle. When finding the value of a trigonometric function by using a calculator, a reference angle is unnecessary. The function value is found as indicated in Sec.
However, when an angle having a given function value is to be found and that angle is to be in a specific quadrant, a reference angle is usually needed even when using a calculator. All the angles that have the same function value also have the same reference angle. The quadrants for the angle are determined by the sign of the function value.
The relationships from Sec. The tangent is positive in quadrants I and III. If n is even, including zero, u is in quadrant I or quadrant 1 1 II and sin u is positive while 2 u is in quadrant I or quadrant III and tan 2 u is positive. There will be two angles see Chap. The magnitude of a function is given by the length of the corresponding segment, and the sign is given by the indicated direction.
The directed segments 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 Fig. The graphs of the trigonomic functions are shown in Fig.
Figure 7. If c is a positive number, then adding it to a trigonometric function results in the graph being shifted up c units [see Fig. For a positive number d, a trigonometric function is shifted left d units when d is added to the angle [see Fig. 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. 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. Then, corresponding to each x value, we find the y value by finding y1 value for that x, the y2 values for that x, and adding the two values together.
Note the position of the y axis. For proofs of the quotient and Pythagorean relationships, see Probs. The reciprocal relation- ships were treated in Chap. See also Probs. Each form is equally useful. In Example 8.
The eight basic relationships in Sec. A trigonometric identity is verified by transforming one member your choice into the other.
In general, one begins with the more complicated side. In some cases each side is transformed into the same new form. General Guidelines for Verifying Identities 1. Know the eight basic relationships and recognize alternative forms of each. Know the procedures for adding and subtracting fractions, reducing fractions, and transforming frac- tions into equivalent fractions.
Know factoring and special product techniques. Use only substitution and simplification procedures that allow you to work on exactly one side of an equation. Select the side of the equation that appears more complicated and attempt to transform it into the form of the other side of the equation. See Example 8. If neither side is uncomplicated, transform each side of the equation, independently, into the same form. Avoid substitutions that introduce radicals.
Use substitutions to change all trigonometric functions into expressions involving only sine and cosine and then simplify. Multiply the numerator and denominator of a fraction by the conjugate of either. Simplify a square root of a fraction by using conjugates to transform it into the quotient of perfect squares. We transform the left side of the possible identity into a simpler form and then transform the right side into that same form.
The conjugate of a two-term expression is the expression determined when the sign between the two terms is replaced by its opposite. The only time we use this procedure is when the product of the expression and its conjugate gives us a form of a Pythagorean relationship.
We will use the conjugate of the numerator since this will make the denominator the square of the value we want in the denominator. The procedures used in Examples 8. For P x, y defined as in Prob. When u is a first-quadrant angle sin u and cos u are both positive, while sin u is positive and cos u is negative when u is a second-quadrant angle. Thus, a and b are valid relations.
Thus, by a finite number of repetitions of the argument, we show that the formulas are valid for any two given positive angles. Then see Prob. From Prob. Such a triangle contains either three acute angles or two acute angles and one obtuse angle. The convention of denoting the angles by A, B, and C and the lengths of the corresponding opposite sides by a, b, and c will be used here. It is possible to have no solution for the angle, one solution for the angle, or two solutions—an angle and its supplement.
See Example Thus, there may be one or two triangles determined. This case is discussed geometrically in Prob. The results obtained may be summarized as follows: When the given angle is acute, there will be a One solution if the side opposite the given angle is equal to or greater than the other given side b 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 d One solution if the side opposite the given angle is greater than the other given side EXAMPLE This, the so-called ambiguous case, is solved by the law of sines.
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. The given angle B is acute. One triangle isosceles is determined.
The given angle is obtuse. In either right triangle ACD of Fig. In the right triangle BCD of Fig. Case II Find the length of AB. In the triangle ABC of Fig. Find the width of the river and the height of the cliff. Case III How far and in what direction must the pilot now fly to reach the intended destination A? Denote the turn-back point as B and the final position as C. Case V Find AC and BC. There could be two solutions in this case. Find the length of the longer diagonal.
State whether the law of sines or cosines should be used to solve for the required part, and then find its value. Can the two ships communicate directly? No; they are km apart. Find the distance and direction of the last position from the first. A ship leaves the dock at 9 A. At what time will it be 8 km from the lighthouse? Find the angle between the directions in which the given forces act. Find the inclination of the hill to a horizontal plane. Find the angles of the triangle formed by joining the centers of the circles.
In general, if enough infor- mation about a triangle is known so that it can be solved, then its area can be found. The area of the triangle equals a side squared times the product of the sines of the angles including the side divided by twice the sine of the angle opposite the side; i. Since there are sometimes two solutions for the second angle, there will be times when the area of two triangles must be found. Case IV Given two sides and the included angle of triangle ABC The area of the triangle is equal to one-half the product of the two sides times the sine of the included angle; i.
For a derivation of the formula, see Prob. What is the area of the gable if it is a tri- angle with two sides of What is the area of the triangle formed by joining their centers? Find a the length of DA and b the area of the field. Find the area of the curvilinear triangle formed by the three circles.
Let the points of tangency of the circles be R, S, and T as in Fig. This is a Case III triangle in which there may be two solutions. After turning through an angle of What is the area of the lot? What is the area of the sign? If they are externally tangent, what is the area of the triangle formed by joining their centers?
What is the area of the triangle formed by her path? But when x is given, the equation may have no solution or many solutions. 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. 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 prin- cipal value is called for, we write Arcsin x, Arccos x, etc. The portions of the graphs on which the principal val- ues of each of the inverse trigonometric relations lie are shown in Fig. The definitions given are the most convenient for calculus. In many calculus textbooks, the inverse of a trigonometric func- tion is defined as the principal-valued inverse, and no capital letter is used in the notation.
This generally causes no problem in a calculus class. Since the value of a trigonometric relation of y is known, two positions are determined in general for the terminal side of the angle y see Chap. Let y1 and y2 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 function.
Hereafter in this chapter we will use the term equation instead of conditional equation. If a given equation has one solution, it has in general an unlimited number of solutions. Several standard procedures are illustrated in the following examples, and other procedures are introduced in the Solved Problems. A The equation may be factorable. B The various functions occurring in the equation may be expressed in terms of a single function.
The necessity of the check is illustrated in Examples C Both members of the equation are squared. D Solutions are approximate values. NOTE: Since we will be using real number properties in solving the equation, the approximate values for the angles will be stated in radians, which can be found using Table 3 in Appendix 2 or a calculator.
These values are not exact and may not yield an exact check when substituted into the given equation. See Chap. Thus, to the nearest hundredth radian the solutions for x are 1. E Equation contains a multiple angle. F Equations containing half angles. Second Solution. In a num- ber of the solutions, the details of the check have been omitted. First Solution. The solution of the first of these equations is 4. All these val- ues are solutions. Thus, x must be positive. Thus, x must be negative.
Use Table 3 in Appendix 2 when find- ing approximate values for x. No solution in given interval The first term a is called the real part of the complex number, and the second term bi is called the imaginary part. Complex numbers may be thought of as including all real numbers and all imaginary numbers. Subtraction To subtract two complex numbers, subtract the real parts and the imaginary parts separately.
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 num- bers 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 rep- resented [see Fig. The vector representation of these numbers [Fig. Division The modulus of the quotient of two complex numbers is the modulus of the dividend divided by the modu- lus 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 Prob. The procedure for determining these roots is given in Example Thus, the five fifth roots are obtained by assigning the values 0, 1, 2, 3, 4 i. See also Prob. The modulus of each of the roots is 22; hence these roots lie on a circle of radius 22 with center at the origin.
The difference in amplitude of two consecutive roots is ; hence the roots are equally spaced on this circle, as shown in Fig. Since each number satisfies the equation, it is a root of the equation.
The purpose of this mate- rial is to provide information useful in solving problems in trigonometry. Two angles are equal when they have the same measure. When two angles have a common ver- tex and a common side between them, the angles are adjacent angles.
If the exterior sides of two adjacent angles form a straight line, the angles form a linear pair. If two lines intersect, they have exactly one point in com- mon. Two lines in a plane are parallel if they have no common point.
When two lines intersect to form equal adjacent angles, the lines are perpendicular. Each of the angles formed by two perpendicular lines is a right angle.
The sides of a right angle are perpendicular. A transversal is a line that intersects two or more coplanar lines in distinct points. When two lines are cut by a transversal, the angles formed are classified by their location.
The angles between the two lines are called interior angles and the angles not between the two lines are called exterior angles. Interior or exterior angles are said to alternate if the two angles have different ver- tices and lie on opposite sides of the transversal.
A pair of corresponding angles are two angles, one an inte- rior angle and one an exterior angle, that have different vertices and lie on the same side of the transversal. The angles numbered 3 and 6 and the angles numbered 4 and 5 are pairs of alternate interior angles. The angles numbered 1 and 8 and those numbered 2 and 7 are pairs of alternate exterior angles. The pairs of corresponding angles are numbered 1 and 5, 2 and 6, 3 and 7, and 4 and 8. Triangles that have no two sides with the same length are called scalene triangles, those with at least two sides having the same length are called isosceles triangles, and those with all three sides having the same length are called equilateral triangles.
Chapter 32 introduces three additional procedures for approximating the real zeros of polynomial equations of degree three or more. Chapter 33 is an informal development of the basic calculus concepts of limit, continuity, and conver-gence using the algebra procedures from the earlier chapters.
Chapter 34 contains additional solved problems and supplementary problems with answers for each of the prior chapters. The choice of whether to use a calculator or not is left to the student. A calculator is not required, but it can be used in conjunction with the book. There are no directions on how to use a graphing calculator to do the problems, but there are several instances of the general procedures to be used and the student needs to consult the manual for the calculator being used to see how to implement the procedures on that particular calculator.
Do you like this book? Please share with your friends, let's read it!! Search Ebook here:. Book Preface In the fifth edition, the comprehensiveness of earlier editions is retained and three new chapters are added so that all of the topics commonly taught in college algebra are contained in a single source.
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