`APPENDIX D.3Review of Trigonometric FunctionsD17APPE N DIXD.3Review of Trigonometric FunctionsAngles and Degree Measure · Radian Measure · The Trigonometric Functions · Evaluating Trigonometric Functions · Solving Trigonometric Equations · Graphs of Trigonometric FunctionsAngles and Degree MeasureAn angle has three parts: an initial ray, a terminal ray, and a vertex (the point of intersection of the two rays), as shown in Figure D.24. An angle is in standard position if its initial ray coincides with the positive x-axis and its vertex is at the origin. It is assumed that you are familiar with the degree measure of an angle.* It is common practice to use (the Greek lowercase theta) to represent both an angle and its measure. Angles between 0 and 90 are acute, and angles between 90 and 180 are obtuse. Positive angles are measured counterclockwise, and negative angles are measured clockwise. For instance, Figure D.25 shows an angle whose measure is 45 . You cannot assign a measure to an angle by simply knowing where its initial and terminal rays are located. To measure an angle, you must also know how the terminal ray was revolved. For example, Figure D.25 shows that the angle measuring 45 has the same terminal ray as the angle measuring 315 . Such angles are coterminal. In general, if is any angle, then n 360 , n is a nonzero integerr TeVertexminalray Initial rayStandard position of an angle Figure D.24is coterminal with . An angle that is larger than 360 is one whose terminal ray has been revolved more than one full revolution counterclockwise, as shown in Figure D.26. You can form an angle whose measure is less than 360 by revolving a terminal ray more than one full revolution clockwise.315° - 45°405°45°Coterminal angles Figure D.25Coterminal angles Figure D.26NOTE It is common to use the symbol to refer to both an angle and its measure. For instance, in Figure D.26, you can write the measure of the smaller angle as 45 .*For a more complete review of trigonometry, see Precalculus, 6th edition, by Larson and Hostetler (Boston, Massachusetts: Houghton Mifflin, 2004).D18APPENDIX DPrecalculus ReviewRadian MeasureTo assign a radian measure to an angle , consider to be a central angle of a circle of radius 1, as shown in Figure D.27. The radian measure of is then defined to be the length of the arc of the sector. Because the circumference of a circle is 2 r, the circumference of a unit circle (of radius 1) is 2 . This implies that the radian 2 radians. measure of an angle measuring 360 is 2 . In other words, 360 Using radian measure for , the length s of a circular arc of radius r is s r , as shown in Figure D.28.The arc length of the sector is the radian measure of  . Arc length is s = r .r=1rUnit circle Figure D.27Circle of radius r Figure D.28You should know the conversions of the common angles shown in Figure D.29. For other angles, use the fact that 180 is equal to radians.30° =  645° =  460° =  390° =  2180° =  360° = 2Radian and degree measure for several common angles Figure D.29EXAMPLE 1 a. 40 b. 540 c. 270Conversions Between Degrees and Radians rad 2 radian 180 deg 9 rad 540 deg 3 radians 180 deg 270 deg40 degrad 3 radians 180 deg 2 180 deg radians rad 90 d. 2 2 rad 180 deg 360 2 rad 114.59 e. 2 radians rad f. 9 radians 2 9 rad 2 180 deg rad 810APPENDIX D.3Review of Trigonometric FunctionsD19The Trigonometric FunctionsHy p n oteAdjacentOppositeuseThere are two common approaches to the study of trigonometry. In one, the trigonometric functions are defined as ratios of two sides of a right triangle. In the other, these functions are defined in terms of a point on the terminal side of an angle in standard position. The six trigonometric functions, sine, cosine, tangent, cotangent, secant, and cosecant (abbreviated as sin, cos, etc.), are defined below from both viewpoints.Sides of a right triangle Figure D.30yDefinition of the Six Trigonometric Functions Right triangle definitions, where 0 &lt;x 2 + y2r= (x, y) y r xsin cscxopposite hypotenuse hypotenuse oppositecos sec(see Figure D.30). 2 adjacent opposite tan hypotenuse adjacent hypotenuse adjacent cot adjacent opposite is any angle (see Figure D.31). x r r , x x 0 tan cot y , x x x , y y 0 0&lt;Circular function definitions, where sin csc y r r , y y 0 cos secAn angle in standard position Figure D.31The following trigonometric identities are direct consequences of the definitions. is the Greek letter phi.Trigonometric IdentitiesPythagorean Identities:[Note that sin2is used to represent sinReduction Formulas:2.]sin2 cos2 1 2 2 1 tan sec 2 1 cot csc2Sum or Difference of Two Angles:sin cos tansin cos tansin cos tansin cos tanHalf­Angle Formulas:Double­Angle Formulas:sin cos tan± ± ±sin cos cos cos± cossin sinsin2 cos21 1sincos 2 2 cos 2 2sin 2 cos 22 sin 2 cos2cos 1tan ± tan 1 tan tanReciprocal Identities:1 2 sin2 cos2 sin2Quotient Identities:Law of Cosines:a2bb2c2a c2bc cos Acsc sec cotA1 sin 1 cos 1 tantan cotsin cos cos sinD20APPENDIX DPrecalculus ReviewEvaluating Trigonometric FunctionsThere are two ways to evaluate trigonometric functions: (1) decimal approximations with a calculator and (2) exact evaluations using trigonometric identities and formulas from geometry. When using a calculator toevaluate a trigonometric function, remember to set the calculator to the appropriate mode--degree mode or radian mode. EXAMPLE 2(x, y) r=1 yExact Evaluation of Trigonometric FunctionsEvaluate the sine, cosine, and tangent of . 3Solution x3 in standard position, as shown in Begin by drawing the angle 3 radians, you can draw an equilateral triangle Figure D.32. Then, because 60 with sides of length 1 and as one of its angles. Because the altitude of this triangle 1 bisects its base, you know that x 2. Using the Pythagorean Theorem, you obtain y r2 x2 1 1 223 43 . 2r=1yNow, knowing the values of x, y, and r, you can write the following. sin cos tan 3 3 3 y r x r y x 3 2 3 1 2 1 2 1 1 2 3 2 3 1 260° x=1 2The angle 3 in standard position Figure D.32NOTE All angles in this text are measured in radians unless stated otherwise. For example, when sin 3 is written, the sine of 3 radians is meant, and when sin 3 is written, the sine of 3 degrees is meant.45° 2 1The degree and radian measures of several common angles are shown in the table below, along with the corresponding values of the sine, cosine, and tangent (see Figure D.33).Common First Quadrant Angles45° 1Degrees Radians30°0 0 0 1 030 6 1 2 3 2 3 345 4 2 2 2 2 160 3 3 2 1 2 390 2 1 0 Undefined32sin cos tan60° 1Common angles Figure D.33APPENDIX D.3Review of Trigonometric FunctionsD21yQuadrant II sin  : + cos  : - tan  : -Quadrant I sin  : + cos  : + tan  : +xThe quadrant signs for the sine, cosine, and tangent functions are shown in Figure D.34. To extend the use of the table on the preceding page to angles in quadrants other than the first quadrant, you can use the concept of a reference angle (see Figure D.35), with the appropriate quadrant sign. For instance, the reference angle for 3 4 is 4, and because the sine is positive in Quadrant II, you can write sin 3 4 sin 4 2 . 2Quadrant III sin  : - cos  : - tan  : +Quadrant IV sin  : - cos  : + tan  : -Similarly, because the reference angle for 330 is 30 , and the tangent is negative in Quadrant IV, you can write tan 330 tan 30 3 . 3Quadrant signs for trigonometric functions Figure D.34Reference angle:  Reference angle:  Reference angle:  Quadrant II  =  -  (radians)  = 180° -  (degrees)Quadrant III  =  -  (radians)  =  - 180°(degrees)Quadrant IV  = 2 -  (radians)  = 360° -  (degrees)Figure D.35EXAMPLE 3Trigonometric Identities and CalculatorsEvaluate each trigonometric expression. a. sinSolution3b. sec 60c. cos 1.2a. Using the reduction formula sin sin sin 3 . 2sin , you can write33b. Using the reciprocal identity sec sec 60 1 cos 60 1 1 2 2.1 cos , you can writec. Using a calculator, you obtain cos 1.2 0.3624.Remember that 1.2 is given in radian measure. Consequently, your calculator must be set in radian mode.D22APPENDIX DPrecalculus ReviewSolving Trigonometric Equations0? You know that 0 is one solution, How would you solve the equation sin but this is not the only solution. Any one of the following values of is also a solution. . . ., 3 , 2 , , 0, , 2 , 3 , . . .You can write this infinite solution set as n : n is an integer . EXAMPLE 4 Solving a Trigonometric EquationSolve the equation sinSolutiony3 . 2To solve the equation, you should consider that the sine is negative in Quadrants III and IV and that sin 3 3 . 2y = sin 1So, you are seeking values of in the third and fourth quadrants that have a reference angle of 3. In the interval 0, 2 , the two angles fitting these criteria are- 2 3 - 3- 2 24 33 2x 235 34 3and235 . 3y=- 3 2By adding integer multiples of 2 to each of these solutions, you obtain the following general solution. 4 3 2n or 5 3 2n , where n is an integer.Solution points of sin u Figure D.363 2See Figure D.36. EXAMPLE 5 Solve cos 2SolutionSolving a Trigonometric Equation 2 3 sin , where 0   2 . 1 2 sin2 , you can rewrite theWrite original equation. Trigonometric identityUsing the double-angle identity cos 2 equation as follows. 1 cos 2 2 sin2 0 0 2 3 sin 2 3 sin 2 sin2 3 sin 2 sin 1 sin1 1Quadratic form Factor.If 2 sin then sin 6 ,1 0, then sin 1 2 and 1 and 2. So, for 0  5 , 6 or 26 or 5 6. If sin 1  2 , there are three solutions.0,APPENDIX D.3Review of Trigonometric FunctionsD23Graphs of Trigonometric Functionsf x A function f is periodic if there exists a nonzero number p such that f x p for all x in the domain of f. The smallest such positive value of p (if it exists) is the period of f. The sine, cosine, secant, and cosecant functions each have a period of 2 , and the other two trigonometric functions, tangent and cotangent, have a period of , as shown in Figure D.37.y 6 5 4 3 2 1 yDomain: all reals Range: [-1, 1] Period: 2y6 5 4 3Domain: all reals Range: [-1, 1] Period: 25 4 3 2 1Domain: all x   + n 2 Range: (-, ) y = tan x Period: y = sin xx2y = cos xx -3x2-12-1y Domain: all x  n 4 3 2 1Range: (-, -1] and [1, ) Period: 2 y Domain: all x  2 + n 4 3 2y 5 4 3 2Range: (-, -1] and [1, ) Period: 2 Domain: all x  n Range: (- , ) Period: -1 2x -1 -2 -32x12x1 y = csc x = sin xy = sec x =1 cos x1 y = cot x = tan xThe graphs of the six trigonometric functions Figure D.37Note in Figure D.37 that the maximum value of sin x and cos x is 1 and the minimum value is 1. The graphs of the functions y a sin bx and y a cos bx oscillate between a and a, and so have an amplitude of a . Furthermore, because bx 0 when x 0 and bx 2 when x 2 b, it follows that the functions y a sin bx and y a cos bx each have a period of 2 b . The table below summarizes the amplitudes and periods for some types of trigonometric functions.Function y y y a sin bx or a tan bx or a sec bx or y y y a cos bx a cot bx a csc bx Period 2 b b 2 b Amplitude a Not applicable Not applicableD24APPENDIX DPrecalculus ReviewEXAMPLE 6ySketching the Graph of a Trigonometric Function 3 cos 2x.f (x) = 3 cos 2x (0, 3)Sketch the graph of f xSolutionAmplitude = 3x3The graph of f x 3 cos 2x has an amplitude of 3 and a period of 2 2 . Using the basic shape of the graph of the cosine function, sketch one period of the function on the interval 0, , using the following pattern. Maximum: 0, 3 Minimum: 2 , ,3 3-1 -2 -3 23 22Period = Maximum:Figure D.38By continuing this pattern, you can sketch several cycles of the graph, as shown in Figure D.38. Horizontal shifts, vertical shifts, and reflections can be applied to the graphs of trigonometric functions, as illustrated in Example 7. EXAMPLE 7 Shifts of Graphs of Trigonometric FunctionsSketch the graph of each function. a. f xSolutionsin x2b. f x2sin xc. f x2sin x4a. To sketch the graph of f x sin x 2 , shift the graph of y sin x to the left 2 units, as shown in Figure D.39(a). b. To sketch the graph of f x 2 sin x, shift the graph of y sin x upward two units, as shown in Figure D.39(b). c. To sketch the graph of f x 2 sin x 4 , shift the graph of y sin x upward two units and to the right 4 units, as shown in Figure D.39(c).y 6 5 4 3 2 y yf (x) = sin x +  2  2((x6 5 4 3 2f (x) = 2 + sin x 2x6 5 4 3 2 f (x) = 2 + sin x - 4  4((2x-2y = sin x-2y = sin x-2y = sin x(a) Horizontal shift to the left(b) Vertical shift upward(c) Horizontal and vertical shiftsTransformations of the graph of y Figure D.39sin xAPPENDIX D.3Review of Trigonometric FunctionsD25EXERCISES FOR APPENDIX D.3In Exercises 1 and 2, determine two coterminal angles in degree measure (one positive and one negative) for each angle. 1. (a) = 36°10. Angular Speed A car is moving at the rate of 50 miles per hour, and the diameter of its wheels is 2.5 feet. (a) Find the number of revolutions per minute that the wheels are rotating.(b) = -120°(b) Find the angular speed of the wheels in radians per minute. In Exercises 11 and 12, determine all six trigonometric functions for the angle . 11. (a)y2. (a) = 300°(b) = - 420°(b)(3, 4)yxxIn Exercises 3 and 4, determine two coterminal angles in radian measure (one positive and one negative) for each angle. 3. (a)= 9(-12, -5)(b)  = 4312. (a)y(b)xyx4. (a) = - 9 4(b) = 8 9(8, -15)(1, -1)In Exercises 13 and 14, determine the quadrant in which 13. (a) sin (b) sec In Exercises 5 and 6, rewrite each angle in radian measure as a multiple of and as a decimal accurate to three decimal places. 5. (a) 30 6. (a) 20 (b) 150 (b) 240 (c) 315 (c) 270 (d) 120 (d) 144 14. (a) sin (b) csc &lt; 0 and cos &gt; 0 and cot &gt; 0 and cos &lt; 0 and tan &lt; 0 &lt; 0 &lt; 0 &gt; 0lies.In Exercises 15­18, evaluate the trigonometric function. 15. sin1 216. sin tan1 3In Exercises 7 and 8, rewrite each angle in degree measure. 3 7. (a) 2 7 8. (a) 3 7 (b) 6 (b) 11 30 (c) 7 12 11 (c) 6 (d) 2.367cos23 11(d) 0.4389. Let r represent the radius of a circle, the central angle (measured in radians), and s the length of the arc subtended by the angle. Use the relationship s r to complete the table. r s 8 ft 12 ft 1.6 3 4 15 in. 85 cm 96 in. 4 8642 mi 2 317. cos cot4 518. sec csc13 55 1345D26APPENDIX DPrecalculus ReviewIn Exercises 19­22, evaluate the sine, cosine, and tangent of each angle without using a calculator. 19. (a) 60 (b) 120 (c) (d) 4 5 4 225 5 3 11 6 20. (a) 3039. Airplane Ascent An airplane leaves the runway climbing at an angle of 18 with a speed of 275 feet per second (see figure). Find the altitude a of the plane after 1 minute.(b) 150 (c) (d) 6 218° a21. (a) 225 (b) (c) (d)22. (a) 750 (b) 510 (c) (d) 10 3 17 3 40. Height of a Mountain In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5 . After you drive 13 miles closer to the mountain, the angle of elevation is 9 . Approximate the height of the mountain.In Exercises 23­26, use a calculator to evaluate each trigonometric function. Round your answers to four decimal places. 23. (a) sin 10 (b) csc 10 25. (a) tan (b) tan 9 10 9 24. (a) sec 225 (b) sec 135 26. (a) cot 1.35 (b) tan 1.353.5° 13 mi9°Not drawn to scaleIn Exercises 41­44, determine the period and amplitude of each function. 41. (a) yy2 sin 2x(b) yy 11 2sin xIn Exercises 27­30, find two solutions of each equation. Give your answers in radians 0  &lt; 2 . Do not use a calculator. 27. (a) cos (b) cos 29. (a) tan (b) cot 1 3 (b) sin In Exercises 31­38, solve the equation for 31. 2 sin 33. tan2 35. sec csc 37. cos2 23 2 1 x 1 -1 22 2 2 228. (a) sec (b) sec2 2 4-3 23 4 5 4x30. (a) sin3 2 3 2 0  3 cos cos 142. (a) yy 33 x cos 2 2(b) yy2 sinx 3&lt; 2.2 x -1 -2 -3 x -1 -21 tan 0 2 csc sin 132.tan234. 2 cos2 36. sin 38. cos 2  232cos1 43. y 44. y 3 sin 4 x 2 x cos 3 10APPENDIX D.3Review of Trigonometric FunctionsD27In Exercises 45­48, find the period of the function. 45. y 47. y 5 tan 2x sec 5x 46. y 48. y 7 tan 2 x csc 4x67. Sales The monthly sales S (in thousands of units) of a seasonal product are modeled by S 58.3 32.5 cos t 6Writing In Exercises 49 and 50, use a graphing utility tograph each function f in the same viewing window for c 2, c 1, c 1, and c 2. Give a written description of the change in the graph caused by changing c. 49. (a) f x (b) f x (c) f x c sin x cos cx cos x c 50. (a) f x (b) f x (c) f x sin x c cos x c cwhere t is the time (in months) with t 1 corresponding to January. Use a graphing utility to graph the model for S and determine the months when sales exceed 75,000 units. 68. Investigation Two trigonometric functions f and g have a period of 2, and their graphs intersect at x 5.35. (a) Give one smaller and one larger positive value of x where the functions have the same value. (b) Determine one negative value of x where the graphs intersect. (c) Is it true that f 13.35 answer. g 4.65 ? Give a reason for yoursin 2 xIn Exercises 51­62, sketch the graph of the function. 51. y 53. y 55. y 57. y 59. y 61. y 62. y x sin 2 sin csc x 2 2 x 3 52. y 54. y 56. y 58. y 60. y 2 cos 2xPattern Recognition In Exercises 69 and 70, use a graphing2 tan x tan 2x csc 2 x cos x 3 utility to compare the graph of f with the given graph. Try to improve the approximation by adding a term to f x . Use a graphing utility to verify that your new approximation is better than the original. Can you find other terms to add to make the approximation even better? What is the pattern? (In Exercise 69, sine terms can be used to improve the approximation and in Exercise 70, cosine terms can be used.) 69. f x 2y2 sec 2x sin x 1 1 cos x sin x4sin x1 sin 3 x 322 1 x 3 -2In Exercises 63 and 64, find a, b, and c such that the graph of the function matches the graph in the figure. 63. yy 4 2 x 1Graphical Reasoninga cos bxc64. yya sin bxc70. f xy1 242cos x1 cos 3 x 9-43-1 2 -1 23 4x2 1 x -1 1 2 3 465. Think About It Sketch the graphs of f x sin x, gx sin x , and h x sin x . In general, how are the graphs of f x and f x related to the graph of f ? 66. Think About It car is h 51 The model for the height h of a Ferris wheel-250 sin 8 twhere t is measured in minutes. (The Ferris wheel has a radius of 50 feet.) This model yields a height of 51 feet when t 0. Alter the model so that the height of the car is 1 foot when t 0.`

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