D.3. Angles and Degree Measure. Review of Trigonometric Functions


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1 APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric Equations Graphs of Trigonometric Functions Verte Terminal ra Initial ra Standard position of an angle Figure D. Angles and Degree Measure An angle has three parts: an initial ra, a terminal ra, and a verte (the point of intersection of the two ras), as shown in Figure D.. An angle is in standard position if its initial ra coincides with the positive ais and its verte is at the origin. We assume that ou 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 80 are obtuse. Positive angles are measured counterclockwise, and negative angles are measured clockwise. For instance, Figure D. shows an angle whose measure is. You cannot assign a measure to an angle b simpl knowing where its initial and terminal ras are located. To measure an angle, ou must also know how the terminal ra was revolved. For eample, Figure D. shows that the angle measuring has the same terminal ra as the angle measuring. Such angles are coterminal. In general, if is an angle, then n is a nonzero integer is coterminal with. An angle that is larger than 0 is one whose terminal ra has been revolved more than one full revolution counterclockwise, as shown in Figure D.. You can form an angle whose measure is less than 0 b revolving a terminal ra more than one full revolution clockwise. n0, 0 Coterminal angles Coterminal angles Figure D. Figure D. NOTE It is common to use the smbol to refer to both an angle and its measure. For instance, in Figure D., ou can write the measure of the smaller angle as. *For a more complete review of trigonometr, see Precalculus, th edition, b Larson and Hostetler (Boston, Massachusetts: Houghton Mifflin, 00).
2 D8 APPENDIX D Precalculus Review Radian Measure To assign a radian measure to an angle, consider to be a central angle of a circle of radius, as shown in Figure D.7. The radian measure of is then defined to be the length of the arc of the sector. Because the circumference of a circle is r, the circumference of a unit circle (of radius ) is. This implies that the radian measure of an angle measuring 0 is. In other words, 0 radians. Using radian measure for, the length s of a circular arc of radius r is s r, as shown in Figure D.8. Arc length is s = r. The arc length of the sector is the radian measure of. r = r Unit circle Circle of radius r Figure D.7 Figure D.8 You should know the conversions of the common angles shown in Figure D.9. For other angles, use the fact that 80 is equal to radians. 0 = = 0 = 90 = 80 = 0 = Radian and degree measure for several common angles Figure D.9 EXAMPLE Conversions Between Degrees and Radians a. 0 0 deg 80 deg 9 radians b deg 80 deg radians c. radians rad 80 deg d. radians 9 rad 80 deg 9 rad rad rad 90 rad 80
3 APPENDIX D Precalculus Review D9 Hpotenuse Adjacent Sides of a right triangle Figure D.0 (, ) r r = + Opposite The Trigonometric Functions There are two common approaches to the stud of trigonometr. 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. We define the si trigonometric functions, sine, cosine, tangent, cotangent, secant, and cosecant (abbreviated as sin, cos, etc.), from both viewpoints. Definition of the Si Trigonometric Functions Right triangle definitions, where 0 < < (see Figure D.0). sin opposite cos adjacent tan opposite hpotenuse hpotenuse adjacent csc hpotenuse opposite Circular function definitions, where sec hpotenuse adjacent cot adjacent opposite is an angle (see Figure D.). sin r cos r tan An angle in standard position Figure D. csc r sec r cot The following trigonometric identities are direct consequences of the definitions. is the Greek letter phi. Trigonometric Identities [Note that sin is used to represent sin.] Pthagorean Identities: Reduction Formulas: sin cos tan sec cot csc Sum or Difference of Two Angles: sin cos tan ± ± cos cos sin sin ± Law of Cosines: sin cos ± cos sin tan ± tan tan tan a b c bc cos A b A c a sin sin cos cos tan tan Half Angle Formulas: sin cos cos cos Reciprocal Identities: csc sin sec cos cot tan sin sin cos cos tan tan Double Angle Formulas: sin sin cos cos cos Quotient Identities: tan sin cos cot cos sin sin cos sin
4 D0 APPENDIX D Precalculus Review Evaluating Trigonometric Functions There are two was to evaluate trigonometric functions: () decimal approimations with a calculator (or a table of trigonometric values) and () eact evaluations using trigonometric identities and formulas from geometr. When using a calculator to evaluate a trigonometric function, remember to set the calculator to the appropriate mode degree mode or radian mode. r = 0 = r = (, ) The angle / in standard position Figure D. EXAMPLE Eact Evaluation of Trigonometric Functions Evaluate the sine, cosine, and tangent of. Solution Begin b drawing the angle in standard position, as shown in Figure D.. Then, because 0 radians, ou can draw an equilateral triangle with sides of length and as one of its angles. Because the altitude of this triangle bisects its base, ou know that. Using the Pthagorean Theorem, ou obtain r. Now, knowing the values of,, and r, ou can write the following. sin r cos r tan NOTE All angles in this tet are measured in radians unless stated otherwise. For eample, when we write sin, we mean the sine of radians, and when we write sin, we mean the sine of degrees. 0 Common angles Figure D. 0 The degree and radian measures of several common angles are given in the table below, along with the corresponding values of the sine, cosine, and tangent (see Figure D.). Common First Quadrant Angles Degrees 0 Radians 0 sin cos tan Undefined 90
5 APPENDIX D Precalculus Review D Quadrant II sin : + cos : tan : Quadrant III sin : cos : tan : + Quadrant I sin : + cos : + tan : + Quadrant IV sin : cos : + tan : Quadrant signs for trigonometric functions Figure D. The quadrant signs for the sine, cosine, and tangent functions are shown in Figure D.. To etend the use of the table on the preceding page to angles in quadrants other than the first quadrant, ou can use the concept of a reference angle (see Figure D.), with the appropriate quadrant sign. For instance, the reference angle for is, and because the sine is positive in the second quadrant, ou can write sin sin. Similarl, because the reference angle for 0 is 0, and the tangent is negative in the fourth quadrant, ou can write tan 0 tan 0. Reference angle: Reference angle: Reference angle: Quadrant II = (radians) = 80 (degrees) Figure D. Quadrant III = (radians) = 80 (degrees) Quadrant IV = (radians) = 0 (degrees) EXAMPLE Trigonometric Identities and Calculators Evaluate the trigonometric epression. a. sin b. sec 0 c. cos. Solution a. Using the reduction formula sin sin, ou can write sin sin. b. Using the reciprocal identit sec cos, ou can write sec 0 cos 0. c. Using a calculator, ou can obtain cos. 0.. Remember that. is given in radian measure. Consequentl, our calculator must be set in radian mode.
6 D APPENDIX D Precalculus Review Solving Trigonometric Equations How would ou solve the equation sin 0? You know that is one solution, but this is not the onl solution. An one of the following values of is also a solution....,,,, 0,,,,... You can write this infinite solution set as n: n is an integer. 0 EXAMPLE Solving a Trigonometric Equation = = sin Solution points of sin Figure D. Solve the equation sin. Solution To solve the equation, ou should consider that the sine is negative in Quadrants III and IV and that sin. Thus, ou are seeking values of in the third and fourth quadrants that have a reference angle of. In the interval 0,, the two angles fitting these criteria are and. B adding integer multiples of to each of these solutions, ou obtain the following general solution. n (See Figure D..) or n, where n is an integer. EXAMPLE Solving a Trigonometric Equation Solve cos sin, where 0. Solution Using the doubleangle identit cos sin, ou can rewrite the equation as follows. sin sin Given equation Trigonometric identit Quadratic form Factor. If sin 0, then sin and or If sin 0, then sin and Thus, for 0, there are three solutions. cos sin, 0 sin sin 0 sin sin,. or.
7 APPENDIX D Precalculus Review D Graphs of Trigonometric Functions A function f is periodic if there eists a nonzero number p such that f p f for all in the domain of f. The smallest such positive value of p (if it eists) is the period of f. The sine, cosine, secant, and cosecant functions each have a period of, and the other two trigonometric functions have a period of, as shown in Figure D.7. Domain: all reals Range: [, ] Period: = sin Domain: all reals Range: [, ] Period: = cos Domain: all + n Range: (, ) = tan Period: Domain: all n Range: (, ] and [, ) Period: Domain: all + n Range: (, ] and [, ) Period: Domain: all n Range: (, ) Period: = csc = sin = sec = cos = cot = tan The graphs of the si trigonometric functions Figure D.7 Note in Figure D.7 that the maimum value of sin and cos is and the minimum value is. The graphs of the functions a sin b and a cos b oscillate between a and a, and hence have an amplitude of a. Furthermore, because b 0 when 0 and b when b, it follows that the functions a sin b and a cos b each have a period of b. The table below summarizes the amplitudes and periods for some tpes of trigonometric functions. a sin b a tan b a sec b Function Period Amplitude or or or a cos b a cot b a csc b b b b a Not applicable Not applicable
8 D APPENDIX D Precalculus Review f( ) = cos (0, ) Period = Figure D.8 Amplitude = EXAMPLE Sketching the Graph of a Trigonometric Function Sketch the graph of f cos. Solution The graph of f cos has an amplitude of and a period of. 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. Maimum: 0, Minimum: Maimum:,, B continuing this pattern, ou can sketch several ccles of the graph, as shown in Figure D.8. Horizontal shifts, vertical shifts, and reflections can be applied to the graphs of trigonometric functions, as illustrated in Eample 7. EXAMPLE 7 Shifts of Graphs of Trigonometric Functions Sketch the graphs of the following functions. a. f sin b. f sin c. f sin Solution a. To sketch the graph of f sin, shift the graph of sin to the left units, as shown in Figure D.9(a). b. To sketch the graph of f sin, shift the graph of sin up two units, as shown in Figure D.9(b). c. To sketch the graph of f sin, shift the graph of sin up two units and to the right units, as shown in Figure D.9(c). f ( ) = sin + ( ( f ( ) = + sin f( ) = + sin ( ( = sin = sin = sin (a) Horizontal shift to the left (b) Vertical shift upward (c) Horizontal and vertical shifts Transformations of the graph of sin Figure D.9
9 APPENDIX D Precalculus Review D E X E R C I S E S F O R APPENDIX D. In Eercises and, determine two coterminal angles (one positive and one negative) for each given angle. Epress our answers in degrees.. (a) (b) =. (a) = 00 (b) = 0 = 0 0. Angular Speed A car is moving at the rate of 0 miles per hour, and the diameter of its wheels is. feet. (a) Find the number of revolutions per minute that the wheels are rotating. (b) Find the angular speed of the wheels in radians per minute. In Eercises and, determine all si trigonometric functions for the angle.. (a) (b) (, ) In Eercises and, determine two coterminal angles (one positive and one negative) for each given angle. Epress our answers in radians.. (a) (b) = 9 =. (a) (b) (, ). (a) (b) = 9 In Eercises and, epress the angles in radian measure as multiples of and as decimals accurate to three decimal places.. (a) 0 (b) 0 (c) (d) 0. (a) 0 (b) 0 (c) 70 (d) = 8 9 In Eercises and, determine the quadrant in which. (a) sin < 0 and cos < 0 (b) sec > 0 and cot < 0. (a) sin > 0 and cos < 0 (b) csc < 0 and tan > 0 In Eercises 8, evaluate the trigonometric function.. sin. sin cos (8, ) tan (, ) lies. In Eercises 7 and 8, epress the angles in degree measure. 7. (a) (b) (c) 7 (d) (a) (b) (c) (d) Let r represent the radius of a circle, the central angle (measured in radians), and s the length of the arc subtended b the angle. Use the relationship s r to complete the table. r 8 ft in. 8 cm 7. cos 8. cot sec csc s ft 9 in. 8 mi.
10 D APPENDIX D Precalculus Review In Eercises 9, evaluate the sine, cosine, and tangent of each angle without using a calculator. 9. (a) 0 0. (a) 0 (b) 0 (b) 0 (c) (c) (d) (d). (a). (a) 70 (b) (b) 0 (c) (d) In Eercises, use a calculator to evaluate the trigonometric functions to four significant digits. (c) (d). (a) sin 0. (a) sec (b) csc 0 (b) sec. (a) tan. (a) cot. 9 (b) tan 0 (b) tan Airplane Ascent An airplane leaves the runwa climbing at 8 with a speed of 7 feet per second (see figure). Find the altitude a of the plane after minute Height of a Mountain In traveling across flat land, ou notice a mountain directl in front of ou. Its angle of elevation (to the peak) is.. After ou drive miles closer to the mountain, the angle of elevation is 9. Approimate the height of the mountain.. 9 mi (not to scale) In Eercises, determine the period and amplitude of each function.. (a) sin (b) sin a In Eercises 7 0, find two solutions of each equation. Epress the results in radians 0 <. Do not use a calculator. 7. (a) cos 8. (a) sec (b) cos (b) sec 9. (a) tan 0. (a) (b) cot (b) sin sin. (a) (b) cos sin In Eercises 8, solve the equation for. sin. tan. tan tan 0. cos cos. sec csc csc. sin cos 7. cos sin 8. cos cos 0 <.. sin. cos 0
11 APPENDIX D Precalculus Review D7 In Eercises 8, find the period of the function.. tan. 7 tan 7. sec 8. csc Writing In Eercises 9 and 0, use a graphing utilit to graph each function f on the same set of coordinate aes for c, c, c, and c. Give a written description of the change in the graph caused b changing c. 9. (a) f c sin 0. (a) f sin c (b) f cosc (b) f sin c (c) f cos c (c) f c cos In Eercises, sketch the graph of the function.. sin. cos 7. Sales Sales S, in thousands of units, of a seasonal product are modeled b t S 8.. cos where t is the time in months (with t corresponding to Januar and t corresponding to December). Use a graphing utilit to graph the model for S and determine the months when sales eceed 7,000 units. 8. Investigation Two trigonometric functions f and g have a period of, and their graphs intersect at.. (a) Give one smaller and one larger positive value of where the functions have the same value. (b) Determine one negative value of where the graphs intersect. (c) Is it true that f. g.? Give a reason for our answer.. sin.. csc. 7. sec sin 0.. cos. sin tan tan csc cos Pattern Recognition In Eercises 9 and 70, use a graphing utilit to compare the graph of f with the given graph. Tr to improve the approimation b adding a term to f. Use a graphing utilit to verif that our new approimation is better than the original. Can ou find other terms to add to make the approimation even better? What is the pattern? (In Eercise 9, sine terms can be used to improve the approimation and in Eercise 70, cosine terms can be used.) 9. f sin sin Graphical Reasoning In Eercises and, find a, b, and c such that the graph of the function matches the graph in the figure.. a cosb c. a sinb c 70. f cos 9 cos. Think About It Sketch the graphs of f sin, g sin and In general, how are the graphs of, h sin f and f. related to the graph of f?. Think About It The model for the height h of a Ferris wheel car is h 0 sin 8t where t is measured in minutes. (The Ferris wheel has a radius of 0 feet.) This model ields a height of feet when t 0. Alter the model so that the height of the car is foot when t 0.
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