C.3 Review of Trigonometric Functions
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1 C. Review of Trigonometric Functions C7 C. Review of Trigonometric Functions Describe angles and use degree measure. Use radian measure. Understand the definitions of the si trigonometric functions. Evaluate trigonometric functions. Solve trigonometric equations. Graph trigonometric functions. Verte Terminal ra Initial ra Standard position of an angle Figure C. 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 C.. An angle is in standard position when its initial ra coincides with the positive -ais and its verte is at the origin. It is assumed that ou are familiar with the degree measure of an angle.* It is common practice to use (the lowercase Greek letter 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 C. 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 C. 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 n0, n is a nonzero integer Coterminal angles Figure C. 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 C.. You can form an angle whose measure is less than 0 b revolving a terminal ra more than one full revolution clockwise. 0 Coterminal angles Figure C. Note that it is common to use the smbol to refer to both an angle and its measure. For instance, in Figure C., ou can write the measure of the smaller angle as. *For a more complete review of trigonometr, see Precalculus, 9th edition,b Larson (Brooks/Cole, Cengage Learning, 0).
2 C8 Appendi C 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 C.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 C.8. r = The arc length of the sector is the radian measure of. Arc length is s = r. r Unit circle Circle of radius r Figure C.7 Figure C.8 You should know the conversions of the common angles shown in Figure C.9. For other angles, use the fact that 80 is equal to radians. 0 = = 0 = 90 = 80 = Radian and degree measures for several common angles Figure C.9 0 = Conversions Between Degrees and Radians a. b. c. d. e. f. 0 0 deg 80 deg 0 0 deg 80 deg radians deg 80 deg radians radians rad 80 deg radians rad 80 deg 9 rad rad 9 radian rad rad rad radians 9 rad 80 deg rad
3 C. Review of Trigonometric Functions C9 Hpotenuse Adjacent Sides of a right triangle Figure C.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. The si trigonometric functions, sine, cosine, tangent, cotangent, secant, and cosecant (abbreviated as sin, cos, tan, cot, sec, and csc, respectivel), are defined below from both viewpoints. Definition of the Si Trigonometric Functions Right triangle definitions, where 0 < sin opposite hpotenuse csc hpotenuse opposite Circular function definitions, where < cos adjacent hpotenuse sec hpotenuse adjacent (see Figure C.0). tan opposite adjacent cot adjacent opposite is an angle (see Figure C.). sin r cos r tan, 0 An angle in standard position Figure C. csc r, 0 sec r, 0 cot, 0 The trigonometric identities listed below are direct consequences of the definitions. Note that is the lowercase Greek letter phi and sin is used to represent sin. TRIGONOMETRIC IDENTITIES Pthagorean Identities: sin cos tan sec cot csc Even/Odd Identities sin sin cos cos tan tan csc csc sec sec cot cot Sum and Difference Formulas Power-Reducing Formulas Double-Angle Formulas sin cos tan ± ± cos cos sin sin ± sin cos ± cos sin tan ± tan tan tan Law of Cosines: Reciprocal Identities Quotient Identities: a b c bc cos A b A c a sin cos tan csc sin sec cos cot tan cos cos cos cos sin sin cos cos cos sin cos sin tan tan tan tan sin cos cot cos sin
4 C0 Appendi C Precalculus Review Evaluating Trigonometric Functions There are two was to evaluate trigonometric functions: () decimal approimations with a calculator 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 = r = 0 = Figure C. (, ) Evaluate the sine, cosine, and tangent of. Eact Evaluation of Trigonometric Functions Solution Because 0 radians, ou can draw an equilateral triangle with sides of length and as one of its angles, as shown in Figure C.. 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 that all angles in this tet are measured in radians unless stated otherwise. For eample, when sin is written, the sine of radians is meant, and when sin is written, the sine of degrees is meant. The 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 C.). Common First-Quadrant Angles 0 Common angles Figure C. 0 Degrees Radians 0 sin cos tan 0 0 Undefined 0
5 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 C. The quadrant signs for the sine, cosine, and tangent functions are shown in Figure C.. 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 C.), with the appropriate quadrant sign. For instance, the reference angle for is, and because the sine is positive in Quadrant II, ou can write sin sin. Similarl, because the reference angle for 0 is 0, and the tangent is negative in Quadrant IV, ou can write tan 0 tan 0. C. Review of Trigonometric Functions C Reference angle: Reference angle: Reference angle: Quadrant II = (radians) = 80 (degrees) Figure C. Quadrant III = (radians) = 80 (degrees) Quadrant IV = (radians) = 0 (degrees) Evaluate each trigonometric epression. a. sin b. sec 0 c. cos. Solution Trigonometric Identities and Calculators 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 obtain cos. 0.. Remember that. is given in radian measure. Consequentl, our calculator must be set in radian mode.
6 C Appendi C Precalculus Review Solving Trigonometric Equations 0 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. Solving a Trigonometric Equation = = sin Solution points of sin Figure C. 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 So, 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 general solution.. n See Figure C... to each of these solutions, ou obtain the following or n, where n is an integer. Solve cos sin, where 0 Solving a Trigonometric Equation Solution Using the double-angle identit cos sin, ou can rewrite the equation as follows. sin sin Write original equation. Trigonometric identit Quadratic form Factor. If sin 0, then sin and or If sin 0, then sin and So, for 0, there are three solutions. cos sin, 0 sin sin 0 sin sin., or..
7 Graphs of Trigonometric Functions C. Review of Trigonometric Functions C A function f is periodic when there eists a nonzero number p such that f p f for all in the domain of f. The least 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, tangent and cotangent, have a period of, as shown in Figure C.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 C.7 Note in Figure C.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 so 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 of some tpes of trigonometric functions. Function Period Amplitude a sin b or a cos b a tan b or a cot b a sec b or a csc b b b b a Not applicable Not applicable
8 C Appendi C Precalculus Review Sketching the Graph of a Trigonometric Function f() = cos (0, ) Period = Figure C.8 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 Amplitude = 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 C.8. Horizontal shifts, vertical shifts, and reflections can be applied to the graphs of trigonometric functions, as illustrated in Eample 7. Shifts of Graphs of Trigonometric Functions Sketch the graph of each function. 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 C.9(a). b. To sketch the graph of f sin, shift the graph of sin upward two units, as shown in Figure C.9. c. To sketch the graph of f sin, shift the graph of sin upward two units and to the right units, as shown in Figure C.9(c). f() = sin + ( ) f() = + sin f() = + sin ( ) = sin = sin = sin (a) Horizontal shift to the left Vertical shift upward (c) Horizontal and vertical shifts Transformations of the graph of sin Figure C.9
9 C. Review of Trigonometric Functions C C. Eercises Coterminal Angles in Degrees In Eercises and, determine two coterminal angles in degree measure (one positive and one negative) for each angle.. (a) =. (a) = 00 = 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. Find the angular speed of the wheels in radians per minute. Finding the Si Trigonometric Functions In Eercises and, determine all si trigonometric functions for the angle.. (a) (, ) Coterminal Angles in Radians In Eercises and, determine two coterminal angles in radian measure (one positive and one negative) for each angle. (, ). (a) = 9 =. (a). (a) 9 = = 8 9 (8, ) (, ) Determining a Quadrant In Eercises and, determine the quadrant in which lies. Degrees to Radians In Eercises and, rewrite each angle in radian measure as a multiple of and as a decimal accurate to three decimal places.. (a) 0 0 (c) (d) 0. (a) 0 0 (c) 70 (d) Radians to Degrees In Eercises 7 and 8, rewrite each angle in degree measure. 7. (a) (c) 7 (d) (a) (c) (d) (a) sin < 0 and cos < 0 sec > 0 and cot < 0. (a) sin > 0 and cos < 0 csc < 0 and tan > 0 Evaluating a Trigonometric Function In Eercises 8, evaluate the trigonometric function.. sin. sin cos tan 9. Completing a Table 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. 7. cos 8. cot sec csc r 8 ft in. 8 cm s ft 9 in. 8 mi.
10 C Appendi C Precalculus Review Evaluating Trigonometric Functions In Eercises 9, evaluate the sine, cosine, and tangent of each angle without using a calculator. 9. (a) 0 0. (a) 0 0 (c) (d) 0 (c) (d). (a). (a) 70 (c) (d) 0 Evaluating Trigonometric Functions In Eercises, use a calculator to evaluate each trigonometric function. Round our answers to four decimal places. (c) (d). (a) sin 0. (a) sec csc 0 sec. (a) tan. (a) cot. 9 tan. Solving a Trigonometric Equation In Eercises 7 0, find two solutions of each equation. Give our answers in radians 0 Do not use a calculator. 7. (a) cos 8. (a) sec 9. (a) tan 0. (a) cot sec Solving a Trigonometric Equation In Eercises 8, solve the equation for. sin. tan. tan tan 0. cos cos. sec csc csc. sin cos 7. cos sin tan 0 9 <. cos 8. cos cos 0 <. 0 7 sin sin 9. Airplane Ascent An airplane leaves the runwa climbing at an angle of 8 with a speed of 7 feet per second (see figure). Find the altitude a of the plane after minute. 0. Height of a Mountain While 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. Period and Amplitude In Eercises, determine the period and amplitude of each function.. (a) sin sin. (a) sin cos. sin. cos 0. 9 mi 8 Not drawn to scale a
11 C. Review of Trigonometric Functions C7 Period 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 in the same viewing window 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 f cosc (c) f cos c f sin c (c) f c cos Sketching the Graph of a Trigonometric Function Eercises, sketch the graph of the function.. sin. cos. sin. tan. csc. tan 7. sec 8. csc 9. sin 0. cos. cos. 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. Think About It Sketch the graphs of f sin, g sin, and h sin In general, how are the graphs of and f. related to the graph of f? f. Think About It The model for the height h of a Ferris wheel car is In 7. Sales The monthl 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. 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 at which the functions have the same value. Determine one negative value of at which the graphs intersect. (c) Is it true that f. g.? Give a reason for our answer. 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 70. f sin sin cos 9 cos 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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