Radian and Degree Measure

Transcription

1 Radian and Degree Measure What ou should learn Describe angles. Use radian measure. Use degree measure. Use angles to model and solve real-life problems. Wh ou should learn it You can use angles to model and solve real-life problems. For instance, in Eercise 08, ou are asked to use angles to find the speed of a biccle. Angles As derived from the Greek language, the word trigonometr means measurement of triangles. Initiall, trigonometr dealt with relationships among the sides and angles of triangles and was used in the development of astronom, navigation, and surveing. With the development of calculus and the phsical sciences in the 7th centur, a different perspective arose one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domains. Consequentl, the applications of trigonometr epanded to include a vast number of phsical phenomena involving rotations and vibrations. These phenomena include sound waves, light ras, planetar orbits, vibrating strings, pendulums, and orbits of atomic particles. The approach in this tet incorporates both perspectives, starting with angles and their measure. Video. Video Verte Terminal side Initial side Terminal side Initial side. Angle Angle in Standard Position FIGURE FIGURE An angle is determined b rotating a ra (half-line) about its endpoint. The starting position of the ra is the initial side of the angle, and the position after rotation is the terminal side, as shown in Figure. The endpoint of the ra is the verte of the angle. This perception of an angle fits a coordinate sstem in which the origin is the verte and the initial side coincides with the positive -ais. Such an angle is in standard position, as shown in Figure. Positive angles are generated b counterclockwise rotation, and negative angles b clockwise rotation, as shown in Figure. Angles are labeled with Greek letters (alpha), (beta), and (theta), as well as uppercase letters A, B, and C. In Figure, note that angles and have the same initial and terminal sides. Such angles are coterminal. Positive angle (counterclockwise) Negative angle (clockwise) α β β α FIGURE FIGURE Coterminal Angles

2 Radian Measure r r s = r The measure of an angle is determined b the amount of rotation from the initial side to the terminal side. One wa to measure angles is in radians. This tpe of measure is especiall useful in calculus. To define a radian, ou can use a central angle of a circle, one whose verte is the center of the circle, as shown in Figure 5. Arc length radius when radian FIGURE 5 Definition of Radian One radian is the measure of a central angle that intercepts an arc s equal in length to the radius r of the circle. See Figure 5. Algebraicall, this means that s r where is measured in radians. Because the circumference of a circle is r units, it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length of radians FIGURE radians r r radians r r radian r r 5 radians radians s r. Moreover, because there are just over si radius lengths in a full circle, as shown in Figure. Because the units of measure for s and r are the same, the ratio sr has no units it is simpl a real number. Because the radian measure of an angle of one full revolution is, ou can obtain the following. revolution radians revolution.8, radians Video revolution radians. These and other common angles are shown in Figure 7. One revolution around a circle of radius r corresponds to an angle of radians because s r r r radians. FIGURE 7 Recall that the four quadrants in a coordinate sstem are numbered I, II, III, and IV. Figure 8 on the net page shows which angles between 0 and lie in each of the four quadrants. Note that angles between 0 and are acute angles and angles between and are obtuse angles.

3 = Quadrant II < < Quadrant I 0 < < = = 0 Quadrant III Quadrant IV < < < < The phrase the terminal side of lies in a quadrant is often abbreviated b simpl saing that lies in a quadrant. The terminal sides of the quadrant angles 0,,, and do not lie within quadrants. FIGURE 8 Two angles are coterminal if the have the same initial and terminal sides. For instance, the angles 0 and are coterminal, as are the angles and. You can find an angle that is coterminal to a given angle b adding or subtracting (one revolution), as demonstrated in Eample. A given angle has infinitel man coterminal angles. For instance, is coterminal with n where n is an integer. = Eample Sketching and Finding Coterminal Angles a. For the positive angle, subtract to obtain a coterminal angle. See Figure 9. b. For the positive angle, subtract to obtain a coterminal angle 5. See Figure 0. c. For the negative angle, add to obtain a coterminal angle. See Figure. = 0 FIGURE 9 FIGURE 0 FIGURE Now tr Eercise 7. = 5 0 = 0

4 Two positive angles and are complementar (complements of each other) if their sum is. Two positive angles are supplementar (supplements of each other) if their sum is. See Figure. β α β α Complementar Angles FIGURE Supplementar Angles Eample Complementar and Supplementar Angles = (0 ) 0 = (0 ) 5 = (0 ) 8 0 = (0 ) FIGURE Video If possible, find the complement and the supplement of (a) 5 and (b) 5. a. The complement of 5 is 5 The supplement of 5 is b. Because 5 is greater than, it has no complement. (Remember that complements are positive angles.) The supplement is Degree Measure Now tr Eercise. A second wa to measure angles is in terms of degrees, denoted b the smbol. A measure of one degree ( ) is equivalent to a rotation of 0 of a complete revolution about the verte. To measure angles, it is convenient to mark degrees on the circumference of a circle, as shown in Figure. So, a full revolution (counterclockwise) corresponds to 0, a half revolution to 80, a quarter revolution to 90, and so on. Because radians corresponds to one complete revolution, degrees and radians are related b the equations 0 rad and 80 rad. From the latter equation, ou obtain 80 and rad 80 rad which lead to the conversion rules at the top of the net page

5 . Video Conversions Between Degrees and Radians. To convert degrees to radians, multipl degrees b. To convert radians to degrees, multipl radians b rad rad. To appl these two conversion rules, use the basic relationship (See Figure.) rad FIGURE When no units of angle measure are specified, radian measure is implied. For instance, if ou write ou impl that radians., Eample Converting from Degrees to Radians Technolog With calculators it is convenient to use decimal degrees to denote fractional parts of degrees. Historicall, however, fractional parts of degrees were epressed in minutes and seconds,using the prime ( ) and double prime ( ) notations, respectivel. That is, one minute 0 one second Consequentl, an angle of degrees, minutes, and 7 seconds is represented b Man calculators have special kes for converting an angle in degrees, minutes, and seconds D M S to decimal degree form, and vice versa a. 5 5 deg 80 deg radians Multipl b 80. b deg 80 deg radians Multipl b 80. c deg 80 deg radians Multipl b 80. Eample Now tr Eercise 7. Converting from Radians to Degrees a. rad rad 80 deg Multipl b 80. b. rad 9 rad 80 deg 80 Multipl b 80. c. rad rad 80 deg 0.59 Multipl b rad rad rad rad 90 rad rad Now tr Eercise 5. If ou have a calculator with a radian-to-degree conversion ke, tr using it to verif the result shown in part (c) of Eample.

6 . s Video = 0 r = Applications The radian measure formula, a circle. Arc Length sr, can be used to measure arc length along For a circle of radius r, a central angle intercepts an arc of length s given b s r Length of circular arc where is measured in radians. Note that if r, then s, and the radian measure of equals the arc length. Eample 5 Finding Arc Length FIGURE 5 A circle has a radius of inches. Find the length of the arc intercepted b a central angle of 0, as shown in Figure 5. To use the formula s r, first convert 0 to radian measure. 0 0 deg 80 deg Then, using a radius of r inches, ou can find the arc length to be s r.7 inches. r rad radians Note that the units for are determined b the units for r because is given in radian measure, which has no units. Now tr Eercise 87. The formula for the length of a circular arc can be used to analze the motion of a particle moving at a constant speed along a circular path. Linear speed measures how fast the particle moves, and angular speed measures how fast the angle changes. B dividing the formula for arc length b t, ou can establish a relationship between linear speed v and angular speed, as shown. s r s r t t v r Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed v of the particle is Linear speed v Moreover, if is the angle (in radian measure) corresponding to the arc length s, then the angular speed (the lowercase Greek letter omega) of the particle is Angular speed arc length time s t. central angle time t.

7 Eample Finding Linear Speed FIGURE 0. cm The second hand of a clock is 0. centimeters long, as shown in Figure. Find the linear speed of the tip of this second hand as it passes around the clock face. In one revolution, the arc length traveled is s r centimeters. Substitute for r. The time required for the second hand to travel this distance is t minute 0 seconds. So, the linear speed of the tip of the second hand is Linear speed s t 0. centimeters 0 seconds.08 centimeters per second. Now tr Eercise ft FIGURE 7 Eample 7 Finding Angular and Linear Speeds A Ferris wheel with a 50-foot radius (see Figure 7) makes.5 revolutions per minute. a. Find the angular speed of the Ferris wheel in radians per minute. b. Find the linear speed of the Ferris wheel. a. Because each revolution generates radians, it follows that the wheel turns.5 radians per minute. In other words, the angular speed is Angular speed t b. The linear speed is Linear speed s t radians minute r t 50 feet minute Now tr Eercise 05. radians per minute. 7. feet per minute.

8 A sector of a circle is the region bounded b two radii of the circle and their intercepted arc (see Figure 8). r FIGURE 8 Area of a Sector of a Circle For a circle of radius r, the area A of a sector of the circle with central angle is given b where A r is measured in radians. Eample 8 Area of a Sector of a Circle A sprinkler on a golf course fairwa is set to spra water over a distance of 70 feet and rotates through an angle of 0 (see Figure 9). Find the area of the fairwa watered b the sprinkler. First convert 0 to radian measure as follows. FIGURE ft 0 0 deg 80 deg radians rad Multipl b 80. Then, using and r 70, the area is A r square feet. Formula for the area of a sector of a circle Substitute for r and. Simplif. Simplif. Now tr Eercise 07.

9 Eercises The smbol indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. Click on to view the complete solution of the eercise. Click on to print an enlarged cop of the graph. Click on to view the Make a Decision eercise. VOCABULARY CHECK: Fill in the blanks. Glossar. means measurement of triangles.. An is determined b rotating a ra about its endpoint.. Two angles that have the same initial and terminal sides are.. One is the measure of a central angle that intercepts an arc equal to the radius of the circle. 5. Angles that measure between 0 and are angles, and angles that measure between and are angles.. Two positive angles that have a sum of are angles, whereas two positive angles that have a sum of are angles The angle measure that is equivalent to of a complete revolution about an angle s verte is one. 8. The speed of a particle is the ratio of the arc length traveled to the time traveled. 9. The speed of a particle is the ratio of the change in the central angle to time. 0. The area of a sector of a circle with radius r and central angle, where is measured in radians, is given b the formula. In Eercises, estimate the angle to the nearest one-half radian..... In Eercises, sketch each angle in standard position.. (a) (b). (a) 7 (b) 5 5. (a) (b). (a) (b) In Eercises 7 0, determine two coterminal angles (one positive and one negative) for each angle. Give our answers in radians (a) (b) = 0 5 = In Eercises 7, determine the quadrant in which each angle lies. (The angle measure is given in radians.) 7. (a) (b) 8. (a) (b) (a) (b) 0. (a) (b) 7 9. (a).5 (b).5. (a).0 (b) (a) (b) 7 = 9. (a) (b) 9 0. (a) (b) 0 5 = 0

10 In Eercises, find (if possible) the complement and supplement of each angle.. (a) (b). (a) (b). (a) (b). (a) (b).5 In Eercises 5 0, estimate the number of degrees in the angle In Eercises, determine the quadrant in which each angle lies. In Eercises 5 8, sketch each angle in standard position. In Eercises 9, determine two coterminal angles (one positive and one negative) for each angle. Give our answers in degrees. 9. (a) (b) = 5. (a) 0 (b) 85. (a) 8. (b) (a) 50 (b). (a) 0 (b). 5. (a) 0 (b) 50. (a) 70 (b) 0 7. (a) 05 (b) (a) 750 (b) (a) (b) = 0 = = 0. (a) (b). (a) (b) In Eercises, find (if possible) the complement and supplement of each angle.. (a) 8 (b) 5. (a) (b) 5. (a) 79 (b) 50. (a) 0 (b) 70 In Eercises 7 50, rewrite each angle in radian measure as a multiple of. (Do not use a calculator.) 7. (a) 0 (b) (a) 5 (b) 0 9. (a) 0 (b) (a) 70 (b) In Eercises 5 5, rewrite each angle in degree measure. (Do not use a calculator.) 5. (a) (b) 5. (a) 7 (b) 9 5. (a) (b) 5. (a) (b) 0 5 In Eercises 55, convert the angle measure from degrees to radians. Round to three decimal places In Eercises 70, convert the angle measure from radians to degrees. Round to three decimal places In Eercises 7 7, convert each angle measure to decimal degree form (a) 5 5 (b) (a) 5 0 (b) 7. (a) (b) (a) 5 (b) 08 0 In Eercises 75 78, convert each angle measure to form. 75. (a) 0. (b) (a) 5. (b) (a).5 (b) (a) 0.55 (b) D M S

11 In Eercises 79 8, find the angle in radians Cit 9. San Francisco, California Seattle, Washington Latitude 7 7 N N In Eercises 8 8, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius r Arc Length s 8. 7 inches inches 8. feet 8 feet centimeters 5 centimeters kilometers 0 kilometers In Eercises 87 90, find the length of the arc on a circle of radius r intercepted b a central angle. Radius r Central Angle inches feet meters radian centimeters radian In Eercises 9 9, find the area of the sector of the circle with radius r and central angle. Radius r 9. inches 9. millimeters 9..5 feet 9.. miles 5 7 Central Angle Distance Between Cities In Eercises 95 and 9, find the distance between the cities. Assume that Earth is a sphere of radius 000 miles and that the cities are on the same longitude (one cit is due north of the other). Cit 95. Dallas, Teas Omaha, Nebraska Latitude 7 9 N 5 50 N Difference in Latitudes Assuming that Earth is a sphere of radius 78 kilometers, what is the difference in the latitudes of Sracuse, New York and Annapolis, Marland, where Sracuse is 50 kilometers due north of Annapolis? 98. Difference in Latitudes Assuming that Earth is a sphere of radius 78 kilometers, what is the difference in the latitudes of Lnchburg, Virginia and Mrtle Beach, South Carolina, where Lnchburg is 00 kilometers due north of Mrtle Beach? 99. Instrumentation The pointer on a voltmeter is centimeters in length (see figure). Find the angle through which the pointer rotates when it moves.5 centimeters on the scale. cm ft FIGURE FOR 99 FIGURE FOR 00 0 in. Not drawn to scale 00. Electric Hoist An electric hoist is being used to lift a beam (see figure). The diameter of the drum on the hoist is 0 inches, and the beam must be raised feet. Find the number of degrees through which the drum must rotate. 0. Angular Speed A car is moving at a rate of 5 miles per hour, and the diameter of its wheels is.5 feet. (a) Find the number of revolutions per minute the wheels are rotating. (b) Find the angular speed of the wheels in radians per minute. 0. Angular Speed A two-inch-diameter pulle on an electric motor that runs at 700 revolutions per minute is connected b a belt to a four-inch-diameter pulle on a saw arbor. (a) Find the angular speed (in radians per minute) of each pulle. (b) Find the revolutions per minute of the saw.

12 0. Linear and Angular Speeds A -inch circular power saw rotates at 500 revolutions per minute. (a) Find the angular speed of the saw blade in radians per minute. (b) Find the linear speed (in feet per minute) of one of the cutting teeth as the contact the wood being cut. 0. Linear and Angular Speeds A carousel with a 50-foot diameter makes revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed of the platform rim of the carousel. 05. Linear and Angular Speeds The diameter of a DVD is approimatel centimeters. The drive motor of the DVD plaer is controlled to rotate precisel between 00 and 500 revolutions per minute, depending on what track is being read. (a) Find an interval for the angular speed of a DVD as it rotates. (b) Find an interval for the linear speed of a point on the outermost track as the DVD rotates. 0. Area A car s rear windshield wiper rotates 5. The total length of the wiper mechanism is 5 inches and wipes the windshield over a distance of inches. Find the area covered b the wiper. 07. Area A sprinkler sstem on a farm is set to spra water over a distance of 5 meters and to rotate through an angle of 0. Draw a diagram that shows the region that can be irrigated with the sprinkler. Find the area of the region. 08. Speed of a Biccle The radii of the pedal sprocket, the wheel sprocket, and the wheel of the biccle in the figure are inches, inches, and inches, respectivel. A cclist is pedaling at a rate of revolution per second. in. Model It 7 Snthesis True or False? In Eercises 09, determine whether the statement is true or false. Justif our answer. 09. A measurement of radians corresponds to two complete revolutions from the initial side to the terminal side of an angle. 0. The difference between the measures of two coterminal angles is alwas a multiple of 0 if epressed in degrees and is alwas a multiple of radians if epressed in radians.. An angle that measures 0 lies in Quadrant III.. Writing In our own words, eplain the meanings of (a) an angle in standard position, (b) a negative angle, (c) coterminal angles, and (d) an obtuse angle.. Think About It A fan motor turns at a given angular speed. How does the speed of the tips of the blades change if a fan of greater diameter is installed on the motor? Eplain.. Think About It Is a degree or a radian the larger unit of measure? Eplain. 5. Writing If the radius of a circle is increasing and the magnitude of a central angle is held constant, how is the length of the intercepted arc changing? Eplain our reasoning.. Proof Prove that the area of a circular sector of radius r with central angle is A r, where is measured in radians. Skills Review Model It (continued) (c) Write a function for the distance d (in miles) a cclist travels in terms of the time t (in seconds). Compare this function with the function from part (b). (d) Classif the tpes of functions ou found in parts (b) and (c). Eplain our reasoning. In Eercises 7 0, simplif the radical epression. in. in. (a) Find the speed of the biccle in feet per second and miles per hour. (b) Use our result from part (a) to write a function for the distance d (in miles) a cclist travels in terms of the number n of revolutions of the pedal sprocket In Eercises, sketch the graphs of 5 and the specified transformation.. f 5. f 5. f 5. f 5

13 Trigonometric Functions: The Unit Circle What ou should learn Identif a unit circle and describe its relationship to real numbers. Evaluate trigonometric functions using the unit circle. Use the domain and period to evaluate sine and cosine functions. Use a calculator to evaluate trigonometric functions. Wh ou should learn it Trigonometric functions are used to model the movement of an oscillating weight. For instance, in Eercise 57, the displacement from equilibrium of an oscillating weight suspended b a spring is modeled as a function of time. The Unit Circle The two historical perspectives of trigonometr incorporate different methods for introducing the trigonometric functions. Our first introduction to these functions is based on the unit circle. Consider the unit circle given b as shown in Figure 0. Unit circle (, 0) (0, ) (0, ) (, 0) FIGURE 0 Simulation. Imagine that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in Figure. t > 0 (, ) t t t < 0 (, 0) (, 0) t (, ) t FIGURE. Video As the real number line is wrapped around the unit circle, each real number t corresponds to a point, on the circle. For eample, the real number 0 corresponds to the point, 0. Moreover, because the unit circle has a circumference of, the real number also corresponds to the point, 0. In general, each real number t also corresponds to a central angle (in standard position) whose radian measure is t. With this interpretation of t, the arc length formula s r (with r ) indicates that the real number t is the length of the arc intercepted b the angle, given in radians.

14 . Video Note in the definition at the right that the functions in the second row are the reciprocals of the corresponding functions in the first row. The Trigonometric Functions From the preceding discussion, it follows that the coordinates and are two functions of the real variable t. You can use these coordinates to define the si trigonometric functions of t. sine cosecant cosine secant tangent cotangent These si functions are normall abbreviated sin, csc, cos, sec, tan, and cot, respectivel. Definitions of Trigonometric Functions Let t be a real number and let, be the point on the unit circle corresponding to t. sin t csc t, 0 cos t sec t, 0 tan t, cot t, 0 0 (0, ),, ( ), (, 0) ( ) FIGURE, (, ) (, ) FIGURE ( ) (, 0) (0, ) (, ) (0, ) (0, ) ( ) (, 0) ( ), (, ) (, 0) (, ) (, ) (, ) In the definitions of the trigonometric functions, note that the tangent and secant are not defined when 0. For instance, because t corresponds to, 0,, it follows that tan and sec are undefined. Similarl, the cotangent and cosecant are not defined when 0. For instance, because t 0 corresponds to,, 0, cot 0 and csc 0 are undefined. In Figure, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, and.,,,,,,, Similarl, in Figure, the unit circle has been divided into equal arcs, corresponding to t-values of 0, and.,,,, 5,,,,,,, To verif the points on the unit circle in Figure, note that also, lies on the line. So, substituting for in the equation of the unit circle produces the following. ± Because the point is in the first quadrant, and because, ou also have You can use similar reasoning to verif the rest of the points in. Figure and the points in Figure. Using the, coordinates in Figures and, ou can easil evaluate the trigonometric functions for common t-values. This procedure is demonstrated in Eamples and. You should stud and learn these eact function values for common t-values because the will help ou in later sections to perform calculations quickl and easil

15 Eample Evaluating Trigonometric Functions Evaluate the si trigonometric functions at each real number. a. t b. t 5 c. t 0 d. t For each t-value, begin b finding the corresponding point, on the unit circle. Then use the definitions of trigonometric functions listed on the previous page. a. t corresponds to the point 5 b. t corresponds to the point sin cos tan 5 sin 5 cos 5 tan,,. c. t 0 corresponds to the point,, 0.,,. csc sec cot csc 5 sec 5 cot 5 sin 0 0 csc 0 is undefined. cos 0 sec 0 tan cot 0 is undefined. d. t corresponds to the point,, 0. sin 0 cos tan 0 0 Now tr Eercise. csc is undefined. sec cot is undefined.

16 Eploration With our graphing utilit in radian and parametric modes, enter the equations XT = cos T and YT = sin T and use the following settings. Tmin = 0, Tma =., Tstep = 0. Xmin = -.5, Xma =.5, Xscl = Ymin = -, Yma =, Yscl =. Graph the entered equations and describe the graph.. Use the trace feature to move the cursor around the graph. What do the t-values represent? What do the - and -values represent?. What are the least and greatest values of and? (0, ) (, 0) (, 0) (0, ) FIGURE t =, +, +,... t =, + t =, +,...,... t =,,... t = 5 5, +, t = 0,,... t =, +, +,... t =, +, +,... Eample Evaluating Trigonometric Functions Evaluate the si trigonometric functions at t. Moving clockwise around the unit circle, it follows that t corresponds to the point,,. Now tr Eercise 5. Domain and Period of Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. To determine the range of these two functions, consider the unit circle shown in Figure. Because r, it follows that sin t and cos t. Moreover, because, is on the unit circle, ou know that and. So, the values of sine and cosine also range between and. and sin t cos t Adding to each value of t in the interval 0, completes a second revolution around the unit circle, as shown in Figure 5. The values of sint and cost correspond to those of sin t and cos t. Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result and sin cos tan sint n sin t cost n cos t csc sec cot for an integer n and real number t. Functions that behave in such a repetitive (or cclic) manner are called periodic. Definition of Periodic Function A function f is periodic if there eists a positive real number c such that ft c f t for all t in the domain of f. The smallest number c for which f is periodic is called the period of f. FIGURE 5

17 Recall that a function f is even if f t f t, and is odd if f t ft. Even and Odd Trigonometric Functions The cosine and secant functions are even. cost cos t sect sec t The sine, cosecant, tangent, and cotangent functions are odd. sint sin t csct csc t tant tan t cott cot t Video Eample Using the Period to Evaluate the Sine and Cosine. From the definition of periodic function, it follows that the sine and cosine functions are periodic and have a period of. The other four trigonometric functions are also periodic, and will be discussed further later in this chapter.. Technolog When evaluating trigonometric functions with a calculator, remember to enclose all fractional angle measures in parentheses. For instance, if ou want to evaluate sin for ou should enter SIN Video, ENTER. These kestrokes ield the correct value of 0.5. Note that some calculators automaticall place a left parenthesis after trigonometric functions. Check the user s guide for our calculator for specific kestrokes on how to evaluate trigonometric functions. a. Because ou have sin sin, b. Because 7 ou have, cos 7 cos c. For sin t sint because the sine function is odd. 5, 5 Now tr Eercise. Evaluating Trigonometric Functions with a Calculator When evaluating a trigonometric function with a calculator, ou need to set the calculator to the desired mode of measurement (degree or radian). Most calculators do not have kes for the cosecant, secant, and cotangent functions. To evaluate these functions, ou can use the ke with their respective reciprocal functions sine, cosine, and tangent. For eample, to evaluate csc8, use the fact that csc 8 sin8 and enter the following kestroke sequence in radian mode. SIN 8 ENTER Displa.59 Eample Using a Calculator cos 0. Function Mode Calculator Kestrokes Displa a. sin Radian SIN ENTER b. cot.5 Radian TAN.5 ENTER Now tr Eercise 5. sin.

18 Eercises The smbol Click on Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph. to view the Make a Decision eercise. VOCABULARY CHECK: Fill in the blanks. Glossar. Each real number t corresponds to a point, on the.. A function f is if there eists a positive real number c such that f t c f t for all t in the domain of f.. The smallest number c for which a function f is periodic is called the of f.. A function f is if f t ft and if f t f t. In Eercises, determine the eact values of the si trigonometric functions of the angle , ( 7 7 ( ( 5, ( 9. t 0.. t. t 5 t In Eercises 8, evaluate (if possible) the si trigonometric functions of the real number... In Eercises 5, find the point, on the unit circle that corresponds to the real number t. 5. t. t 7. t 7 8. t 5 9. t 0. t 5. t. t In Eercises, evaluate (if possible) the sine, cosine, and tangent of the real number.. t. t 5. t. t 7. t 7 8. t 5 ( ( ( 5, 5 (. t. 5. t. 7. t 8. In Eercises 9, evaluate the trigonometric function using its period as an aid. 9. sin 5 0. cos 5. cos 8. sin 9. cos 5. sin 9 5. sin 9. In Eercises 7, use the value of the trigonometric function to evaluate the indicated functions. 7. sin t 8. sint 8 (a) sint (a) sin t (b) csct (b) csc t 9. cost 5 0. cos t (a) cos t (a) cost (b) sect (b) sect. sin t 5. cos t 5 (a) (a) (b) sint (b) cost sin t t 5 t t 7 cos 8 cos t

19 In Eercises 5, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. (Be sure the calculator is set in the correct angle mode.). sin. tan 5. csc.. cot 7. cos.7 8. cos.5 9. csc sec.8 5. sec.8 5. sin0.9 Estimation In Eercises 5 and 5, use the figure and a straightedge to approimate the value of each trigonometric function. 5. (a) sin 5 (b) cos 5. (a) sin 0.75 (b) cos FIGURE FOR Estimation In Eercises 55 and 5, use the figure and a straightedge to approimate the solution of each equation, where 0 t <. 55. (a) sin t 0.5 (b) cos t (a) sin t 0.75 (b) cos t 0.75 Model It 57. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended b a spring and subject to the damping effect of friction is given b t e t cos t where is the displacement (in feet) and t is the time (in seconds). 58. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended b a spring is given b t cos t, where is the displacement (in feet) and t is the time (in seconds). Find the displacement when (a) t 0, (b) t and (c) t,. Snthesis True or False? In Eercises 59 and 0, determine whether the statement is true or false. Justif our answer. 59. Because sint sin t, it can be said that the sine of a negative angle is a negative number. 0. tan a tana. Eploration Let, and, be points on the unit circle corresponding to t t and t t, respectivel. (a) Identif the smmetr of the points, and,. (b) Make a conjecture about an relationship between sin t and (c) Make a conjecture about an relationship between cos t and. Use the unit circle to verif that the cosine and secant functions are even and that the sine, cosecant, tangent, and cotangent functions are odd. Skills Review Model It (continued) (a) Complete the table. t 0 (b) Use the table feature of a graphing utilit to approimate the time when the weight reaches equilibrium. (c) What appears to happen to the displacement as t increases? sin t. cos t. In Eercises, find the inverse function one-to-one function f.. f. f 5. f,. f f of the In Eercises 7 70, sketch the graph of the rational function b hand. Show all asmptotes f 8. f f f

20 Right Triangle Trigonometr What ou should learn Evaluate trigonometric functions of acute angles. Use the fundamental trigonometric identities. Use a calculator to evaluate trigonometric functions. Use trigonometric functions to model and solve real-life problems. Wh ou should learn it Trigonometric functions are often used to analze real-life situations. For instance, in Eercise 7, ou can use trigonometric functions to find the height of a helium-filled balloon. The Si Trigonometric Functions Our second look at the trigonometric functions is from a right triangle perspective. Consider a right triangle, with one acute angle labeled, as shown in Figure. Relative to the angle, the three sides of the triangle are the hpotenuse, the opposite side (the side opposite the angle ), and the adjacent side (the side adjacent to the angle ). FIGURE Hpotenuse Side adjacent to Side opposite Using the lengths of these three sides, ou can form si ratios that define the si trigonometric functions of the acute angle. sine cosecant cosine secant tangent cotangent In the following definitions, it is important to see that 0 < < 90 lies in the first quadrant) and that for such angles the value of each trigonometric function is positive.. Video Right Triangle Definitions of Trigonometric Functions Let be an acute angle of a right triangle. The si trigonometric functions of the angle are defined as follows. (Note that the functions in the second row are the reciprocals of the corresponding functions in the first row.) sin opp hp csc hp opp cos adj hp sec hp adj The abbreviations opp, adj, and hp represent the lengths of the three sides of a right triangle. opp the length of the side opposite adj the length of the side adjacent to hp the length of the hpotenuse tan opp adj cot adj opp

21 Eample Evaluating Trigonometric Functions FIGURE 7 Hpotenuse Use the triangle in Figure 7 to find the values of the si trigonometric functions of. B the Pthagorean Theorem, hp opp adj, it follows that hp 5 5. So, the si trigonometric functions of sin opp hp 5 are csc hp opp 5 cos adj hp 5 sec hp adj 5 Historical Note Georg Joachim Rhaeticus (5 57) was the leading Teutonic mathematical astronomer of the th centur. He was the first to define the trigonometric functions as ratios of the sides of a right triangle. 5 FIGURE 8 5 tan opp adj Now tr Eercise. In Eample, ou were given the lengths of two sides of the right triangle, but not the angle. Often, ou will be asked to find the trigonometric functions of a given acute angle. To do this, construct a right triangle having as one of its angles. Eample Evaluating Trigonometric Functions of 5 Find the values of sin 5, cos 5, and tan 5. Construct a right triangle having 5 as one of its acute angles, as shown in Figure 8. Choose the length of the adjacent side to be. From geometr, ou know that the other acute angle is also 5. So, the triangle is isosceles and the length of the opposite side is also. Using the Pthagorean Theorem, ou find the length of the hpotenuse to be. sin 5 opp hp cos 5 adj hp tan 5 opp adj Now tr Eercise 7. cot adj opp.

22 Eample Evaluating Trigonometric Functions of 0 and 0 Because the angles 0, 5, and 0,, and occur frequentl in trigonometr, ou should learn to construct the triangles shown in Figures 8 and 9. Use the equilateral triangle shown in Figure 9 to find the values of cos 0, sin 0, and cos 0. 0 sin 0, Technolog You can use a calculator to convert the answers in Eample to decimals. However, the radical form is the eact value and in most cases, the eact value is preferred. FIGURE 9 Use the Pthagorean Theorem and the equilateral triangle in Figure 9 to verif the lengths of the sides shown in the figure. For ou have adj, opp, and hp. So, sin 0 opp hp 0, and For adj, opp, and hp. So, sin 0 opp hp and Now tr Eercise , cos 0 adj hp. cos 0 adj hp. Sines, Cosines, and Tangents of Special Angles sin 0 sin cos 0 cos tan 0 tan sin 5 sin sin 0 sin cos 5 cos cos 0 cos tan 5 tan tan 0 tan In the bo, note that sin 0 cos 0. This occurs because 0 and 0 are complementar angles. In general, it can be shown from the right triangle definitions that cofunctions of complementar angles are equal. That is, if is an acute angle, the following relationships are true. sin90 cos cos90 sin tan90 cot cot90 tan sec90 csc csc90 sec

23 Trigonometric Identities In trigonometr, a great deal of time is spent studing relationships between trigonometric functions (identities).. Video Fundamental Trigonometric Identities Reciprocal Identities sin csc csc sin Quotient Identities tan sin cos Pthagorean Identities sin cos cos sec sec cos cot cos sin tan sec cot csc tan cot cot tan Note that sin represents sin, cos represents cos, and so on. Eample Appling Trigonometric Identities FIGURE Let be an acute angle such that sin 0.. Find the values of (a) cos and (b) tan using trigonometric identities. a. To find the value of cos, use the Pthagorean identit sin cos. So, ou have 0. cos Substitute 0. for sin. Subtract 0. from each side. Etract the positive square root. b. Now, knowing the sine and cosine of, ou can find the tangent of to be tan sin cos cos cos Use the definitions of cos and tan, and the triangle shown in Figure 0, to check these results. Now tr Eercise 9.

24 Eample 5 Appling Trigonometric Identities 0 Let be an acute angle such that tan. Find the values of (a) cot and (b) sec using trigonometric identities. a. cot Reciprocal identit tan cot FIGURE You can also use the reciprocal identities for sine, cosine, and tangent to evaluate the cosecant, secant, and cotangent functions with a calculator. For instance, ou could use the following kestroke sequence to evaluate sec 8. COS 8 ENTER The calculator should displa.570. b. sec tan Pthagorean identit sec sec 0 sec 0 Use the definitions of cot and sec, and the triangle shown in Figure, to check these results. Now tr Eercise. Evaluating Trigonometric Functions with a Calculator To use a calculator to evaluate trigonometric functions of angles measured in degrees, first set the calculator to degree mode and then proceed as demonstrated in the previous section. For instance, ou can find values of cos 8 and sec 8 as follows. Function Mode Calculator Kestrokes Displa a. cos 8 Degree COS 8 ENTER b. sec 8 Degree COS 8 ENTER.570 Throughout this tet, angles are assumed to be measured in radians unless noted otherwise. For eample, sin means the sine of radian and sin means the sine of degree. Eample Using a Calculator.. Video Video Use a calculator to evaluate sec5 0. Begin b converting to decimal degree form. [Recall that Then, use a calculator to evaluate sec 5.7. Function Calculator Kestrokes Displa sec5 0 sec 5.7 COS 5.7 ENTER.009 Now tr Eercise 7. and

25 . Observer Observer FIGURE Video Object Angle of elevation Horizontal Horizontal Angle of depression Object Angle of elevation 78. Applications Involving Right Triangles Man applications of trigonometr involve a process called solving right triangles. In this tpe of application, ou are usuall given one side of a right triangle and one of the acute angles and are asked to find one of the other sides, or ou are given two sides and are asked to find one of the acute angles. In Eample 7, the angle ou are given is the angle of elevation, which represents the angle from the horizontal upward to an object. For objects that lie below the horizontal, it is common to use the term angle of depression, as shown in Figure. Eample 7 Using Trigonometr to Solve a Right Triangle A surveor is standing 5 feet from the base of the Washington Monument, as shown in Figure. The surveor measures the angle of elevation to the top of the monument as 78.. How tall is the Washington Monument? From Figure, ou can see that tan 78. opp adj where 5 and is the height of the monument. So, the height of the Washington Monument is tan feet. Now tr Eercise. FIGURE = 5 ft Not drawn to scale Eample 8 Using Trigonometr to Solve a Right Triangle An historic lighthouse is 00 ards from a bike path along the edge of a lake. A walkwa to the lighthouse is 00 ards long. Find the acute angle between the bike path and the walkwa, as illustrated in Figure. 00 d 00 d FIGURE From Figure, ou can see that the sine of the angle sin opp 00 hp Now ou should recognize that Now tr Eercise 5. is

26 0 B now ou are able to recognize that is the acute angle that satisfies the equation sin. Suppose, however, that ou were given the equation sin 0. and were asked to find the acute angle. Because sin 0 and sin ou might guess that lies somewhere between 0 and 5. In a later section, ou will stud a method b which a more precise value of can be determined. Eample 9 Solving a Right Triangle Find the length c of the skateboard ramp shown in Figure c ft FIGURE 5 From Figure 5, ou can see that sin 8. opp hp c. So, the length of the skateboard ramp is c sin feet. Now tr Eercise 7.

27 Eercises The smbol Click on Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph. to view the Make a Decision eercise. VOCABULARY CHECK:. Match the trigonometric function with its right triangle definition. (a) Sine (b) Cosine (c) Tangent (d) Cosecant (e) Secant (f) Cotangent hpotenuse adjacent hpotenuse adjacent opposite opposite (i) (ii) (iii) (iv) (v) (vi) adjacent opposite opposite hpotenuse hpotenuse adjacent In Eercises and, fill in the blanks. Glossar. Relative to the angle, the three sides of a right triangle are the side, the side, and the.. An angle that measures from the horizontal upward to an object is called the angle of, whereas an angle that measures from the horizontal downward to an object is called the angle of. In Eercises, find the eact values of the si trigonometric functions of the angle shown in the figure. (Use the Pthagorean Theorem to find the third side of the triangle.) In Eercises 5 8, find the eact values of the si trigonometric functions of the angle for each of the two triangles. Eplain wh the function values are the same In Eercises 9, sketch a right triangle corresponding to the trigonometric function of the acute angle. Use the Pthagorean Theorem to determine the third side and then find the other five trigonometric functions of. 9. sin 0. cos 5 7. sec. cot 5. tan. sec 5. cot. csc 7 In Eercises 7, construct an appropriate triangle to complete the table. 0 90, 0 / Function (deg) (rad) Function Value 7. sin 0 8. cos 5 9. tan 0. sec. cot. csc. cos. sin 5. cot. tan

28 In Eercises 7, use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions. 7. sin 0, cos 0 (a) tan 0 (b) sin 0 (c) cos 0 (d) cot 0 8. sin 0 tan 0, (a) csc 0 (b) cot 0 (c) cos 0 (d) cot 0 9. (a) sin (b) cos (c) tan (d) sec90 0. sec 5, tan (a) cos (b) cot (c) cot90 (d) sin. cos (a) sec (b) sin (c) cot (d) sin90. tan 5 (a) cot (b) cos (c) tan90 (d) csc In Eercises, use trigonometric identities to transform the left side of the equation into the right side 0 < < /.. tan cot. cos sec 5. tan cos sin. cot sin cos 7. cos cos sin 8. sin sin cos 9. sec tan sec tan 0. sin cos sin.. csc, sec sin cos csc sec cos sin tan cot csc tan In Eercises 5, use a calculator to evaluate each function. Round our answers to four decimal places. (Be sure the calculator is in the correct angle mode.). (a) sin 0 (b) cos 80. (a) tan.5 (b) cot.5 5. (a) sin.5 (b) csc.5. (a) cos 8 (b) sin (a) sec (b) csc (a) cos 50 5 (b) sec (a) cot 5 (b) tan (a) sec (b) cos (a) csc 0 (b) tan 8 5. (a) sec (b) cot In Eercises 5 58, find the values of in degrees 0 < < 90 and radians 0 < < / without the aid of a calculator. 5. (a) sin (b) csc 5. (a) cos (b) tan 55. (a) sec (b) cot 5. (a) tan (b) cos 57. (a) csc (b) sin 58. (a) cot (b) sec In Eercises 59, solve for,, or r as indicated. 59. Solve for. 0. Solve for. 0. Solve for.. Solve for r Empire State Building You are standing 5 meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 8th floor (the observator) is 8. If the total height of the building is another meters above the 8th floor, what is the approimate height of the building? One of our friends is on the 8th floor. What is the distance between ou and our friend? r 0

29 . Height A si-foot person walks from the base of a broadcasting tower directl toward the tip of the shadow cast b the tower. When the person is feet from the tower and feet from the tip of the shadow, the person s shadow starts to appear beond the tower s shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the tower. (b) Use a trigonometric function to write an equation involving the unknown quantit. (c) What is the height of the tower? 5. Angle of Elevation You are skiing down a mountain with a vertical height of 500 feet. The distance from the top of the mountain to the base is 000 feet. What is the angle of elevation from the base to the top of the mountain?. Width of a River A biologist wants to know the width w of a river so in order to properl set instruments for studing the pollutants in the water. From point A, the biologist walks downstream 00 feet and sights to point C (see figure). From this sighting, it is determined that How wide is the river? 5. w C = 5 A 00 ft 7. Length A steel cable zip-line is being constructed for a competition on a realit television show. One end of the zip-line is attached to a platform on top of a 50-foot pole. The other end of the zip-line is attached to the top of a 5-foot stake. The angle of elevation to the platform is (see figure). 8. 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.5. After ou drive miles closer to the mountain, the angle of elevation is 9. Approimate the height of the mountain. 9. Machine Shop Calculations A steel plate has the form of one-fourth of a circle with a radius of 0 centimeters. Two two-centimeter holes are to be drilled in the plate positioned as shown in the figure. Find the coordinates of the center of each hole mi (, ) (, ) Machine Shop Calculations A tapered shaft has a diameter of 5 centimeters at the small end and is 5 centimeters long (see figure). The taper is. Find the diameter d of the large end of the shaft. Not drawn to scale 5 cm d 5 ft = 50 ft 5 cm (a) How long is the zip-line? (b) How far is the stake from the pole? (c) Contestants take an average of seconds to reach the ground from the top of the zip-line. At what rate are contestants moving down the line? At what rate are the dropping verticall?

30 7. Height A 0-meter line is used to tether a heliumfilled balloon. Because of a breeze, the line makes an angle of approimatel 85 with the ground. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write an equation involving the unknown quantit. (c) What is the height of the balloon? (d) The breeze becomes stronger and the angle the balloon makes with the ground decreases. How does this affect the triangle ou drew in part (a)? (e) Complete the table, which shows the heights (in meters) of the balloon for decreasing angle measures. Angle, Height Angle, Height Model It 7. Geometr Use a compass to sketch a quarter of a circle of radius 0 centimeters. Using a protractor, construct an angle of 0 in standard position (see figure). Drop a perpendicular line from the point of intersection of the terminal side of the angle and the arc of the circle. B actual measurement, calculate the coordinates, of the point of intersection and use these measurements to approimate the si trigonometric functions of a 0 angle (f) As the angle the balloon makes with the ground approaches 0, how does this affect the height of the balloon? Draw a right triangle to eplain our reasoning. Snthesis True or False? In Eercises 7 78, determine whether the statement is true or false. Justif our answer. 7. sin 0 csc 0 7. sec 0 csc sin 5 cos 5 7. cot 0 csc 0 sin sin 78. tan5 tan 5 sin Writing In right triangle trigonometr, eplain wh sin 0 regardless of the size of the triangle. 80. Think About It You are given onl the value tan. Is it possible to find the value of sec without finding the measure of? Eplain. 8. Eploration (a) Complete the table. (b) Is or sin greater for in the interval 0, 0.5? (c) As approaches 0, how do and sin compare? Eplain. 8. Eploration (a) Complete the table. sin cos sin (b) Discuss the behavior of the sine function for range from 0 to 90. (c) Discuss the behavior of the cosine function for range from 0 to 90. in the in the (d) Use the definitions of the sine and cosine functions to eplain the results of parts (b) and (c). Skills Review In Eercises 8 8, perform the operations and simplif. 0 cm 0 (, ) t 5t t 9 t t t

31 Trigonometric Functions of An Angle What ou should learn Evaluate trigonometric functions of an angle. Use reference angles to evaluate trigonometric functions. Evaluate trigonometric functions of real numbers. Wh ou should learn it You can use trigonometric functions to model and solve real-life problems. For instance, in Eercise 87, ou can use trigonometric functions to model the monthl normal temperatures in New York Cit and Fairbanks, Alaska. Introduction In the previous section, the definitions of trigonometric functions were restricted to acute angles. In this section, the definitions are etended to cover an angle. If is an acute angle, these definitions coincide with those given in the preceding section. Definitions of Trigonometric Functions of An Angle Let be an angle in standard position with, a point on the terminal side of and r 0. sin cos r r (, ) tan cot, 0, 0 sec r, 0 csc r, 0 r Because r cannot be zero, it follows that the sine and cosine functions are defined for an real value of. However, if 0, the tangent and secant of are undefined. For eample, the tangent of 90 is undefined. Similarl, if 0, the cotangent and cosecant of are undefined. Video Eample Evaluating Trigonometric Functions. (, ) r Let, be a point on the terminal side of. Find the sine, cosine, and tangent of. Referring to Figure, ou can see that,, and r 5 5. So, ou have the following. sin r 5 cos r 5 FIGURE tan Now tr Eercise.