6.2 Trigonometric Functions: Unit Circle Approach

Transcription

1 SECTION. Trigonometric Functions: Unit Circle Aroach [Note: There is a 90 angle between the two foul lines. Then there are two angles between the foul lines and the dotted lines shown. The angle between the two dotted lines outside the 00-foot foul lines is 9.] Elaining Concets: Discussion and Writing 8. Do ou refer to measure angles using degrees or radians? rovide justification and a rationale for our choice. 9. What is radian? What is degree? 0. Which angle has the larger measure: degree or radian? Or are the equal?. Elain the difference between linear seed and angular seed.. For a circle of radius r,a central angle of degrees subtends an arc whose length s is s = u. Discuss whether this is a 80 ru true or false statement. Give reasons to defend our osition. 7. ulles Two ulles, one with radius r and the other with radius r, are connected b a belt. The ulle with radius r rotates at v revolutions er minute, whereas the ulle with radius r rotates at revolutions er minute. Show that r = v. r v v. Discuss wh shis and airlanes use nautical miles to measure distance. Elain the difference between a nautical mile and a statute mile.. Investigate the wa that seed biccles work. In articular, elain the differences and similarities between 5-seed and 9-seed derailleurs. Be sure to include a discussion of linear seed and angular seed. 5. In Eamle, we found that the distance between Albuquerque, New Meico, and Glasgow, Montana, is aroimatel 90 miles. According to maquest.com, the distance is aroimatel 00 miles. What might account for the difference? Are You reared? Answers. ; A = r C = r. r # t. Trigonometric Functions: Unit Circle Aroach REARING FOR THIS SECTION Before getting started, review the following: Geometr Essentials (Aendi A, Section A.,. A A9) Unit Circle (Section.,. 5) Now Work the Are You reared? roblems on age 75. Smmetr (Section.,. ) Functions (Section.,. 5) OBJECTIVES Find the Eact Values of the Trigonometric Functions Using a oint on the Unit Circle (. 5) Find the Eact Values of the Trigonometric Functions of Quadrantal Angles (. ) Find the Eact Values of the Trigonometric Functions of (. 8) = 5 Find the Eact Values of the Trigonometric Functions of and = 0 (. 9) = 0 5 Find the Eact Values of the Trigonometric Functions for Integer Multiles of and (. 7) = 0, = 5, = 0 Use a Calculator to Aroimate the Value of a Trigonometric Function (. 7) 7 Use a Circle of Radius r to Evaluate the Trigonometric Functions (. 7) We now introduce the trigonometric functions using the unit circle. The Unit Circle Recall that the unit circle is a circle whose radius is and whose center is at the origin of a rectangular coordinate sstem. Also recall that an circle of radius r has

2 CHATER Trigonometric Functions circumference of length r. Therefore, the unit circle radius = has a circumference of length. In other words, for revolution around the unit circle the length of the arc is units. The following discussion sets the stage for defining the trigonometric functions using the unit circle. Let t be an real number. We osition the t-ais so that it is vertical with the ositive direction u. We lace this t-ais in the -lane so that t = 0 is located at the oint, 0 in the -lane. If t Ú 0, let s be the distance from the origin to t on the t-ais. See the red ortion of Figure 8(a). Now look at the unit circle in Figure 8(a). Beginning at the oint, 0 on the unit circle, travel s = t units in the counterclockwise direction along the circle, to arrive at the oint =,. In this sense, the length s = t units is being wraed around the unit circle. If t 0, we begin at the oint, 0 on the unit circle and travel s = ƒtƒ units in the clockwise direction to arrive at the oint =,. See Figure 8. Figure 8 (, ) s t units t (, 0) s t units 0 (, 0) 0 s t units t (a) s t units (, ) If t 7 or if t -, it will be necessar to travel around the unit circle more than once before arriving at the oint. Do ou see wh? Let s describe this rocess another wa. icture a string of length s = ƒtƒ units being wraed around a circle of radius unit. We start wraing the string around the circle at the oint, 0. If t Ú 0, we wra the string in the counterclockwise direction; if t 0, we wra the string in the clockwise direction. The oint =, is the oint where the string ends. This discussion tells us that, for an real number t, we can locate a unique oint =, on the unit circle. We call the oint on the unit circle that corresonds to t. This is the imortant idea here. No matter what real number t is chosen, there is a unique oint on the unit circle corresonding to it.we use the coordinates of the oint =, on the unit circle corresonding to the real number t to define the si trigonometric functions of t. DEFINITION Let t be a real number and let =, be the oint on the unit circle that corresonds to t. The sine function associates with t the -coordinate of and is denoted b sin t = The cosine function associates with t the -coordinate of and is denoted b cos t =

3 SECTION. Trigonometric Functions: Unit Circle Aroach 5 In Words The sine function takes as inut a real number t that corresonds to a oint (, )on the unit circle and oututs the -coordinate. The cosine function takes as inut a real number t that corresonds to a oint (, )on the unit circle and oututs the -coordinate. If Z 0, the tangent function associates with t the ratio of the -coordinate to the -coordinate of and is denoted b tan t = If Z 0, the cosecant function is defined as csc t = If Z 0, the secant function is defined as sec t = If Z 0, the cotangent function is defined as cot t = Notice in these definitions that if = 0, that is, if the oint is on the -ais, then the tangent function and the secant function are undefined. Also, if = 0, that is, if the oint is on the -ais, then the cosecant function and the cotangent function are undefined. Because we use the unit circle in these definitions of the trigonometric functions, the are sometimes referred to as circular functions. Find the Eact Values of the Trigonometric Functions Using a oint on the Unit Circle Figure 9, t (, 0) WARNING When writing the values of the trigonometric functions, do not forget the argument of the function. sin t = sin = EXAMLE correct incorrect Finding the Values of the Si Trigonometric Functions Using a oint on the Unit Circle Let t be a real number and let = a - be the oint on the unit circle that, b corresonds to t. Find the values of sin t, cos t, tan t, csc t, sec t, and cot t. See Figure 9. We follow the definition of the si trigonometric functions, using = a - Then, with = - and = we have, b =,., sin t = = cos t = = - csc t = = = sec t = = = - cot t = - = - Now Work ROBLEM tan t = = = - - = -

4 CHATER Trigonometric Functions Trigonometric Functions of Angles Let =, be the oint on the unit circle corresonding to the real number t. See Figure 0(a). Let ube the angle in standard osition, measured in radians, whose terminal side is the ra from the origin through and whose arc length is t. See Figure 0. Since the unit circle has radius unit, if s = t units, then from the arc length formula s = ru, we have u = t radians. See Figures 0(c) and (d). Figure 0 (, ) (a) t (, 0) (, ) t (, 0) (, ) (c) s t units, t 0 t radians (, 0) t radians (, 0) s t units, t 0 (, ) (d) The oint =, on the unit circle that corresonds to the real number t is also the oint on the terminal side of the angle u = t radians. As a result, we can sa that sin t = sin u c c Real number u = t radians and so on. We can now define the trigonometric functions of the angle u. DEFINITION If u = t radians, the si trigonometric functions of the angle U are defined as sin u = sin t cos u = cos t tan u = tan t csc u = csc t sec u = sec t cot u = cot t Even though the trigonometric functions can be viewed both as functions of real numbers and as functions of angles, it is customar to refer to trigonometric functions of real numbers and trigonometric functions of angles collectivel as the trigonometric functions. We shall follow this ractice from now on. If an angle uis measured in degrees, we shall use the degree smbol when writing a trigonometric function of u, as, for eamle, in sin 0 and tan 5. If an angle uis measured in radians, then no smbol is used when writing a trigonometric function of u, as, for eamle, in cos and sec. Finall, since the values of the trigonometric functions of an angle uare determined b the coordinates of the oint =, on the unit circle corresonding to u, the units used to measure the angle u are irrelevant. For eamle, it does not matter whether we write u = radians or u = 90. The oint on the unit circle corresonding to this angle is = 0,. As a result, sin = sin 90 = and cos = cos 90 = 0 Find the Eact Values of the Trigonometric Functions of Quadrantal Angles To find the eact value of a trigonometric function of an angle uor a real number t requires that we locate the oint =, on the unit circle that corresonds to t. This is not alwas eas to do. In the eamles that follow, we will evaluate the

5 SECTION. Trigonometric Functions: Unit Circle Aroach 7 trigonometric functions of certain angles or real numbers for which this rocess is relativel eas. A calculator will be used to evaluate the trigonometric functions of most other angles. EXAMLE Finding the Eact Values of the Si Trigonometric Functions of Quadrantal Angles Find the eact values of the si trigonometric functions of: (a) u = 0 = 0 u = = 90 (c) u = = 80 (d) u = = 70 Figure (a) The oint on the unit circle that corresonds to u = 0 = 0 is =, 0. See Figure (a). Then sin 0 = sin 0 = = 0 cos 0 = cos 0 = = (, 0) 0 0 (a) (0, ) π 90 tan 0 = tan 0 = = 0 sec 0 = sec 0 = = Since the -coordinate of is 0, csc 0 and cot 0 are not defined. The oint on the unit circle that corresonds to u = is = 0,. See Figure. Then = 90 sin = sin 90 = = cos = cos 90 = = 0 csc = csc 90 = = cot = cot 90 = = 0 Since the -coordinate of is 0, tan and sec are not defined. (c) The oint on the unit circle that corresonds to u = = 80 is = -, 0. See Figure (c). Then sin = sin 80 = = 0 cos = cos 80 = = - (, 0) θ π 80 (c) π 70 tan = tan 80 = = 0 sec = sec 80 = = - Since the -coordinate of is 0, csc and cot are not defined. (d) The oint on the unit circle that corresonds to u = is = 0, -. See Figure (d). Then = 70 sin = sin 70 = = - cos = cos 70 = = 0 csc = csc 70 = = - cot = cot 70 = = 0 (0, ) (d) Since the -coordinate of is 0, tan and sec are not defined. Table on the net age summarizes the values of the trigonometric functions found in Eamle.

6 8 CHATER Trigonometric Functions Table Quadrantal Angles U (Radians) U (Degrees) sin U cos U tan U csc U sec U cot U Not defined Not defined 90 0 Not defined Not defined Not defined - Not defined 70-0 Not defined - Not defined 0 There is no need to memorize Table. To find the value of a trigonometric function of a quadrantal angle, draw the angle and al the definition, as we did in Eamle. EXAMLE Finding Eact Values of the Trigonometric Functions of Angles That Are Integer Multiles of Quadrantal Angles Find the eact value of: (a) sin cos- 70 (a) See Figure (a). The oint on the unit circle that corresonds to u = is = -, 0, so sin = = 0. See Figure. The oint on the unit circle that corresonds to u = - 70 is = 0,, so cos- 70 = = 0. Figure (0, ) (, 0) 70 (a) Now Work ROBLEMS AND Find the Eact Values of the Trigonometric Functions of 5 EXAMLE Finding the Eact Values of the Trigonometric Functions of 5 Find the eact values of the si trigonometric functions of = 5. We seek the coordinates of the oint =, on the unit circle that corresonds to u = See Figure. First, observe that lies on the line =. (Do ou = 5.

7 SECTION. Trigonometric Functions: Unit Circle Aroach 9 Figure (, ) 5 see wh? Since u = 5 = # 90, must lie on the line that bisects quadrant I.) Since =, also lies on the unit circle, + =, it follows that + = + = = =, 7 0, 7 0 = = = sin Then = sin 5 = cos csc = csc 5 = = sec = sec 5 = = cos 5 = tan = cot = tan 5 = = cot 5 = = = EXAMLE 5 Finding the Eact Value of a Trigonometric Eression Find the eact value of each eression. (a) sin 5 cos 80 tan - sin (c) asec b + csc (a) (c) sin 5 cos 80 = # - = - From Eamle From Table tan - sin = - - = c c From Eamle From Table c asec b + csc = A B + = + = c Now Work ROBLEM 5 Find the Eact Values of the Trigonometric Functions of 0 and 0 Consider a right triangle in which one of the angles is. It then follows that = 0 the third angle is. Figure (a) illustrates such a triangle with hotenuse = 0 of length. Our roblem is to determine a and b.

8 70 CHATER Trigonometric Functions We begin b lacing net to this triangle another triangle congruent to the first, as shown in Figure. Notice that we now have a triangle whose three angles each equal 0. This triangle is therefore equilateral, so each side is of length. Figure c b c b c 0 b 0 a (a) 0 0 a a 0 a (c) This means the base is a =, and so a = B the thagorean Theorem,. b satisfies the equation a + b = c, so we have a + b = c + b = b = - b = = a =, c = b 7 0 because b is the length of the side of a triangle. This results in Figure (c). EXAMLE Finding the Eact Values of the Trigonometric Functions of 0 Find the eact values of the si trigonometric functions of = 0. Figure 5 (, ), 0 ( ) osition the triangle in Figure (c) so that the 0 angle is in standard osition. See Figure 5. The oint on the unit circle that corresonds to u = is = 0 = a Then, b. sin csc = csc 0 = tan = sin 0 = cos = cos 0 = = tan 0 = = = sec = sec 0 = = cot = cot 0 = = = =

9 SECTION. Trigonometric Functions: Unit Circle Aroach 7 Figure 0 EXAMLE 7 (, ), ( ) Finding the Eact Values of the Trigonometric Functions of 0 Find the eact values of the trigonometric functions of = 0. osition the triangle in Figure (c) so that the 0 angle is in standard osition. See Figure. The oint on the unit circle that corresonds to u = is = a Then, = 0 b. sin = sin 0 = cos csc = csc 0 = tan = tan 0 = = sec = sec 0 = = = cot = cos 0 = = cot 0 = Table summarizes the information just derived for and = 0, = 5, Until ou memorize the entries in Table, ou should draw an aroriate = 0. diagram to determine the values given in the table. = = = Table U (Radians) U (Degrees) sin U cos U tan U csc U sec U cot U Now Work ROBLEM Figure 7 EXAMLE 8 in Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets inches wide. After marking off a length of inches from each edge, this length is bent u at an angle u. See Figure 7. The area A of the oening ma be eressed as a function of uas in in in Au = sin ucos u + in θ in θ Find the area A of the oening for u = 0, u = 5, and u = 0. For u = 0 : A0 = sin 0 cos 0 + = a ba + b = + 8 L.9 The area of the oening for u = 0 is about.9 square inches.

10 7 CHATER Trigonometric Functions For u = 5 : A5 = sin 5 cos 5 + = a ba The area of the oening for u = 5 is about 9. square inches. For u = 0 : A0 = sin 0 cos 0 + = a ba + b = L 9. + b = L 0.8 The area of the oening for u = 0 is about 0.8 square inches. Figure 8 (, ) 5 7 (, (, ) (, ) ) 5 Find the Eact Values of the Trigonometric Functions for Integer Multiles of and 0, 5, We know the eact values of the trigonometric functions of Using = 5. smmetr, we can find the eact values of the trigonometric functions of 7 and = 5, 5 = 5, = 5. See Figure 8. Using smmetr with resect to the -ais, the oint a -, b is the oint on the unit circle that corresonds to the angle Similarl, using smmetr with resect to the origin, the oint a - the oint on the unit circle that corresonds to the angle smmetr with resect to the -ais, the oint circle that corresonds to the angle 7 = 5. a, - b 5 = 5. 0 = 5., - b is Finall, using is the oint on the unit EXAMLE 9 Finding Eact Values for Multiles of Find the eact value of each eression. (a) cos 5 sin 5 (c) tan 5 (d) sina- (e) cos b (a) From Figure 8, we see the oint a- 5 corresonds to so cos 5, - b, = =-. Since 5 = the oint a- corresonds to 5, so sin 5 =,, b. (c) Since 5 = 7 the oint a corresonds to 5, so, -, b tan 5 = - = -. 5

11 SECTION. Trigonometric Functions: Unit Circle Aroach 7 (e) The oint a- corresonds to so, b, Now Work ROBLEMS 5 AND 55 cos (d) The oint a corresonds to - so sina-, - b, b =-. =-. The use of smmetr also rovides information about certain integer multiles of the angles and See Figures 9 and 0. = 0 = 0. Figure 9 Figure 0 (, ) (, ) (, ) (, ) 7 5 ( (, ), ) 5 (, ) (, ) EXAMLE 0 WARNING On our calculator the second functions sin, cos, and tan do not reresent the recirocal of sin, cos, and tan. EXAMLE Finding Eact Values for Multiles of Based on Figures 9 and 0, we see that (a) (c) cos 0 = cos 7 = - tan 5 = - = - Now Work ROBLEM 7 0 or Use a Calculator to Aroimate the Value of a Trigonometric Function Before getting started, ou must first decide whether to enter the angle in the calculator using radians or degrees and then set the calculator to the correct MODE. Check our instruction manual to find out how our calculator handles degrees and radians. Your calculator has kes marked sin, cos, and tan. To find the values of the remaining three trigonometric functions, secant, cosecant, and cotangent, we use the fact that, if =, is a oint on the unit circle on the terminal side of u, then (d) sin- 0 = sina- b =- 8 = + T cos 8 = cos =- sec u = = cos u csc u = = sin u cot u = = Using a Calculator to Aroimate the Value of a Trigonometric Function Use a calculator to find the aroimate value of: 0 = tan u (a) cos 8 csc (c) tan Eress our answer rounded to two decimal laces.

12 7 CHATER Trigonometric Functions Figure (a) First, we set the MODE to receive degrees. Rounded to two decimal laces, cos 8 = 0.90 L 0.7 Most calculators do not have a csc ke. The manufacturers assume that the user knows some trigonometr. To find the value of csc, use the fact that csc = Rounded to two decimal laces, sin. csc L.79 (c) Set the MODE to receive radians. Figure shows the solution using a TI-8 lus grahing calculator. Rounded to two decimal laces, tan L 0.7 Now Work ROBLEM 5 Figure (, ) * (*, *) * A A* * O r r 7 Use a Circle of Radius r to Evaluate the Trigonometric Functions Until now, finding the eact value of a trigonometric function of an angle urequired that we locate the corresonding oint =, on the unit circle. In fact, though, an circle whose center is at the origin can be used. Let ube an nonquadrantal angle laced in standard osition. Let =, be the oint on the circle + = r that corresonds to u, and let * = *, * be the oint on the unit circle that corresonds to u. See Figure, where uis shown in quadrant II. Notice that the triangles OA** and OA are similar; as a result, the ratios of corresonding sides are equal. * = * r = * r * = * = r * = r * * = These results lead us to formulate the following theorem: THEOREM For an angle uin standard osition, let =, be the oint on the terminal side of uthat is also on the circle + = r. Then sin u = r cos u = r tan u = Z 0 csc u = r Z 0 sec u = r Z 0 cot u = Z 0 Figure EXAMLE r 5 (, ) 5 Finding the Eact Values of the Si Trigonometric Functions Find the eact values of each of the si trigonometric functions of an angle u if, - is a oint on its terminal side in standard osition. See Figure. The oint, - is on a circle of radius r = + - = + 9 = 5 = 5 with the center at the origin. For the oint, =, -, we have = and = -. Since r = 5, we find sin u = r = - cos u = csc u = r 5 r = tan u = = - 5 sec u = r 5 = - = 5 cot u = = - Now Work ROBLEM 77

13 SECTION. Trigonometric Functions: Unit Circle Aroach 75 Historical Feature The name sine for the sine function is due to a medieval confusion. The name comes from the Sanskrit word jıva (meaning chord), first used in India b Arabhata the Elder (AD 50). He reall meant half-chord, but abbreviated it. This was brought into Arabic as jiba, which was meaningless. Because the roer Arabic word jaib would be written the same wa (short vowels are not written out in Arabic), jıba was ronounced as jaib, which meant bosom or hollow, and jıba remains as the Arabic word for sine to this da. Scholars translating the Arabic works into Latin found that the word sinus also meant bosom or hollow, and from sinus we get the word sine. The name tangent, due to Thomas Finck (58), can be understood b looking at Figure. The line segment DC is tangent to the circle at C. If d(o, B) = d(o, C) =, then the length of the line segment DC is d(d, C) = d(d, C) = d(d, C) d(o, C) = tan a The old name for the tangent is umbra versa (meaning turned shadow), referring to the use of the tangent in solving height roblems with shadows. The names of the remaining functions came about as follows. If a and b are comlementar angles, then cos a = sin b. Because b is the comlement of a, it was natural to write the cosine of a as sin co a. robabl for reasons involving ease of ronunciation, the co migrated to the front, and then cosine received a three-letter abbreviation to match sin, sec, and tan. The two other cofunctions were similarl treated, ecet that the long forms cotan and cosec survive to this da in some countries. Figure O B A D C. Assess Your Understanding Are You reared? Answers are given at the end of these eercises. If ou get a wrong answer, read the ages listed in red.. In a right triangle, with legs a and b and hotenuse c, the thagorean Theorem states that. (. A). The value of the function f = - 7 at 5 is. (. 5). True or False For a function = f, for each in the domain, there is eactl one element in the range. (. 5). If two triangles are similar, then corresonding angles are and the lengths of corresonding sides are. (. A A9) Concets and Vocabular 7. The function takes as inut a real number t that corresonds to a oint = (, ) on the unit circle and oututs the -coordinate. 8. The oint on the unit circle that corresonds to u = is =. 9. The oint on the unit circle that corresonds to u = is = Skill Building. 5. What oint is smmetric with resect to the -ais to the oint a (. ), b?. If, is a oint on the unit circle in quadrant IV and if = what is? (. 5), 0. The oint on the unit circle that corresonds to u = is =.. For an angle u in standard osition, let = (, ) be the oint on the terminal side of u that is also on the circle + = r. Then, sin u = and cos u =.. True or False Eact values can be found for the sine of an angle. In roblems 0, =, is the oint on the unit circle that corresonds to a real number t. Find the eact values of the si trigonometric functions of t.. a, b. a, - b 5. a - 5, 5 b. a - 5, 5 b 7. a -, b 8. a, b 9. a, - b 0. a - 5, - b

14 7 CHATER Trigonometric Functions In roblems 0, find the eact value. Do not use a calculator.. sin. sec8 7. cosa-. cos7. tan. cot 7 5. csc b 8. sin- 9. sec- 0. tan- In roblems, find the eact value of each eression. Do not use a calculator.. sin 5 + cos 0. sin 0 - cos 5. sin 90 + tan 5. cos 80 - sin sin 5 cos 5. tan 5 cos 0 7. csc 5 tan 0 8. sec 0 cot 5 9. sin 90 - tan cos 90-8 sin 70. sin - tan. sin + tan. sec + cot. csc + cot 5. csc + cot. sec - csc In roblems 7, find the eact values of the si trigonometric functions of the given angle. If an are not defined, sa not defined. Do not use a calculator sin 8. cos 7. sec 8. cot tan sin In roblems 5 7, use a calculator to find the aroimate value of each eression rounded to two decimal laces. 7. cot 7. csc 5 7. sin 7. tan 75. sin 7. tan In roblems 77 8, a oint on the terminal side of an angle u in standard osition is given. Find the eact value of each of the si trigonometric functions of u , 78. 5, - 79., , , , 8. a, b 8. 0., Find the eact value of: sin 5 + sin 5 + sin 5 + sin 5 8. Find the eact value of: tan 0 + tan Find the eact value of: sin 0 + sin 0 + sin 0 + sin Find the eact value of: tan 0 + tan If fu = sin u = 0., find fu If fu = cos u = 0., find fu If fu = tan u =, find fu If fu = cot u = -, find fu If sin u = find csc u. 5, 9. If cos u = find sec u., In roblems 95 0, fu = sin uand gu = cos u. Find the eact value of each function below if u = 0. Do not use a calculator. 95. fu 9. gu 97. fa u b 98. ga u b 99. fu 00. gu 0. fu 0. gu 0. fu 0. gu 05. f- u 0. g- u

15 SECTION. Trigonometric Functions: Unit Circle Aroach Mied ractice In roblems 07, f() = sin, g() = cos, h() =, and () = 07. (f + g)(0 ) 08. (f - g)(0 ) 0. (f # g) a. (f # h) a b b 77. Find the value of each of the following: 09. (f # g) a b. (g # )(0 ). (h # f) a. ( # g)(5 ) 5 b. (a) Find ga b. What oint is on the grah of g? 5. (a) Find fa b. What oint is on the grah of f? Assuming g is one-to-one*, use the result of art (a) to Assuming f is one-to-one*, use the result of art (a) to find a oint on the grah of g -. find a oint on the grah of f -. (c) What oint is on the grah of = g a (c) What oint is on the grah of = fa + b- b if =? if =? Alications and Etensions 7. Find two negative and three ositive angles, eressed in radians, for which the oint on the unit circle that corresonds to each angle is a, b. 8. Find two negative and three ositive angles, eressed in radians, for which the oint on the unit circle that corresonds to each angle is a - 9. Use a calculator in radian mode to comlete the following table. sin u What can ou conclude about the value of fu = as u aroaches 0? u 0.5 U , b sin u f u = sin u u 0. Use a calculator in radian mode to comlete the following table. cos u - What can ou conclude about the value of gu = as u aroaches 0? u 0.5 U cos u - gu = cos u - u For roblems, use the following discussion. rojectile Motion The ath of a rojectile fired at an inclination u to the horizontal with initial seed v0 is a arabola (see the figure). The range R of the rojectile, that is, the horizontal distance that the rojectile travels, is found b using the function Ru = v0 = Initial seed Height, H θ v0 sinu g where g L. feet er second er second L 9.8 meters er second er second is the acceleration due to gravit. The maimum height H of the rojectile is given b the function Range, R *In Section 7., we discuss the necessar domain restriction so that the function is one-to-one. Hu = v0 sin u g

16 78 CHATER Trigonometric Functions In roblems, find the range R and maimum height H.. The rojectile is fired at an angle of 5 to the horizontal with an initial seed of 00 feet er second.. The rojectile is fired at an angle of 0 to the horizontal with an initial seed of 50 meters er second.. The rojectile is fired at an angle of 5 to the horizontal with an initial seed of 500 meters er second.. The rojectile is fired at an angle of 50 to the horizontal with an initial seed of 00 feet er second. 5. Inclined lane See the figure. distance of mile from a aved road that arallels the ocean. See the figure. Ocean Beach aved ath mi River mi mi θ a If friction is ignored, the time t (in seconds) required for a block to slide down an inclined lane is given b the function a tu = A g sin u cos u where a is the length (in feet) of the base and g L feet er second er second is the acceleration due to gravit. How long does it take a block to slide down an inclined lane with base a = 0 feet when: (a) u = 0? u = 5? (c) u = 0?. iston Engines In a certain iston engine, the distance (in centimeters) from the center of the drive shaft to the head of the iston is given b the function u = cos u cosu where uis the angle between the crank and the ath of the iston head. See the figure. Find when u = 0 and when u = 5. Sall can jog 8 miles er hour along the aved road, but onl miles er hour in the sand on the beach. Because of a river directl between the two houses, it is necessar to jog in the sand to the road, continue on the road, and then jog directl back in the sand to get from one house to the other.the time T to get from one house to the other as a function of the angle ushown in the illustration is Tu = + sin u -, 0 u 90 tan u (a) Calculate the time T for u = 0. How long is Sall on the aved road? Calculate the time T for u = 5. How long is Sall on the aved road? (c) Calculate the time T for u = 0. How long is Sall on the aved road? (d) Calculate the time T for u = 90. Describe the ath taken. Wh can t the formula for T be used? 8. Designing Fine Decorative ieces A designer of decorative art lans to market solid gold sheres encased in clear crstal cones. Each shere is of fied radius R and will be enclosed in a cone of height h and radius r. See the illustration. Man cones can be used to enclose the shere, each having a different slant angle u. The volume V of the cone can be eressed as a function of the slant angle uof the cone as Vu = + sec u R tan u, 0 u 90 What volume V is required to enclose a shere of radius centimeters in a cone whose slant angle is 0? 5? 0? u θ h R O r θ 7. Calculating the Time of a Tri Two oceanfront homes are located 8 miles aart on a straight stretch of beach, each a 9. rojectile Motion An object is roelled uward at an angle u, 5 u 90, to the horizontal with an initial

17 SECTION. roerties of the Trigonometric Functions 79 velocit of v 0 feet er second from the base of an inclined lane that makes an angle of 5 with the horizontal. See the illustration. If air resistance is ignored, the distance R that it travels u the inclined lane as a function of uis given b Ru = v 0 sinu - cosu - (a) Find the distance R that the object travels along the inclined lane if the initial velocit is feet er second and u = 0. Grah R = Ru if the initial velocit is feet er second. (c) What value of umakes R largest? θ 0. If u, 0 u, is the angle between the ositive -ais and a nonhorizontal, nonvertical line L, show that the sloe m of L equals tan u. The angle uis called the inclination of L. R [Hint: See the illustration, where we have drawn the line M arallel to L and assing through the origin. Use the fact that M intersects the unit circle at the oint cos u, sin u. ] Elaining Concets: Discussion and Writing. Write a brief aragrah that elains how to quickl comute the trigonometric functions of 0, 5, and 0.. Write a brief aragrah that elains how to quickl comute the trigonometric functions of 0, 90, 80, and 70. Are You reared? Answers 5 (cos, sin ) M O In roblems and, use the figure to aroimate the value of the si trigonometric functions at t to the nearest tenth. Then use a calculator to aroimate each of the si trigonometric functions at t. b (a) t = t = 5.. (a) t = t = 5 L Unit Circle 5. How would ou elain the meaning of the sine function to a fellow student who has just comleted college algebra? a.. 8. True. equal; roortional 5. a -. -, c = a + b b. roerties of the Trigonometric Functions REARING FOR THIS SECTION Before getting started, review the following: Functions (Section.,. 5) Identit (Aendi A, Section A.,. A) Now Work the Are You reared? roblems on age 90. Even and Odd Functions (Section.,. 9 70) OBJECTIVES Determine the Domain and the Range of the Trigonometric Functions (. 80) Determine the eriod of the Trigonometric Functions (. 8) Determine the Signs of the Trigonometric Functions in a Given Quadrant (. 8) Find the Values of the Trigonometric Functions Using Fundamental Identities (. 8) 5 Find the Eact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle (. 8) Use Even Odd roerties to Find the Eact Values of the Trigonometric Functions (. 89)