Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of

Transcription

1 Oiginall TRIGONOMETRY was that banch of mathematics concened with solving tiangles using tigonometic atios which wee seen as popeties of tiangles athe than of angles. The wod Tigonomet comes fom the Geek wods: Teis thee, Goniaangle, and Metonmeasue. Theefoe, the wod tigonomet means measuement of tiangles. This unit will eploe tigonomet as it elates to ight tiangles. Math 99 Intemediate lgeba Whatcom ommunit ollege evised b H Ypma Winte 00

2 Intoduction to Tigonomet Tigonomet is the banch of mathematics that studies the elationship between angles and sides of a tiangle. Using special tig functions, ou will be able to calculate the height of a tee fom its shadow o the distance acoss a wide lake b walking and measuing along its shoe. Let s begin b woking with a RIGHT TRINGLE (one with a 90 angle in it) and using it to define those special tig functions we mentioned ealie. Fist of all, we will use standad notation when labeling ou ight tiangles. apital lettes will epesent the vetices o angles;,, and will be ou standad lettes. will be used to epesent the ight angle. We will use lowe case lettes to epesent the sides; a, b, and c will be the standad choices hee. The side opposite the angle labeled will be side a, the side opposite angle will be b and the side opposite angle will be c. b a c In a Right Tiangle, the ight angle will alwas measue 90 and the two othe angles will add up to 90. You could have, fo eample, a ight tiangle with angles measuing 90, 4, and 4, o ou could have a ight tiangle with angles measuing 90,, and. Notice that all thee angles in a tiangle ( ight tiangle o not) will alwas add up to 0 degees. If ou stood at the 90 angle, (the ight angle) and pointed at the longest side of the tiangle (which is opposite the 90 angle) ou would be looking at the HYPOTENUSE. The hpotenuse alwas has the longest measuement of the thee sides. Now let s talk biefl about the othe two angles. In the dawing at the ight, we also have angle and angle. If ou stand at angle, and point to the side diectl in font of ou, ou would see the side OPPOSITE the angle. Since ou have found the hpotenuse side alead, and ou know the side opposite the angle, what do we call the emaining side of the tiangle? It is called the side DJENT to the angle. This side is alwas net to the angle. opposite adjacent hpotenuse

3 Since we ae woking with a ight tiangle, the Pthagoean theoem applies: a + b c OR (side adjacent) + (side opposite) (hpotenuse) HEK FOR UNDERSTNDING: Daw a tiangle and label the angles simila to the one illustated in the eample. Once ou have dawn the tiangle, wite the given measuements net to the appopiate side. Find the measuements of the hpotenuse, the side opposite angle, and the side adjacent to angle. Do the same fo angle. ( the hpotenuse will sta the same, but the othe two sides change in thei oientation to angle ) Veif that the Pthagoean theoem woks. (one of the questions doesn t wok) EXMPLE: In tiangle :,, 4 Find the measuements of the hpotenuse, the side opposite angle, and the side adjacent to. Do the same fo angle. Veif that this is a ight tiangle using the Pthagoean Theoem. Solution: 4 Hpotenuse: Side opposite : 4 Side opposite : Side adjacent : Side adjacent : 4 () + (4) () QUESTIONS:.,,.,,.,, 4. 6, 4,

4 Right Tiangle Tigonomet The basic tigonometic functions ae the atios of the lengths of the sides of ight tiangles. The thee fundamental tigonometic functions ae: sine which is abbeviated as sin cosine which is abbeviated as cos tangent which is abbeviated as tan Let us once again conside the standad ight tiangle and look at the atios that these basic tig functions epesent in elationship to. opposite opp sin hpotenuse hp adjacent hpotenuse adjacent adj cos hpotenuse hp opposite opposite opp tan adjacent adj Notice that in the pevious diagam, the sides have been labeled in efeence to angle. We could also look at these thee elationships in the tiangle with elationship to angle. In the eample below, we will use the standad notation to identif both the sides and the angles of a ight tiangle. We can look at both angles and and the values of thei tigonometic functions. a sin opp sin b c hp c b c b cos adj cos a c hp c a a tan opp tan b b adj a Once we undestand the elationships given b this standad tiangle and the thee basic tig functions, we can then eplace the vaiables with actual numbes.

5 Eample ; Given the tiangle below, find (a) sin nswes: a) 4 4 (b) sin b) (c) cos c) 4 (d) tan d) 4 Thee ae actuall si tigonometic functions. The ecipocals of the atios that define the sine, cosine, and tangent functions ae used to define the emaining thee tigonometic functions. These ae: cosecant which is abbeviated as csc secant which is abbeviated as sec cotangent which is abbeviated as cot The elationships of these tigonometic functions in a standad ight tiangle ae shown below. c hp csc a opp b c c hp sec b adj a b adj cot a opp Eample : Given the tiangle below, find (a) cot nswes: a) 4 4 (b) csc b) (c) sec c) (d) csc d) 4 4

6 nothe View of Right Tiangle Tigonomet When woking with ight tiangles we do not alwas label ou vetices,, and. Sometimes we just use a smbol to indicate the angle of the tiangle we wish to use. The most common labels used fo angles in tig ae the: ponounced theta, α ponounced alpha, and β ponounced beta. Often in tigonomet we view the tiangle in the fist quadant of a ectangula coodinate sstem. In this wa, the tigonometic functions can be viewed in tems of and. (, ) sin cos tan Notice in this epesentation, the acute angle is between the -ais and a line segment fom the oigin to a point (, ) in the fist quadant. Notice that the legs (anothe wod used to descibe the sides foming the ight angle) of the ight tiangle take on the value of the coodinates of the given point. The hpotenuse is enamed as because this is actuall the adius of a cicle. Let s look at an eample using the notation that we have just discussed. Eample : In this tiangle find the si tigonometic functions of. nswes: One leg, Othe leg, (,) sin cos tan csc sec cot

7 Evaluating Tigonometic Functions Now that we have looked at the si diffeent tigonometic functions and was in which we can epesent a ight tiangle, let s look at a couple moe eamples of woking with these new functions. Eample : Find the values of the si tigonometic functions of angleα in the ight tiangle shown below. Solution: Notice that ou have been given onl two sides of the ight tiangle. You fist need to find the thid side. the Pthagoean Theoem, (hp) (opp) + (adj), it follows that (hp) + α (hp) + 64 (hp) 9 hp 9 hp 7 We now have the following elationships: adj α, opp α, and hp 7. The si tigonometic elationships ae: sinα cosα tanα cscα secα cotα Eample : Given the ight tiangle below, find the value of each tigonometic function. (a) sin (b) cos (c) tan Solution: Fist, we appl the Pthagoean theoem to find the length of the leg opposite angle. Letting a be the length of this leg, we find a + b c a + a a 0 a 0 4 Now using the tigonometic atios, we obtain the values of the indicated tigonometic functions. (Notice that values ae given in educed adical fom to peseve accuac.) 4 4 a) sin b) cos c) tan 6

8 Finding ngle Measues In the following eample we ae confonted with a poblem of the following tpe: sin 0.6 Up until now we ve looked onl at the atio of a specific angle, now lets find the actual angle measue the atio epesents. Fist, we note that the poblem can be witten smbolicall as eithe sin 0.6 o acsin 0.6 oth of these epessions ae ead as is the angle whose sine is 0.6. Notice on ou calculato that ou have the smbol sin located ight above the sin button. Like man of the othe functions on the calculato the function and its invese can be found using the same button. You will see that above cos is cos, and above tan is tan. (Since these functions ae located above the actual buttons, be sue to fist pess the nd ke.) These function kes will take us fom a tigonometic atio back to the coesponding acute angle in degee measue when ou calculato is set in degee mode. Theefoe to solve the pevious poblem, we have: sin 0.6 sin aution: It is impotant to emembe that sin 0.6 does not mean The supescipt is pat of a function smbol, and invese sin function. sin sin 0.6 epesents the Eample: Find the measue of. 7.9 and. Solution: We know the side adjacent to and also the hpotenuse. This suggests the use of the cosine atio. We get. cos 7.9. cos Now, since the measue of and must total 90, we can find b subtacting:

9 Solving Right Tiangles To solve a ight tiangle means to find the lengths of all of its sides and the measues of all of its angles, povided that the ae not alead known. To solve a tiangle, we use the tigonometic atios, as needed to find the missing pats. We will look at some eamples of diffeent situations that involve finding missing infomation in a tiangle. Eample : Find the m ( ead as the measue of angle ) Solution: Since the angle measues of an tiangle add up to 0 and the ight angle measues 90, the acute angles must add up to 90 : m + m 90 so, m b 90 Eample : In this tiangle, find a and b. Solution: To solve fo a, we ecognize that the infomation we have is the angle, and the hpotenuse. The side we ae looking fo is the side 6. opposite angle. The tig function that deals with b. this elationship is the sin function. a sin 6.. Solving fo a, we have a a.sin 6. Using ou calculatos *, we find a 6.77 *Note: In ode to use a calculato to solve these poblems, make sue that ou calculato is set in DEGREE mode. We use a simila stateg to solve fo b. When solving a ight tiangle, it is best to use the oiginal infomation that was given wheneve possible to avoid ounding eos in ou answe. If we look at the oiginal poblem, b is the side adjacent to the angle given and we also know the hpotenuse. The tig function that deals with this elationship is the cos function. b cos 6.. Solving fo b, we have b.cos6. b.67 Notice we can also easil find the measue of the angle :

10 We have now successfull solved this ight tiangle. 6. a b c. Eample : Solve the ight tiangle with a. and b 6.7. Solution: In this poblem we ae not given a diagam so the fist thing that needs to be done is to daw a tiangle. We assume that standad notation is used. Net, place the infomation given in the coect locations. It now is appaent that to solve this ight tiangle, we need to find side c,, and. a. c Let s stat b finding c: c a + b c (.) + (6.7) b 6.7 c c c 6.7 Now let s find ounding. We can use this esult to find. To do so, use the oiginal infomation given, to avoid eo b 6.7 tan. 6.7 tan. 7.4 : We could have also found this angle b using a tigonometic elationship and the oiginal infomation given.. tan 6.7. tan 6.7 The tiangle is now solved a. b 6.7 c 6.7 9

11 pplications of Right Tiangles Right tiangles have man applications. To solve a poblem, we daw a ight tiangle and fo the pat associated with the application question solve the tiangle. To solve a tiangle poblem: ) Daw a sketch of the poblem situation. ) Look fo tiangles and sketch them in. ) Mak the known and unknown sides and angles. 4) Epess the desied side o angle in tems of known tigonometic atios. ) Solve. 6) nswe the question being asked. (use appopiate labels) Eample : n obseve stands on level gound, 00 metes fom the base of a TV towe, and looks up at an angle of 6. to see the top of the towe. How high is the towe above the obseve s ee level? Solution: We daw a diagam and see that a ight tiangle is fomed. The infomation we have is an h angle and the adjacent side to this angle. We need to find the opposite side. To elate these pieces of 6. infomation, we can use the tangent function. 00 m h tan 6. (solve fo h ) 00 h 00 tan tan 6. h 99.7 h The height of the towe is about 99.7 metes. Eample : kite flies at a height of 60 feet when 0 feet of sting is out. ssuming that the sting is in a staight line, what is the angle that it makes with the gound? 0 ft. 60 ft. Solution: We daw a diagam of the situation pesented. The infomation we have is the side opposite the angle we ae looking fo and the hpotenuse. To elate this infomation we use the sine function. 60 sin 0 60 sin

12 The tigonometic functions ae often used in calculating distances that ae difficult to measue diectl. In man of these instances, an angle is detemined b a hoizontal line and a line-of-sight. If a peson is looking up at an object, the acute angle measued fom the hoizontal to a line-of-sight obsevation is called the angle of elevation. If a peson is looking down at an object, the acute angle made b the line-of-sight obsevation of the object and the hoizontal is called the angle of depession. angle of depession angle of elevation Eample : suveo is standing 0 feet fom the base of a lage tee, as shown below. The suveo measues the angle of elevation to the top of the tee as 7.. How tall is the tee if the suveo is 6 feet tall? h 6 ft ft. Solution: The infomation given suggests the use of the tangent function. h tan 7. 0 h 0 tan 7. h 49.4 feet ut when we find h we have to add in the suveo s height, since we want the height of the entie tee. The height of the tee is about.4 feet. Eample : Fom the fie towe in Flatlands National Pak, a foest ange sighted a fie. To measue the angle of depession of the fie, the ange used an instument that was known to be metes above the gound. The angle of depession fom the towe to the fie was.. What was the distance between the fie and the base of the towe? m Solution: In this poblem the angle given is not pat. of the ight tiangle we ae going to solve. Howeve, we can use the given angle to detemine one of the acute angles of the tiangle we need to solve since the angle of depession alwas uses a hoizontal line. The angle inside the ight tiangle d at the top of the towe will be: We now have an angle, the side adjacent to the angle and ae looking fo the side opposite the given angle. We use the tangent function to solve. d tan 7.9 d tan metes ( )

13 eakdown of Degees When woking with degees, we have used decimal epesentations to designate pat of a degee. In the pevious eample, we looked at., which epesents an angle of degee measue and one-tenth of a degee. degee can also be divided futhe using minutes and seconds just as an hou is divided into minutes and seconds. Each degee is divided into 60 equal pats called minutes, and each minute is divided into 60 equal pats called seconds. Smbolicall, minutes ae epesented b ' and seconds b ". nothe wa to think about it is that one minute, denoted b ' is defined as 60 degee. One second, denoted b " is defined as 60 minute o, equivalentl, 600 degee. Thus, '4" is a concise wa of witing degees, minutes, and 4 seconds. Decimal degees (DD) ae useful in some instances and degees-minutes-seconds (DMS) ae useful in othes. You should be able to go fom one fom to the othe as demonstated in the eample below. Eample : onveting between Degees, Minutes, Seconds, and Decimal Foms (a) onvet 0 6'" to a decimal in degees. (b) onvet.6 to the DM ' S" fom. Round off ou answe to the neaest second. Solution: (a) ecause ' and " ', we convet as follows: '" ( ) 0.0 (b) We stat with the decimal pat of.6, that is 0.6 : 0.6 ( 0.6)( ) ( 0.6)( 60' ).6' since 60' Now we wok with the decimal pat of.6', that is, 0.6' : 0.6' ( 0.6)( ' ) ( 0.6)( 60" ).6" " Thus, ' + ' + 0.6' + ' +.6" '" It is also possible to do these convesions on the calculato.

14 SET : Evaluating Tigonometic Functions Fo each tiangle, find (a) sin (b) cos (c) tan (d) sin (e) cos (f) tan... 6 In eecises 4-, use the Pthagoean Theoem to find the thid side of the tiangle. Then find the eact values of the si tigonometic functions of the angle. (Leave answes in eact simplified fom In eecises -4, use the ight tiangle shown in the sketch to find the values of the indicated tigonometic functions.. (a) sin α (b) cos α (c) cot β (d) tan α (e) csc β (f) sec β β α. (a) tan β (b) csc α (c) sec α (d) sin β (e) cos β (f) cotα 7 α 9 β 4. (a) sin β (b) cosα (c) tan β (d) cotα (e) sec β (f) cscα α 0 β

15 SET : Evaluating Tigonometic Functions (anothe view) Refe to the figue below to solve poblems -. (, ) Fo each atio in poblems -6 identif the tigonometic function epesented in elation to Wite the atios of sides coesponding to each tigonometic function in poblems sin. cot 9. csc 0. cos. sec. tan Find each side o angle specified. (ound answes to the neaest tenth). 0, 4, find. 4., 6, find.., 0, find. 6..,., find. 7. 4, 7, find.. 60,., find ,, find. 0..7, 0, find..,., find.. 97, 7, find.. 6, 446, find. 4. 4, 6, find. Fo poblems -0 find each acute angle (in degees) to two decimal places using ou calculato.. sin cos tan 9.. sin accos actan

16 SET : pplications Involving Right Tiangles. Solve fo.. Solve fo.. Solve fo. 4. Solve fo Solve fo. 6. Solve fo. 7. Solve fo.. Solve fo Height 6-foot peson standing feet fom a steetlight casts an -foot shadow (see figue). What is the height of the steetlight? 0. Length 0-foot ladde leaning against the side of a house makes a 7 angle with the gound (see figue). How fa up the side of the house does the ladde each?. Height 6-foot man walked fom the base of a boadcasting towe diectl towad the tip of the shadow cast b the towe. When he was feet fom the towe, his shadow stated to appea beond the towe s shadow. What is the height of the towe if the man s shadow is feet long?. Length 4-foot ladde is leaning against the side of on office building. It makes a 60 angle with the sidewalk. How fa fom the building is the base of the ladde?

17 . Width of a Rive biologist wants to know the width w of a ive in ode to popel set instuments fo studing the pollutants in the wate. Fom point, the biologist walks downsteam 00 feet and sights to point. Fom this sighting, it is detemined that 0 (see figue). How wide is the ive? w 00 ft Distance Fom a 0-foot obsevation towe on the coast, a oast Guad office sights a boat in difficult. The angle of depession of the boat is 4 (see figue). How fa is the boat fom the shoeline?. ngle of elevation amp 7 feet in length ises to a loading platfom that is feet off the gound. (see figue). Find the angle 7 SET 4: Moe Tig pplications. The hpotenuse of a ight tiangle is feet. If one leg is foot, find the degee measue of each angle.. ight tiangle contains a angle. If one leg is of length inches, what is the length of the hpotenuse? (Thee ae two answes). Suppose ou ae headed towad a plateau 0 metes high. If the angle of elevation to the top of the plateau is 0, how fa ae ou fom the base of the plateau? 4. -foot etension ladde leaning against a building makes a 70 angle with the gound. How fa up the building does the ladde touch?. t 0 am on pil 6, 99 a building 00 feet high cast a shadow 0 feet long. What was the angle of elevation of the sun? 6. To measue the height of Lincoln s caicatue on Mt. Rushmoe, two sightings 00 feet fom the base of the mountain ae taken. If the angle of elevation to the bottom of Lincoln s face (his chin) is and the angle of elevation to the top (his foehead) is, how long is Lincoln s face on the mountain? 6

18 SET : Using Degees, Minutes, Seconds Solve these tiangles. Standad letteing has been used.. 6 0', a ', a ', b ', b ', b ', b ', c 90. 0', c a.0, b.0 0. a 6.0, c ', c b.0, c 4.00 Solve these tiangles.. 4. b 0' c. 7. c 4 0' b Solve the following poblems.. device fo measuing cloud height at night consists of a vetical beam of light, which makes a spot on the clouds. The spot is viewed fom a point feet awa fom the device. The angle of elevation is 67 40'. Find the height of the clouds. 6. If a jet ailine climbs at an angle of 0' with a constant speed of miles pe hou, how long will it take (to the neaest minute) to each an altitude of.00 miles? ssume thee is no wind. 7. Hoizontal distances must often be measued even though teain is not level. One wa of doing it is as follows. Distance down a slope is measued with a suveo s tape, and the distance d is measued b making a level sighting fom to a pole held veticall at, o the angle α is measued b an instument placed at. Suppose that a slope distance L is measued to be. feet and the angle α is measued to be '. Find the hoizontal distance H. α H L d 7

19 nswes Set I. a) b) c) d) e) f). a) b) c) 4 d) e) f). a) 4 b) c) 4 d) e) 4 f) 4 sin cos tan csc sec cot a) 7 b) 7 c) d) e) 7 7 f). a) 7 b) 9 c) 7 9 d) 9 7 e) 4 9 f) 7 4. a) 0 0 b) 0 0 c) d) e) 0 f) 0

20 SET II. sin 7.. csc.. cos cot 0.. tan. 6. sec SET III feet. 9. feet feet feet feet. feet. SET IV. 9., in,. in. 7.4 m ft ft SET V. 0' b 7. c ' b. c ' b c ' b. 9 c ' a 46. c feet ' a c minutes. 7 40' a 4. c feet 6. 0' a 66. c ' a 949. b ' a 0. b c. 6 o 4' o 6 9' b. 0 o ' o 6 '. 0' a b a. 7 o 6 ' o 6 4' 9