UNIT CIRCLE TRIGONOMETRY


 Gary Holmes
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1 UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = We have peviousl applied tigonomet to tiangles that wee dawn with no efeence to an coodinate sstem. Because the adius of the unit cicle is, we will see that it povides a convenient famewok within which we can appl tigonomet to the coodinate plane. Dawing Angles in Standad Position We will fist lean how angles ae dawn within the coodinate plane. An angle is said to be in standad position if the vete of the angle is at (0, 0 and the initial side of the angle lies along the positive ais. If the angle measue is positive, then the angle has been ceated b a counteclockwise otation fom the initial to the teminal side. If the angle measue is negative, then the angle has been ceated b a clockwise otation fom the initial to the teminal side. θ in standad position, whee θ is positive: Teminal side θ in standad position, whee θ is negative: θ Initial side Initial side θ Teminal side Unit Cicle Tigonomet Dawing Angles in Standad Position
2 Eamples The following angles ae dawn in standad position:. θ = 40. θ = 60 θ θ. θ = 0 θ Notice that the teminal sides in eamples and ae in the same position, but the do not epesent the same angle (because the amount and diection of the otation in each is diffeent. Such angles ae said to be coteminal. Eecises Sketch each of the following angles in standad position. (Do not use a potacto; just daw a bief sketch.. θ = 0. θ = 4. θ = 0 4. θ = 70 θ = θ = 70 Unit Cicle Tigonomet Dawing Angles in Standad Position
3 Labeling Special Angles on the Unit Cicle We ae going to deal pimail with special angles aound the unit cicle, namel the multiples of 0 o, 4 o, 60 o, and 90 o. All angles thoughout this unit will be dawn in standad position. Fist, we will daw a unit cicle and label the angles that ae multiples of 90 o. These angles, known as quadantal angles, have thei teminal side on eithe the ais o the  ais. (We have limited ou diagam to the quadantal angles fom 0 o to 60 o. Multiples of 90 o : 90 o 80 o o 0 o 60 o Note: The 0 o angle is said to be coteminal with the 60 o angle. (Coteminal angles ae angles dawn in standad position that shae a teminal side. Net, we will epeat the same pocess fo multiples of 0 o, 4 o, and 60 o. (Notice that thee is a geat deal of ovelap between the diagams. Multiples of 0 o : Multiples of 4 o : 0 o 0 o 90 o 60 o 0 o o 90 o 4 o 80 o  0 o 60 o 80 o  0 o 60 o 0 o o 040 o 00 o 70 o o  o 70 o Multiples of 60 o : 0 o 60 o 80 o 0 o  60 o  40 o 00 o Unit Cicle Tigonomet Labeling Special Angles on the Unit Cicle
4 Putting it all togethe, we obtain the following unit cicle with all special angles labeled: o 0 o 90 o 60 o 4 o 0 o 0 o 80 o  0 o 60 o 0 o 0 o o  40 o 00 o 70 o o Coodinates of Quadantal Angles and Fist Quadant Angles We want to find the coodinates of the points whee the teminal side of each of the quadantal angles intesects the unit cicle. Since the unit cicle has adius, these coodinates ae eas to identif; the ae listed in the table below. (, 0 80 o  90 o (0,  70 o (0,  0 o (, 0 60 o Angle Coodinates 0 o (, 0 90 o (0, 80 o (, 0 70 o (0,  60 o (, 0 We will now look at the fist quadant and find the coodinates whee the teminal side of the 0 o, 4 o, and 60 o angles intesects the unit cicle. Unit Cicle Tigonomet Coodinates of Quadantal Angles and Fist Quadant Special Angles
5 Fist, we will daw a ight tiangle that is based on a 0 o efeence angle. (When an angle is dawn in standad position, its efeence angle is the positive acute angle measued fom the ais to the angle s teminal side. The concept of a efeence angle is cucial when woking with angles in othe quadants and will be discussed in detail late in this unit.  0 o  Notice that the above tiangle is a 0 o 60 o 90 o tiangle. Since the adius of the unit cicle is, the hpotenuse of the tiangle has length. Let us call the hoizontal side of the tiangle, and the vetical side of the tiangle, as shown below. (Onl the fist quadant is shown, since the tiangle is located in the fist quadant. 0 o (, 60 o We want to find the values of and, so that we can ultimatel find the coodinates of the point (, whee the teminal side of the 0 o angle intesects the unit cicle. Recall ou theoem about 0 o 60 o 90 o tiangles: In a 0 o 60 o 90 o tiangle, the length of the hpotenuse is twice the length of the shote leg, and the length of the longe leg is times the length of the shote leg. Unit Cicle Tigonomet Coodinates of Quadantal Angles and Fist Quadant Special Angles
6 Since the length of the hpotenuse is and it is twice the length of the shote leg,, we can sa that =. Since the longe leg,, is times the length of the shote leg, we can sa that =, o equivalentl, =. Based on the values of the sides of the tiangle, we now know the coodinates of the point (, whee the teminal side of the 0 o angle intesects the unit cicle. This is the point (,, as shown below. 0 o 60 o (, We will now epeat this pocess fo a 60 o efeence angle. We fist daw a ight tiangle that is based on a 60 o efeence angle, as shown below. (, 60 o We again want to find the values of and. The tiangle is a 0 o 60 o 90 o tiangle. Since the length of the hpotenuse is and it is twice the length of the shote leg,, we can sa that =. Since the longe leg,, is times the length of the shote leg, we can sa that =, o equivalentl, =. Unit Cicle Tigonomet Coodinates of Quadantal Angles and Fist Quadant Special Angles
7 Based on the values of the sides of the tiangle, we now know the coodinates of the point (, whee the teminal side of the 60 o angle intesects the unit cicle. This is the point (,, as shown below. (, 0 o 60 o We will now epeat this pocess fo a 4 o efeence angle. We fist daw a ight tiangle that is based on a 4 o efeence angle, as shown below. (, 4 o This tiangle is a 4 o 4 o 90 o tiangle. We again want to find the values of and. Recall ou theoem fom the pevious unit: In a 4 o 4 o 90 o tiangle, the legs ae conguent, and the length of the hpotenuse is times the length of eithe leg. Since the length of the hpotenuse is times the length of eithe leg, we can sa that the hpotenuse has length. But we know alead that the hpotenuse has length, so =. Solving fo, we find that =. Rationalizing the denominato, = =. Since the legs ae conguent, =, so =. Unit Cicle Tigonomet Coodinates of Quadantal Angles and Fist Quadant Special Angles
8 Based on the values of the sides of the tiangle, we now know the coodinates of the point (, whee the teminal side of the 4 o angle intesects the unit cicle. This is the point (,, as shown below. (, 4 o 4 o Putting togethe all of the infomation fom this section about quadantal angles as well as special angles in the fist quadant, we obtain the following diagam: ( 0, 90 o (, o( 60 o, 4 0 ( o, (, 0 80 o  0 o 60 o (, 070 o ( 0, We will use these coodinates in late sections to find tigonometic functions of special angles on the unit cicle. Unit Cicle Tigonomet Coodinates of Quadantal Angles and Fist Quadant Special Angles
9 Definitions of the Si Tigonometic Functions We will soon lean how to appl the coodinates of the unit cicle to find tigonometic functions, but we want to peface this discussion with a moe geneal definition of the si tigonometic functions. Definitions of the Si Tigonometic Functions: Geneal Case Let θ be an angle dawn in standad position, and let P (, epesent the point whee the teminal side of the angle intesects the cicle tigonometic functions ae defined as follows: + =. The si sin ( θ = csc( θ = = ( 0 sin ( θ cos( θ = sec( θ = cos( θ = ( 0 tan ( θ = ( 0 cot ( θ = = ( 0 tan ( θ Altenative definitions fo the tangent and the cotangent functions ae as follows: tan sin ( θ ( θ = cot ( θ cos( θ cos = sin ( θ ( θ The above functions ae not eall new to us if we elate them to ou pevious unit on ight tiangle tigonomet. Fo the pupose of emembeing the fomulas, we will choose to daw an angle θ in standad position in the fist quadant, and then daw a ight tiangle in the fist quadant which contains that angle, inscibed in the cicle + =. (Remembe that the cicle + = is centeed at the oigin with adius. We label the hoizontal side of the tiangle, the vetical side, and the hpotenuse (since it epesents the adius of the cicle. A diagam of the tiangle is shown below. Unit Cicle Tigonomet Definitions of the Si Tigonometic Functions
10 + = θ Notice that the fomula + = (the equation of the cicle is simpl the Pthagoean Theoem as it elates to the sides of the tiangle. Recall the fomulas fo the basic tigonometic atios which we leaned in the pevious unit on ight tiangle tigonomet, shown below in abbeviated fom: Mnemonic: SOHCAHTOA sin( = cos( = tan( = Opposite Adjacent Opposite θ Hpotenuse θ Hpotenuse θ Adjacent and the ecipocal tigonometic atios: csc( θ = = sin( θ Hpotenuse Opposite sec( θ = = cos( θ Hpotenuse Adjacent cot( θ = = tan( θ Adjacent Opposite Using these fomulas in the tiangle fom the diagam above, we obtain ou si tigonometic functions which we appl to the coodinate plane: sin( θ = = Opposite Hpotenuse csc( θ = = = sin( θ Hpotenuse Opposite cos( θ = = Adjacent Hpotenuse sec( θ = = = cos( θ Hpotenuse Adjacent tan( θ = = Opposite Adjacent cot( θ = = = tan( θ Adjacent Opposite Unit Cicle Tigonomet Definitions of the Si Tigonometic Functions
11 Note that even though we dew the ight tiangle in the fist quadant in ode to easil elate these fomulas to ight tiangle tigonomet, these definitions appl to an angle. (We will discuss late how to popel daw ight tiangles in othe quadants. Right tiangles can not be dawn to illustate the quadantal angles, but the above fomulas still appl. Finall, let us justif the new fomulas fo tangent and cotangent. (We will do this algebaicall, not with ou ight tiangle pictue fom above. Ou altenative definition sin( θ fo the tangent atio is tan ( θ = cos( θ. If we substitute sin( θ = and cos( θ = into this equation, we can see that ou altenative definition is equivalent to ou definition tan( θ =. The algebaic justification is shown below: tan ( θ sin ( θ = = = = cos( θ Similal, ou altenative definition fo the cotangent atio is cot ( θ = cos sin ( θ ( θ. If we substitute sin( θ = and cos( θ = into this equation, we can see that ou altenative definition fo cotangent is equivalent to the fomula cot( θ = tan( θ =. The algebaic justification is shown below: cot ( θ cos( θ = = = = sin ( θ Let us now appl ou tigonometic definitions to the unit cicle. Since the unit cicle has adius, we can see that ou tigonometic functions ae geatl simplified: sin( θ = = = csc( θ = =, 0 cos( θ = = = sec( θ = =, 0 tan ( θ ( θ ( θ sin = =, 0 cos cot ( θ ( θ ( θ cos = =, 0 sin Unit Cicle Tigonomet Definitions of the Si Tigonometic Functions
12 Definitions of the Si Tigonometic Functions: Special Case of the Unit Cicle Let θ be an angle dawn in standad position, and let P (, epesent the point whee the teminal side of the angle intesects the unit cicle + =. The si tigonometic functions ae defined as follows: sin cos tan ( θ = csc( θ = = ( 0 sin ( θ ( θ = sec( θ = = ( 0 cos( θ sin ( θ cos( θ ( θ = = ( 0 cot ( θ cos( θ = tan ( θ = = sin ( θ ( 0 Note that these definitions appl ONLY to the unit cicle! Note that the value of a tigonometic function fo an given angle emains constant egadless of the adius of the cicle. Fo eample, let us suppose that we want to find sin ( 60 in two diffeent was, using the following two diagams. (Onl the fist quadant is shown, since that is all that is needed fo the poblem. Diagam : Find sin ( o P ( 6, 6 Solution: Since the cicle shown has adius, we must use the geneal definition fo the sine function, θ = sin (. Since the point given is ( 6, 6, the value is 6. We know fom the diagam that =. Theefoe, sin ( 60 6 = = =. Unit Cicle Tigonomet Definitions of the Si Tigonometic Functions
13 Diagam : Find sin ( 60. (, Solution: Since the cicle shown has adius, we can use the shotened definition of the sine function sin,, the ( θ =. Since the point given is ( value is. 60 o sin 60 = = Theefoe, (. We will use the unit cicle (and thus the shotened definitions to evaluate the tigonometic functions of special angles. (We will see some eamples late in the unit whee the geneal definitions of the tigonometic functions can be useful. Eamples Find the eact values of the following:. cos( 90. sin ( 4. tan ( 0 4. csc( 80. sec( cot ( 0 7. tan ( 4 8. sin ( sec( 0 Solutions: Fist, ecall ou unit cicle with the coodinates which we have filled in so fa: ( 0, 90 o (, o( 60 o, 4 0 ( o, (, 0 80 o  0 o 60 o (, 070 o ( 0, Unit Cicle Tigonomet Definitions of the Si Tigonometic Functions
14 . cos( 90 The teminal side of a 90 o angle intesects the unit cicle at the point ( 0,. Using the definition cos( cos 90 = 0. θ =, we conclude that (. sin ( 4 The teminal side of a 4 o angle intesects the unit cicle at the point ( Using the definition sin ( sin 4 = θ =, we conclude that (. tan ( 0 The teminal side of a 0 o angle intesects the unit cicle at the point ( Using the definition tan ( θ =, we conclude that ( tan 0 = = Rationalizing the denominato, ( 4. csc( 80.,.,. tan 0 = = = The teminal side of a 80 o angle intesects the unit cicle at the point (, 0. csc Using the definition ( sin( θ undefined, since 0 θ = =, we conclude that csc( 80 is undefined.. sec( 60 The teminal side of a 60 o angle intesects the unit cicle at the point ( Using the definition sec( 6. cot ( 0 θ =, we conclude that (. is,. sec 60 = = =. The teminal side of the 0 o angle intesects the unit cicle at the point (, 0. Using the definition cot ( cot 0 0 is undefined. θ =, we conclude that ( is undefined, since. Unit Cicle Tigonomet Definitions of the Si Tigonometic Functions
15 7. tan ( 4 The teminal side of a 4 o angle intesects the unit cicle at the point ( Using the definition tan ( 8. sin ( 70 θ =, we conclude that (,. tan 4 = = =. The teminal side of a 70 o angle intesects the unit cicle at the point ( 0,. Using the definition sin ( sin 70 =. 9. sec( 0 θ =, we conclude that ( The teminal side of a 0 o angle intesects the unit cicle at the point(,. sec θ = =, we conclude that Using the definition ( cos( θ sec 0 ( = = = ( sec 0 = =.. Rationalizing the denominato, Eample Find tan ( 90 and cot ( 90. Solution: The teminal side of a 90 o angle intesects the unit cicle at the point ( 0,. tan 90, we use the definition tan ( To find ( is undefined, since 0 To find ( 0 cot ( 90 = = 0 is undefined. cot 90, we use the definition cot (. Impotant Note: If we want to find cot ( 90 cot θ =, knowing that tan tan θ ( 90 θ = and conclude that tan ( 90 θ = and conclude that and we instead use the definition ( (. In this case, we need to instead use the diect fomula cot ( undefined we have done above. is undefined, we cannot compute θ = as Unit Cicle Tigonomet Definitions of the Si Tigonometic Functions
16 Coodinates of All Special Angles on the Unit Cicle We will now label the coodinates of all of the special angles in the unit cicle so that we can appl the tigonometic functions to angles in an quadant. Fist, ecall ou unit cicle with the coodinates we have filled in so fa. ( 0, 90 o (, o( 60 o, 4 0 ( o, (, 0 80 o  0 o 60 o (, 070 o ( 0, We will now daw all of the angles (between 0 o and 60 o that have a 0 o efeence angle. Definition of Refeence Angle When an angle is dawn in standad position, its efeence angle is the positive acute angle measued fom ais to the angle s teminal side. Angles with Refeence angles of 0 o : Points A, B, C, and D each epesent the point whee the teminal side of an angle intesects the unit cicle. Fo each of these points, we wish to detemine the angle measue (in standad position fom 0 o to 60 o as well as the coodinates of that paticula angle. B C  0 o 0 o  0 o 0 o A D Unit Cicle Tigonomet Coodinates of All Special Angles on the Unit Cicle
17 Point A is located along the teminal side of a 0 o angle. This point should alead,. be familia to us; the coodinates of point A ae ( Point B is located along the teminal side of a 0 o angle, since 80 0 = 0. The coodinates of point B ae almost identical to those of point A, ecept that the value is negative in the second quadant, so the coodinates of point B ae,. ( Point C is located along the teminal side of a 0 o angle, since = 0. The coodinates of point C ae almost identical to those of point A, ecept that the value and value ae negative in the thid quadant, so the coodinates of point,. C ae ( Point D is located along the teminal side of a 0 o angle, since 60 0 = 0. The coodinates of point D ae almost identical to those of point A, ecept that the value is negative in the fouth quadant, so the coodinates of point D ae,. ( Below is a diagam with all of the 0 o efeence angles between 0 o and 60 o, along with thei espective coodinates: ( (, 0, 00 o 0 (, 0 o  0 o 0 o 0 (, We have gone into geat detail descibing the justification fo the angles and coodinates elated to 0 o efeence angles. Since the eplanations fo the 4 o and 60 o efeence angles ae so simila, we will skip the details and simpl give the final diagams with the angles and coodinates labeled. Unit Cicle Tigonomet Coodinates of All Special Angles on the Unit Cicle
18 Angles with Refeence angles of 4 o : Below is a diagam with all of the 4 o efeence angles between 0 o and 60 o, along with thei espective coodinates: (,  4 (, 4 o 4 o 4 o 4 o (,  (, Angles with Refeence angles of 60 o : Below is a diagam with all of the 60 o efeence angles between 0 o and 60 o, along with thei espective coodinates: (, 060 (, 60 o 60 o 60 o 60 o (, (, We will now put all of the pevious infomation togethe in one diagam which includes the coodinates of all special angles on the unit cicle. Unit Cicle Tigonomet Coodinates of All Special Angles on the Unit Cicle
19 Coodinates of All Special Angles on the Unit Cicle ( (, (, 0, 0 90 ( 0, ( 4 (, 60, ( 0, (, o 0 (, 0 (, 0 0 (, (, (,  (, (, ( 70 0, Evaluating Tigonometic Functions of An Special Angle Now that we know the coodinates of all the special angles on the unit cicle, we can use these coodinates to find tigonometic functions of an special angle (i.e. an multiple of 0 o, 4 o, o 60 o. Fist, let us eview the concept of coteminal angles. Definition of Coteminal Angles Coteminal Angles ae angles dawn in standad position that shae a teminal side. Fo an angle θ, an angle coteminal with θ can be obtained b using the fomula θ + k 60, whee k is an intege. ( Unit Cicle Tigonomet Evaluating Tigonometic Functions of An Special Angle
20 We alead have discussed the fact that a 0 o angle is coteminal with a 60 o angle. Below ae moe eamples of coteminal angles. Eample Find fou angles that ae coteminal with a 0 o angle dawn in standad position. Solution: To find angles coteminal with 0 o, we add to it an multiple of 60 o, since adding 60 o means adding eactl one evolution, which keeps the teminal side of the angle in the same position. Thee ae an infinite numbe of coteminal angles fo an given angle, so the following solutions ae not unique. ( ( ( ( ( ( = = = = 690 Theefoe, fou angles coteminal with 0 o ae 90 o, 70 o, 0 o, and 690 o. Since coteminal angles shae the same teminal side, the intesect the same point on the,, the angles unit cicle. Theefoe, since 0 o intesects the unit cicle at the point ( 90 o, 70 o, 0 o, and 690 o also intesect the unit cicle at the point (,. This allows us to find tigonometic functions of special angles othe than those that we have dawn on ou unit cicle between 0 o and 60 o. Net, ecall the definitions of the tigonometic functions as the appl to the unit cicle: sin cos tan ( θ = csc( θ = sin( θ = ( 0 ( θ = sec( θ = cos( θ = ( 0 sin( θ cos( θ ( θ = = cos( θ ( 0 cot ( θ = tan( = θ = sin( θ ( 0 We can now use these definitions to find tigonometic functions of an special angle. Unit Cicle Tigonomet Evaluating Tigonometic Functions of An Special Angle
21 Eamples Find the eact values of the following:. sin ( 40. cos(. cot ( 0 4. sec( 0. tan ( 6. csc( sin ( tan ( 0 9. sec( 80 Solutions:. sin ( 40 The teminal side of a 40 o angle intesects the unit cicle at the point (,. Using the definition sin ( θ =, we conclude that sin ( 40 =.. cos( The teminal side of a o angle intesects the unit cicle at the point (,. Using the definition cos( θ =, we conclude that cos( =.. cot ( 0 The teminal side of a 0 o angle intesects the unit cicle at the point (,. Using the definition cot ( θ =, we conclude that cot 0 = = =.. ( 4. sec( 0 The teminal side of a 0 o angle intesects the unit cicle at the point ( Using the definition sec( cos( θ ( = = = ( sec 0 sec 0 = =. θ = =, we conclude that. Rationalizing the denominato,,. Unit Cicle Tigonomet Evaluating Tigonometic Functions of An Special Angle
22 . tan ( The teminal side of a o angle intesects the unit cicle at the point (,. Using the definition tan ( θ =, we conclude that ( tan = = =. 6. csc( 60 A 60 o angle is coteminal with a 70 o angle (since = 70 and 0,. Using the its teminal side intesects the unit cicle at the point ( csc definition ( sin( θ 7. sin ( 40 θ = =, we conclude that ( csc 60 = =. A 40 o angle is coteminal with a 0 o angle (since = 0 and,. Using the its teminal side intesects the unit cicle at the point ( θ =, we conclude that ( definition sin ( 8. tan ( 0 sin 40 = A 0 o angle is coteminal with a 0 o angle (since = 0 and,. Using the its teminal side intesects the unit cicle at the point ( definition tan ( θ =, we conclude that (. tan 0 = = = tan 0 = = Rationalizing the denominato, we obtain ( 9. sec( = 90 and its teminal side intesects the unit cicle at the point( 0,. Using the sec sec 80 is undefined, since A 80 o angle is coteminal with a 90 o angle (since ( definition ( cos( θ 0 is undefined. θ = =, we conclude that (. Unit Cicle Tigonomet Evaluating Tigonometic Functions of An Special Angle
23 Eecises Find the eact values of the following:. cos( 0. tan ( 0. sin ( 4. cot ( 60. csc( 6. tan ( 0 7. cot ( cos( sin ( 60 Methods of Finding Tigonometic Functions of Special Angles We can see that the unit cicle assists us geatl in finding tigonometic functions of special angles. But what if we don t have a diagam of the unit cicle, and we don t wish to daw one? It is assumed that the quadantal angles (multiples of 90 o  on the aes ae fail eas to visualize without a diagam. One option is to memoize the coodinates of the special angles in the fist quadant of the unit cicle (0 o, 4 o, and 60 o, and use those values to find the tigonometic functions of angles in othe quadants. We will discuss a few was of finding the basic tigonometic functions of 0 o, 4 o, and 60 o (othe than memoizing the coodinates of these special angles in the fist quadant of the unit cicle. One method is to use a basic 4 o 4 o 90 o tiangle and a 0 o 60 o 90 o tiangle to deive the tigonometic functions of 0 o, 4 o, and 60 o. We leaned ealie in the unit that the sin 0 is alwas tigonometic functions ae constant fo an given angle; fo eample, (, egadless of the size of the adius of the cicle o the size of the tiangle dawn. So we can choose an 4 o 4 o 90 o tiangle and 0 o 60 o 90 o tiangle to wok fom. Fo simplicit, let us choose both tiangles to have a shote leg of length, as shown below: 4 o 60 o 4 o 0 o Recall that in a 4 o 4 o 90 o tiangle, the legs ae conguent, and the length of the hpotenuse is times the length of eithe leg. In a 0 o 60 o 90 o tiangle, the length of the hpotenuse is twice the length of the shote leg, and the length of the longe leg is times the length of the shote leg. Unit Cicle Tigonomet Methods of Finding Tigonometic Functions of Special Angles
24 Filling in the missing sides fom the diagam above, we obtain the following tiangles: 4 o 60 o 4 o 0 o Now ecall the tigonometic atios that we leaned fo ight tiangles (shown below in abbeviated fom: SOHCAHTOA sin( = cos( = tan( = Opposite Adjacent Opposite θ Hpotenuse θ Hpotenuse θ Adjacent Using these atios on the tiangles above, we obtain the following: sin(4 = = = = Opposite Hpotenuse cos(4 = = = = Adjacent Hpotenuse (Note: This is the coodinate of 4 o on the unit cicle. (Note: This is the coodinate of 4 o on the unit cicle. tan(4 = = = Opposite Adjacent Opposite sin(0 = Hpotenuse = Adjacent cos(0 = Hpotenuse = (Note: This is the coodinate of 0 o on the unit cicle. (Note: This is the coodinate of 0 o on the unit cicle. tan(0 = = = = Opposite Adjacent Opposite sin(60 = Hpotenuse = Adjacent cos(60 = Hpotenuse = (Note: This is the coodinate of 60 o on the unit cicle. (Note: This is the coodinate of 60 o on the unit cicle. Opposite tan(60 = Adjacent = The ecipocal tigonometic atios cosecant, secant, and cotangent can be obtained b simpl taking the ecipocals of the sine, cosine, and tangent atios above. Unit Cicle Tigonomet Methods of Finding Tigonometic Functions of Special Angles
25 Anothe method of emembeing the tigonometic functions of 0 o, 4 o, and 60 o, along with 0 o and 90 o, is shown below. (This is moe of a tick fo emembeing the atios, so no mathematical justification is given. Step : Label the columns at the top of the chat fo all of the special angles between 0 o and 90 o in ascending ode, as shown. Also label the ows with the wods sine and cosine. sine cosine 0 o 0 o 4 o 60 o 90 o Step : This chat is not et complete! Wite the numbes 0,,,, 4 in the sine ow, and the numbes 4,,,, 0 in the cosine ow, as shown. sine cosine 0 o 0 o 4 o 60 o 90 o 0 4 This chat is not et complete! 4 0 This chat is not et complete! Step : This chat is coect, but not et simplified Fo each of the numbes in the sine and cosine ows, take the squae oot of the numbe and then divide b. sine cosine 0 o 0 o 4 o 60 o 90 o We then simplif each of the numbes in the chat above. Unit Cicle Tigonomet Methods of Finding Tigonometic Functions of Special Angles
26 Step 4: This is the final vesion of the chat! Simplif each of the values fom the table above, and we obtain ou final chat: sine 0 cosine 0 o 0 o 4 o 60 o 90 o 0 values on the unit cicle values on the unit cicle Eamples Use the methods leaned in this section (special ight tiangles o the chat above to find the eact values of the following tigonometic functions. Note that these eamples wee also included in the pevious section, but will now be solved using a diffeent method.. sin ( 40. cos(. cot ( 0 4. sec( 0. tan ( Solutions: sin 40. ( The teminal side of a 40 o angle measues 60 o to the ais, so we use a 60 o efeence angle. Using eithe a 0 o 60 o 90 o tiangle o ou chat fom above, we find that sin ( 60 =. Since a 40 o angle is in the thid quadant, and  values in the thid quadant ae negative, sin ( 40 sin ( 60 =, we conclude that sin ( 40 =. must be negative. Since. cos( The teminal side of a o angle measues 4 o to the ais, so we use a 4 o efeence angle. Using eithe a 4 o 4 o 90 o tiangle o ou chat fom above, we find that cos( 4 =. Since a o angle is in the fouth quadant, and values in the fouth quadant ae positive, cos( Since (, we conclude that ( cos 4 = cos =. must be positive. Unit Cicle Tigonomet Methods of Finding Tigonometic Functions of Special Angles
27 . cot ( 0 The teminal side of a 0 o angle measues 0 o to the ais, so we use a 0 o efeence angle. Remembe that cot ( tan( θ tan 0 = cos sin ( θ ( θ θ = =. If we use the 0 o 60 o  90 o tiangle, we find that (, so ( tan( θ instead use ou chat fom above, we find that ( cot 0 = =. If we cos 0 = cos( 0 sin ( 0 =, so cot ( 0 = = ( = sin 0 =. Since a 0 o angle is in the thid quadant, and both the and values in the thid quadant ae negative, this means that both cos( 0 and sin ( 0 ae negative, cos( 0 theefoe thei quotient cot ( 0 = is positive. Since cot sin( 0 ( 0 =, cot 0 =. we conclude that ( 4. sec( 0 The teminal side of a 0 o angle measues 0 o to the ais, so we use a 0 o efeence angle. (A efeence angle is not eall necessa in this case, since an angle which is located in the fist quadant can be used asis. Remembe sec θ =. Using eithe a 0 o 60 o 90 o tiangle o ou chat fom that ( cos ( θ cos 0 = above, we find that (, so ( cos( 0 and sec 0 = = = = Rationalizing the denominato, ( sec 0 = =. tan ( The teminal side of a o angle measues 4 o to the ais, so we use a 4 o θ = efeence angle. Remembe that tan ( tiangle, we find that ( sin cos ( θ ( θ.. If we use the 4 o 4 o 90 o tan 4 = =. If we instead use ou chat fom above, sin( 4 sin ( 4 = and sin ( 4 =, so tan ( 4 = = ( = cos 4 =. Since a o angle is in the second quadant, the value is negative and the value is positive, which means that cos( is negative and sin ( is positive; sin( theefoe thei quotient tan ( = is negative. Since tan cos( ( 4 =, we tan =. conclude that (. Unit Cicle Tigonomet Methods of Finding Tigonometic Functions of Special Angles
28 Evaluating Tigonometic Functions of Othe Angles Now that we have eploed the tigonometic functions of special angles, we will biefl look at how to find tigonometic functions of othe angles. Fist, we will deive an impotant tigonometic identit, known as a Pthagoean Identit. It can be elated to the Pthagoean Theoem; conside the following ight tiangle (dawn in the fist quadant fo simplicit. + = θ We can see that the equation + = (the equation of a cicle with adius is also the Pthagoean Theoem as it elates to the ight tiangle above. Let us now diect ou attention to the unit cicle. Since the adius is, an point on the cicle itself satisfies the equation + = (the equation of a cicle with adius. On the unit cicle, we know that = cos( θ and = sin ( θ. Substituting these into the equation + =, we obtain the equation ( cos( θ ( sin ( θ individual tems in shotened fom. We wite ( cos( θ as cos ( sin θ as sin ( ( ( + =. It is standad pactice in tigonomet to wite these θ, and we wite θ. We then obtain the following tigonometic identit: ( θ ( θ cos + sin = The identit ( θ ( θ cos + sin = applies to an angle dawn in an cicle of adius, not just the unit cicle. A shot justification is shown below. Unit Cicle Tigonomet Evaluating Tigonometic Functions of Othe Angles
29 We begin with the geneal equation of a cicle of adius : + = Dividing both sides b, we obtain the equation + =. Simplifing the equation, + =. The equation can then be ewitten as + =. Fo a cicle of adius, we know fom ou geneal tigonometic definitions that cos( θ = and sin ( θ =. We substitute these into the above equation and conclude that ( θ ( θ cos sin + =. We will now use this Pthagoean identit to help us to find tigonometic functions of angles. Eample sin θ = and 90 < θ < 80 If ( Unit Cicle Tigonomet, find the eact values of cos ( θ and ( tan θ. Solution: Since 90 < θ < 80, the teminal side of θ is in the second quadant. Looking at ou unit cicle, we can easil see that thee is no special angle with a value of. Thee ae two methods b which we can solve this poblem, both of which ae shown below. Method : Ou fist method is to use the Pthagoean identit cos θ + sin θ =. ( ( Since sin ( θ =, we plug it into the equation cos ( θ + sin ( θ =. cos ( θ + ( = Evaluating Tigonometic Functions of Othe Angles
30 We then simplif the equation and solve fo cos( θ : 9 cos ( θ + = 9 cos ( θ = 9 cos ( θ = 6 ( cos θ = 4 4 cos( θ = OR ( cos θ = (We need to choose; see below... 4 cos( θ = The teminal side of θ is in the second quadant. We know that values in the second quadant ae negative, theefoe cos( θ is negative. We now want to find tan ( θ. Using the definition tan ( tan ( sin( θ cos( θ θ = = = = tan ( θ = 4 4 We conclude that cos( θ = and (. tan θ =. 4 sin cos ( θ ( θ θ =, Method : Ou second method is to daw a ight tiangle in a cicle of adius. Since 90 < θ < 80, we daw the ight tiangle in the second quadant, as shown below. (When dawing the ight tiangle, we must make sue that the ight tiangle has one leg on the ais and that one acute angle of the tiangle  the efeence angle of θ  has its vete at the oigin. + = (, P θ Unit Cicle Tigonomet Evaluating Tigonometic Functions of Othe Angles
31 Since sin ( θ =, and we know that sin ( θ =, we can label the diagam with the values = and =, as shown below. (The sin θ = can be emembeed b using the atio fomula ( opposite, hpotenuse using the efeence angle of θ which is inside the tiangle. Note that when choosing how to label the sides of the tiangle, the adius of the cicle is alwas chosen to be positive. + = θ To find the value of, we then use the equation fo the cicle (o equivalentl, the Pthagoean Theoem + = : + = + 9= = 6 = 4 OR = 4 (We need to choose; see below. = 4 The teminal side of θ is in the second quadant. We know that values in the second quadant ae negative, theefoe we choose the negative value fo. To find cos( θ, we now use the fomula cos( fomula cos( θ =. (The θ = can be emembeed b using the atio using the efeence angle of θ which is inside the tiangle. Note cos θ =, since that that we can NOT use the fomula ( tigonometic definition onl applies to the unit cicle, and the cicle that we ceated has adius. cos ( θ = = = 4 4 adjacent, hpotenuse Unit Cicle Tigonomet Evaluating Tigonometic Functions of Othe Angles
32 To find tan ( θ, we now use the fomula tan ( that the fomula tan ( θ =. (Remembe θ = can be emembeed b using the atio opposite adjacent, using the efeence angle of θ which is inside the tiangle. tan ( θ = = = 4 4 Note that we could have also used the fomula tan ( θ sin( θ cos( θ = = = = 4 We conclude that cos( θ = and ( 4 tan θ =. We now wish to deive two othe Pthagoean identities. Both identities can be easil cos θ + sin θ =. deived fom the identit ( ( Fist, begin with the identit ( θ ( θ Now divide each tem b cos ( θ : cos ( θ sin ( θ + = cos ( θ cos ( θ cos ( θ sin ( θ cos + sin =. Since tan θ = sec θ, we can simplif the above equation and cos θ cos θ obtain the following Pthagoean identit: ( = ( and ( ( ( θ ( θ + tan = sec In a simila fashion, we again begin with the identit ( θ ( θ Now divide each tem b sin ( θ : cos ( θ sin ( θ + = sin ( θ sin ( θ sin ( θ cos ( θ ( = ( and ( ( cos + sin =. Since cot θ = csc θ, we can simplif the above equation and sin θ sin θ obtain the following Pthagoean identit: ( θ + = ( θ cot csc Unit Cicle Tigonomet Evaluating Tigonometic Functions of Othe Angles
33 Pthagoean Identities ( θ + ( θ = + tan ( θ = sec ( θ cot ( θ + = csc ( θ cos sin Eample cot θ = and 90 < θ < 70 If (, find the eact values of sin ( θ and ( cos θ. Solution: Since 90 < θ < 70, the teminal side of θ is eithe in the second o the thid cot θ = ; we ae given that quadant. Recall the definition fo cotangent ( cot ( θ = and is positive, so the quotient cot (, which is a positive numbe. In the second quadant, is negative θ = is negative; theefoe the teminal side of θ can not lie in the second quadant. In the thid quadant, on the othe cot θ = is positive, which is hand, both and ae negative, so the quotient ( consistent with the given infomation. We then conclude that the teminal side of θ is in the thid quadant. Thee ae two methods b which we can solve this poblem, both of which ae shown below. Method : Ou fist method is to use the Pthagoean identit cot θ + = csc θ. ( ( Since cot ( θ = cot ( θ csc ( θ + =. ( + = csc ( θ, we plug it into the equation We then simplif the equation and solve fo csc( θ : ( + = csc ( θ ( ( ( + = csc θ + = csc θ 6 = csc θ Unit Cicle Tigonomet Evaluating Tigonometic Functions of Othe Angles
34 6 6 csc( θ = OR ( csc θ = (We need to choose; see below. 6 csc( θ = The teminal side of θ is in the thid quadant. We know that values in the thid quadant ae sin θ is negative. negative, which means that ( Since csc( θ = sin( θ, we conclude that ( negative. We now want to find sin ( θ. Since the cosecant and sine csc θ is functions ae ecipocals of each othe, we can eaange the csc sin θ =. equation ( θ = sin ( θ to sa that ( csc( θ sin ( θ = ( = = = θ csc Ou final step is to find cos( θ. Using the identit cos ( θ + sin ( θ =, cos ( θ + ( 6 = cos ( θ + 6 = 6 cos ( θ = 6 = 6 6 cos ( θ = 6 cos( θ = 6 OR ( 6. cos θ = (We need to choose; see below. cos( θ = 6 The teminal side of θ is in the thid quadant. We know that values in the thid quadant ae negative, which means that cos( θ is negative. We conclude that sin ( θ = 6 and ( cos θ =. 6 Method : Ou second method is to daw a ight tiangle in a cicle of adius. Since we have detemined that the teminal side of θ is in the thid quadant, we daw the ight tiangle in the thid quadant, as shown below. (When dawing the ight tiangle, we must make sue that the ight tiangle has one leg on the ais and that one acute angle of the tiangle  the efeence angle of θ  has its vete at the oigin. Unit Cicle Tigonomet Evaluating Tigonometic Functions of Othe Angles
35 + = θ (, P Since cot ( θ =, and we know that cot ( θ =, we can label the diagam with the values = and =, as shown below. We can NOT label the sides of the tiangle with the values = and =, since the tiangle is in the thid quadant, cot θ = can be whee both and ae negative. (The fomula ( emembeed b using the atio adjacent opposite, using the efeence angle of θ which is inside the tiangle. + = θ  (, P To find the value of, we then use the equation fo the cicle (o equivalentl, the Pthagoean Theoem + = : ( + ( =, so + = 6 = = 6 (Note that the adius of the cicle is alwas positive. Unit Cicle Tigonomet Evaluating Tigonometic Functions of Othe Angles
36 To find sin ( θ, we now use the fomula sin ( sin ( θ =. (The fomula opposite θ = can be emembeed b using the atio, using hpotenuse the efeence angle of θ which is inside the tiangle. Note that we sin θ =, since that tigonometic can NOT use the fomula ( definition onl applies to the unit cicle, and the cicle that we ceated has adius 6. sin ( θ = = = 6 6 To find cos( θ, we now use the fomula cos( fomula cos( θ =. (The θ = can be emembeed b using the atio using the efeence angle of θ which is inside the tiangle. Note cos θ =, since that that we can NOT use the fomula ( tigonometic definition onl applies to the unit cicle, and the cicle that we ceated has adius 6. cos ( θ = = = 6 6 We conclude that sin ( θ = 6 and ( 6 cos θ =. adjacent, hpotenuse Eecises Answe the following, using eithe of the two methods descibed in this section.. If ( tan ( θ. cos θ = and 80 < θ < 60. If ( 7 cot ( θ. csc θ = and 90 < θ < 70. If ( sin ( θ. tan θ = and 80 < θ < 60 sin θ and, find the eact values of ( cos θ and, find the eact values of ( cos θ and, find the eact values of ( Unit Cicle Tigonomet Evaluating Tigonometic Functions of Othe Angles
37 Gaphs of the Sine and Cosine Functions In this section, we will lean how to gaph the sine and cosine functions. To do this, we will once again use the coodinates of the special angles fom the unit cicle. We will fist make a chat of values fo = f( = sin(, whee epesents the degee measue of the angle. In the column fo the values, the eact value has also been witten as a decimal, ounded to the neaest hundedth fo gaphing puposes. = sin( = sin( 0 o 0 80 o 0 0 o = 0. 0 o = 0. 4 o 0.7 o o o o 70 o 0 o o 0.87 o 0.7 o o = 0. 0 o = o 0 We now plot the above and values on the coodinate plane, as shown: Unit Cicle Tigonomet Gaphs of the Sine and Cosine Functions
38 Dawing a smooth cuve though the points which we have plotted, we obtain the following gaph of = f( = sin( : = f( = sin( o 0 o 60 o 90 o 0 o 0 o 80 o 0 o 40 o 70 o 00 o 0 o 60 o Since 60 o is coteminal with 0 o, thei values ae the same. This is the case with an coteminal angles; 40 o is coteminal with 90 o, 40 o is coteminal with 80 o, etc. Fo this eason, the above gaph will epeat itself ove and ove again, as shown below: = f( = sin( 60 o 70 o 80 o 90 o 0 o 90 o 80 o 70 o 60 o 40 o 40 o 60 o 70 o  Unit Cicle Tigonomet Gaphs of the Sine and Cosine Functions
39 We will now epeat the same pocess to gaph the cosine function. Fist, we will make a chat of values fo = f( = cos(, whee epesents the degee measue of the angle. In the column fo the values, the eact value has also been witten as a decimal, ounded to the neaest hundedth fo gaphing puposes. = cos( = cos( 0 o 80 o 0 o o o 0.7 o o = o = o 0 70 o 0 0 o = o = 0. o 0.7 o o o o We now plot the above and values on the coodinate plane, as shown: o 0 o 60 o 90 o 0 o 0 o 80 o 0 o 40 o 70 o 00 o 0 o 60 o Unit Cicle Tigonomet Gaphs of the Sine and Cosine Functions
40 Dawing a smooth cuve though the points which we have plotted, we obtain the following gaph of = f( = cos( :.4. = f( = cos( o 0 o 60 o 90 o 0 o 0 o 80 o 0 o 40 o 70 o 00 o 0 o 60 o Since coteminal angles occu eve 60 o, the cosine gaph will epeat itself ove and ove again, as shown below: = f( = cos( 60 o 70 o 80 o 90 o 0 o 90 o 80 o 70 o 60 o 40 o 40 o 60 o 70 o  Unit Cicle Tigonomet Gaphs of the Sine and Cosine Functions
41 The gaphs can easil be used to detemine the sine, cosine, secant, and cosecant of quadantal angles. Eamples Use the gaphs of the sine and cosine functions to find eact values of the following:. sin ( 90 Solution: If we look at the gaph of = sin(, whee = 90, we find that the value sin 90 =. is. Theefoe, we conclude that (. cos( 70 Solution: If we look at the gaph of = cos(, whee = 70, we find that the  cos 70 = 0. value is 0. Theefoe, we conclude that (. sec( 80 Solution: If we look at the gaph of = cos(, whee = 80, we find that the value cos 80 = sec θ =, we conclude is . This means that ( that ( cos( csc( 60 Solution: sec 80 = = =.. Since ( cos( θ If we look at the gaph of = sin(, whee = 60, we find that the value is 0. This means that sin(60 = 0 csc θ =, we conclude that csc( 60 is undefined, since 0. Since ( sin ( θ is undefined. Eecises. Use the gaphs of the sine and cosine functions to find eact values of the following: a ( cos 70 b sin ( 90 c sec( 90 d csc( 40. Sketch the gaphs of the following functions. Label all intecepts. a = f( = sin(, whee b = f( = cos(, whee Unit Cicle Tigonomet Gaphs of the Sine and Cosine Functions
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