Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008
Tigonometic Functions of An Angle Statement of Peequisite Skills Complete all pevious TLM modules befoe completing this module. Requied Suppoting Mateials Access to the Wold Wide Web. Intenet Eploe 5.5 o geate. Macomedia Flash Plae. Rationale Wh is it impotant fo ou to lean this mateial? When eploing intoducto tigonomet concepts the leane is geneall eposed to angles that ae less than 90º. This keeps things simple and allows the leane to concentate on the topic. The leane will encounte man situations involving angles geate than 90º in applied situations. This module will povide the guidance necessa to appl the tigonomet skills that have been leaned to angles geate than 90º. Leaning Outcome When ou complete this module ou will be able to Evaluate tigonometic functions of an angle. Leaning Objectives. Detemine whethe the value of a given tigonometic function is positive o negative. 2. Detemine the efeence angle of a given angle. 3. Detemine the si tigonometic function values fo an angle in standad position when the coodinates of a point on the teminal side ae given. 4. Evaluate tigonometic functions of an angle. 5. Evaluate invese tigonometic functions. Module T2 Tigonometic Functions of An Angle
Connection Activit Conside the following diagam. Given values fo and ou ae able to figue out the tigonometic atios of α. What is ou estimate of the angle epesented b? Is it 35º? Could it be 495º? Could it be 205º? Without some indicato as to how man times otated aound the oigin we do not know the measue of the angle. What we do know is that the tigonometic atios have not changed no matte which measue tuns out to be. We can see that the tigonometic atios fo and α will be tue fo man diffeent angles. This module will help ou undestand this concept and appl it to seveal situations. P(,) Note: is measued in standad position α 2 Module T2 Tigonometic Functions of An Angle
OBJECTIVE ONE When ou complete this objective ou will be able to Detemine whethe the value of a given tigonometic function is positive o negative. Eploation Activit REVIEW The stud of ealie modules has intoduced one set of definitions of tigonometic functions: METHOD : Tiangle Method The following ae the si basic tigonometic functions deived using the tiangle method. sin = cos = opposite hpotenuse adjacent hpotenuse csc = sec = hpotenuse opposite hpotenuse adjacent hpotenuse opposite tan = opposite adjacent cot = adjacent opposite adjacent METHOD 2: Cicle Method Anothe wa of defining tigonometic functions is the cicle method. Fo the cicle method emembe that: a) Positive angles otate counteclockwise. b) Negative angles otate clockwise. The following ae the si basic tigonometic functions deived using the cicle method. sin = cos = tan = csc = sec = cot = P(,) Whee epesents the adius of the cicle and is alwas a positive value. Module T2 Tigonometic Functions of An Angle 3
USING THE CIRCLE METHOD IN ALL QUADRANTS When using the cicle method of defining tigonometic functions, the algebaic sign of a tigonometic function ma be detemined b noting the quadant which contains the teminal side of angle (o the point P (,)). Quadant : is positive ( > 0) is positive ( > 0) is positive ( is alwas positive) Theefoe, all tigonometic functions of in quadant will be positive. P(,) Quadant 2: is negative ( < 0) is positive ( > 0) is positive P(,) Note: is measued in standad position Theefoe, cos, sec, tan and cot will have negative values fo quadant 2 angles. sin and csc will have positive values. Eample: Detemine the algebaic sign of cos 00º. cos 00º = = negative value positive value = negative value cos 00º = negative value 4 Module T2 Tigonometic Functions of An Angle
Quadant 3: is negative ( < 0) is negative ( < 0) is positive Theefoe, sin, csc, cos, and sec will have negative values fo quadant 3 angles. tan and cot will have positive values. Eample: P(,) Detemine the algebaic sign of sin 20º. sin 20º = = negative value positive value = negative sin 20º = negative value Quadant 4: is positive ( > 0) is negative ( < 0) is positive Theefoe, sin, csc, tan, and cot will have negative values fo quadant 4 angles. cos and sec will have positive values. Eample: P(,) Detemine the algebaic sign of sec 35º. sec 35º = = positive value positive value = positive sec 35º = positive value Module T2 Tigonometic Functions of An Angle 5
CAST SSTEM The CAST sstem can be used fo quick ecall of algebaic signs of pima tigonometic functions. CAST C is cosine A is all S is sine T is tangent Quadant II S sine is positive Quadant III T tangent is positive Quadant I A all functions positive Quadant IV C cosine is positive Note: Knowing the CAST ule will be ve useful when detemining invese tig functions (of an angle) late in this module. Note: The cosecant, secant, and cotangent functions have the same algebaic signs as thei ecipocals. 6 Module T2 Tigonometic Functions of An Angle
Epeiential Activit One Ente positive, negative, undefined, o 0 fo the following questions:. Detemine the algebaic sign of the following epessions: a) sin 60º b) cos( 20º) c) 5 tan 4 π d) csc 30º e) 7 sec π Show Me. 4 f) cot 70º g) 2 tan π 3 h) sin 3 4 π 2. Identif the quadant(s) in which is located fo each of the following conditions: a) sin is positive b) cos is positive c) sin is negative d) tan is negative e) cos is negative f) sin is positive, cos is negative g) tan and sin both positive h) cot negative, cos negative Show Me. i) tan negative, cos positive j) all tigonometic functions of ae positive Epeiential Activit One Answes. a) positive b) negative c) positive d) negative e) positive f) negative g) positive h) positive 2. a), 2 b),4 c) 3,4 d) 2,4 e) 2,3 f) 2 g) h) 2 i) 4 j) Module T2 Tigonometic Functions of An Angle 7
OBJECTIVE TWO When ou complete this objective ou will be able to Detemine the efeence angle of a given angle. Eploation Activit DEFINITION OF REFERENCE ANGLE The efeence angle of a given angle is the positive acute angle fomed b the teminal side of the given angle and the -ais. In this book we will use the geek lette alpha (α) to epesent the efeence angle and the geek lette theta () to epesent angles measued in standad position. Eample : Detemine the efeence angle of 50º. P(,) The angle between P(,) and the ais is 30º α = 80º α = 80º 50º α = 30º α = = 50º Theefoe, 30º is the efeence angle of 50º Eample 2: Detemine the efeence angle of 260º. The angle between P(,) and the ais is 80º = 260º α = 80º α = 260º 80º α = 80º Theefoe, 80º is the efeence angle of 260º α = P(,) NOTE:. The efeence angle is alwas positive. 2. The efeence angle is alwas acute. 3. The efeence angle is alwas measued between the teminal am of the angle and the neaest -ais. 4. The tigonometic atio of α is the absolute value of the tigonometic atio of. 8 Module T2 Tigonometic Functions of An Angle
Epeiential Activit Two. Detemine the efeence angle of the following: Angle Refeence Angle a) 4º b) 3π 4 c) 473º d) 22º e) 5π 4 f) 87º g) 320º h) 35º i) j) 8π 7 7π 3 Show Me. 2. Detemine the efeence angles (α) of the following angles measued in standad position () and then daw and label both α and on the diagam povided. a) = 65º α = b) = 25º α = Module T2 Tigonometic Functions of An Angle 9
c) = 305º α = d) = 285º α = 0 Module T2 Tigonometic Functions of An Angle
Epeiential Activit Two Answes. a) 66º b) π/4 c) 67º d) 32º e) π/4 f) 87º g) 40º h) 45º i) π/7 j) π/3 2. a) = 65º α = 5º b) = 25º α = 55º α α c) = 305º α = 55º d) = 285º α = 75º α α Module T2 Tigonometic Functions of An Angle
OBJECTIVE THREE When ou complete this objective ou will be able to Detemine the si tigonometic function values fo an angle in standad position when the coodinates of a point on the teminal side ae given. Eploation Activit FOUR STEPS Thee ae fou steps needed to detemine the tigonometic function values (atios) when the coodinates of a point on the teminal side of the angle ae given. Step : Daw a ight tiangle using the teminal am and the neaest -ais. Eample: P( 3,4) P( 3,4) α Step 2: Use the point to label the sides. P( 3,4) Eample: 4 α 3 2 Module T2 Tigonometic Functions of An Angle
Step 3: Detemine the length of the hpotenuse using the Pthagoean theoem. P(-3,4) Eample: 2 2 2 c = a + b Hpotenuse 5 2 2 c = ( 3 ) 2 + 4 4 c 2 = 25 c = 5 α Step 4: Identif the hpotenuse, the side opposite angle α and the side adjacent to angle α. Use the sides to detemine the si tigonometic atios. P( 3,4) 3 Eample: Opposite 4 5 α Hpotenuse Adjacent 3 Since the efeence angle (α) and the angle measued in standad position () have the same tigonometic atios, we can now detemine the si tigonometic atios of. Si Tig Ratios in eact fom opp adj opp sin = cos = tan = hp hp adj 4 3 4 sin = cos = tan = 5 5 3 hp csc = opp 5 csc = 4 hp sec = adj 5 sec = 3 adj cot = opp 3 cot = 4 Module T2 Tigonometic Functions of An Angle 3
We can now compae the CAST ule to tigonometic atios of. Tig. Functions sin cos tan csc sec cot Signs of tig. atios of in Quadant 2 accoding to the CAST Rule Positive Negative Negative Positive Negative Negative Calculated atio in eact fom Signs of calculated tig. atios of in Quadant 2 4 sin = 5 Positive 3 cos = 5 Negative 4 tan = 3 Negative 5 csc = 4 Positive 5 sec = 3 Negative 3 cot = 4 Negative The signs of the tigonometic atios we calculated fo in quadant 2 ae consistent with the CAST ule. 4 Module T2 Tigonometic Functions of An Angle
Epeiential Activit Thee. Find the si tigonometic atios fo the following angles given a point on the teminal side. a) ( 5,8) Leave ou answes in eact fom. b) ( 7.2, 4.) Round ou answes to 4 significant digits. 2. Find all si tigonometic atios fo the angles that have teminal ams passing though the points given below. a) (4,6) (Leave ou answes in eact fom.) b) (0.8, 6.7) (Round ou answes to 4 decimal places.) 3. Give the value of sin A and cos A fo the angles, which have a teminal am that passes though the points. (Round ou answes to 4 decimal places.) a) (3.0, 4.2) b) ( 2.45, 7.32) Show Me. c) ( 4.82,.83) Epeiential Activit Thee Answes. a) sin = csc = 8 5, cos =, tan = 89 89 89, sec =, cot = 8 589 8 5 5 8 b) sin = 0.4948, cos = 0.8690, tan = 0.5694 csc = 2.0209, sec =.508, cot =.756 6 4 3 52 52 2 2. a) sin =, cos =, tan =, csc =, sec =, cot = 52 52 2 6 4 3 b) sin = 0.9929, cos = 0.86, tan = 8.3750 csc =.007, sec = 8.4345, cot = 0.94 3. a) sin A = 0.837, cos A = 0.582 b) sin A = 0.9483, cos A = 0.374 c) sin A = 0.3549, cos A = 0.9349 Module T2 Tigonometic Functions of An Angle 5
OBJECTIVE FOUR When ou complete this objective ou will be able to Evaluate tigonometic functions of an angle. Eploation Activit Calculato Comment In the pe-calculato ea, evaluating tigonometic functions of angles geate than 90 equied the use of efeence angles. Using calculatos, the 'efeence angle' step is usuall left out, howeve, it is still necessa to use the efeence angle when evaluating invese tigonometic functions (the net objective). DETERMINING THE TRIGONOMETRIC RATIOS OF AN ANGLE. Acute Angles In module 7 we evaluated tigonometic atios of acute angles Eample : Evaluate tan 35º Ensue ou calculato is in degee mode To evaluate tan 35º simpl ente tan 35º into ou calculato and pess the equal ke. tan 35º = 0.7002 Eample 2: Evaluate csc 62º csc 62º = sin 62 csc 62º = 0.8829 csc 62º =.326 Note: Recall fom the CAST ule that all acute angles have positive tigonometic atios. 6 Module T2 Tigonometic Functions of An Angle
Tig Ratios of an Angle Evaluate tig atios of an angle in the same manne ou evaluate acute angles. Eample : Evaluate cos 45º To evaluate cos 45º simpl ente cos 45º into ou calculato and pess the equal ke. cos 45º = -0.892 Note: Notice that the tigonometic atio fo cos 45º is negative. 45º has a teminal am in quadant 2, and the CAST ule tells us that the tigonometic atio of cosine is negative in quadant 2. Eample 2: Evaluate cot 265º cot 265º = tan 265 cot 265º =.430 cot 265º = 0.0875 Note: Notice that the tig atio fo cot 265º is positive. 265º has a teminal am in quadant 3, and the CAST ule tells us that the tigonometic atios fo tangent and cotangent ae positive in quadant 3. Tig Ratios of Refeences Angles We know that efeences angles ae alwas acute and theefoe the tigonometic atio of an efeence angle will alwas be positive. How do the tigonometic atios of efeence angles elate to the tigonometic atios of angles measued in standad position? In the diagam to the ight = 50º and its efeence angle is 30º Since both angles shae the same teminal am the have the same tigonometic atios. Howeve, the tigonometic atio of the efeence angle is alwas positive because it is acute and we use the CAST ule to detemine the sign of the tigonometic atio of. α = 30º = 50º Tig. atios fo α: sin 30 = 0. 500 cos 30 = 0. 8660 tan 30 = 0. 5774 Tig. atios fo sin 50 = 0. 500 cos50 = 0. 8660 tan50 = 0. 5774 The teminal am of 50º is in quadant 2. The CAST ule tells us that the tigonometic atio of sine will be positive, tangent will be negative, and cosine will be negative. Module T2 Tigonometic Functions of An Angle 7
Epeiential Activit Fou. Find the efeence angle, and the sine, cosine, tangent, cotangent, cosecant and secant of the following (If undefined, answe undefined): Angle () a) 07.0º b) 47.5º c) 83.0º d) 80.9º e) 208.0º f) 349.9º g) 46.0º h) 539.3º i) 905.0º j) 7.º k) 940.7º l) 362.6º Ref. (α) sin cos tan cot csc sec m) 260.2º Note the following angles ae measued in adians. Ensue ou calculato is in adian mode to poceed. n).4 o) 6.2 p) 0.65 q).6 ).4 s) 4.9 8 Module T2 Tigonometic Functions of An Angle
Epeiential Activit Fou Answes Angle () Ref. (α) sin cos tan cot csc sec a) 07.0º 73.0º 0.9563 0.2924 3.2709 0.3057.0457 3.4203 b) 47.5º 32.5º 0.5373 0.8434 0.637.5697.862.857 c) 83.0º 3.0º 0.0523 0.9986 0.0524 9.08 9.073.004 d) 80.9º 0.9º 0.057 0.9999 0.057 63.6567 63.6646.000 e) 208.0º 28.0º 0.4695 0.8829 0.537.8807 2.30.326 f) 349.9º 0.º 0.754 0.9845 0.78 5.640 5.7023.057 g) 46.0º 79.0º 0.986 0.908 5.446 0.944.087 5.2408 h) 539.3º 0.7º 0.022 0.9999 0.022 8.8470 8.853.000 i) 905.0º 5.0º 0.0872 0.9962 0.0875.430.4737.0038 j) 7.º 7.º 0.2940 0.9558 0.3076 3.2506 3.4009.0463 k) 940.7º 40.7º 0.652 0.758 0.860.626.5335.390 l) 362.6º 2.6º 0.0454 0.9990-0.0454 22.0444.000 22.027 286.477 286.479 m) 260.2º 0.2º 0.0035.0000 0.0035.0000 7 5 n).4.4 0.9854 0.700 5.7979 0.725.048 5.8835.993 o) 6.2 0.0832 0.083 0.9965 0.0834 2.0352.0035 6 p) 0.65 0.65 0.6052 0.796 0.7602.354.6524.256 34.232 q).6.546 0.9996 0.0292 0.0292.0004 34.247 5 ).4.4 0.9854 0.700 5.7979 0.725.048 5.8835 s) 4.9.3832 0.9825 0.865 5.2675 0.898.079 5.366 Module T2 Tigonometic Functions of An Angle 9
OBJECTIVE FIVE When ou complete this objective ou will be able to Evaluate invese tigonometic functions. Eploation Activit CONDITIONS FOR EVALUATING INVERSE TRIGONOMETRIC FUNCTIONS We have seen that evaluating epessions like sin 85º esults in one answe. ie: sin 85º = 0.9962 Howeve, if we wish to find the angle that has a sine equal to 0.9962 then we would wite: sin = 0.9962 Solving fo we would get: = ac sin 0.9962 o = sin 0.9962 = 85º = 85º This method woks well when all of the angles fo ae acute. Howeve, thee ae man othe values of that also have a sine equal to 0.9962. Eample: Accoding to the CAST ule, sine is positive in quadant 2 as well. The following cicle diagam show that sin 85º and an angle with a efeence angle of 85º in quadant 2 will have the same tigonometic atio. 85º is in quadant sin 85º = 0.9962 = 85º 95º is in quadant 2 sin 95º = 0.9962 The efeence angle of 95º is 85º. The atio is positive because sine is positive in quadant 2 α = 85º = 95º 20 Module T2 Tigonometic Functions of An Angle
265º is in quadant 3 sin 265º = 0.9962 The efeence angle of 265º is 85º. The atio is negative because sine is negative in quadant 3 Although 265º has a efeence angle of 85º it is not a solution because the sine of 265º is negative. 275º is in quadant 4 sin 275º = 0.9962 The efeence angle of 275º is 85º. The atio is negative because sine is negative in quadant 4 Although 275º has a efeence angle of 85º it is not a solution because the sine of 275º is negative. = 265º α = 85º = 275º α = 85º Summa An angle that has a efeence angle of 85º and is in quadants and 2 will have a sine of positive 0.9962. In the eample the solutions wee 85º and 95º. Howeve, an angle coteminal with 85º and 95º will also be solutions: 445º and 275º ae all coteminal with 85º 455º and 265º ae all coteminal with 95º Theefoe, sin 445º = sin 275º = sin 455º = sin 265º All of these evaluate to 0.9962. To get aound this poblem of man possible answes, we will include a condition with each question. 0º < 360º This condition indicates that the answe fo must be geate than o equal to 0º and less than 360º. Conclusion Find fo sin = 0.9962 given 0º < 360º With this condition, the values of fo the question sin = 0.9962 would be 85º and 95º onl. Othe angles coteminal with 85º and 95º would not satisf the condition of 0º < 360º Module T2 Tigonometic Functions of An Angle 2
EVALUATING INVERSE TRIGONOMETRIC FUNCTIONS Thee ae thee steps needed to evaluate invese tigonometic functions. We will evaluate the following question to demonstate the thee steps. Find the value fo: sin = 0.5976 given 0º < 360º Step : Detemine the quadant(s) will be in given the sign of the tig atio and the condition. The condition 0º < 360º indicates that must be geate than o equal to 0º and less than 360º. The tigonometic atio of sine of is positive. The CAST ule shows that sine is positive onl in quadants and 2. Quadant II Sine Positive Quadant III Tangent Positive Quadant I All Positive Quadant IV Cosine Positive Theefoe, we will have two solutions fo. One solution in quadant and one solution in quadant 2. Step 2: Detemine the efeence angle. The tig atio of the efeence angle α is alwas the absolute value of the tig atio of. sin = 0.5976 sin α = 0.5976 solving fo α α = ac sin 0.5976 o α = sin 0.5976 α = 36.7º Step 3: Use the efeence angle to detemine. Fom step we know we will have answes in quadants and 2. Answe in Quadant : The efeence angle α and angle ae the same in quadant Theefoe, one solution is =36.7º Check: Ente sin 36.7º into ou calculato to check. 22 Module T2 Tigonometic Functions of An Angle
Answe in Quadant 2: We know that the efeence angle α is 36.7º. We need to find the angle in quadant 2 that has a efeence angle of 36.7º. If we gaph the efeence angle in quadant 2 we can detemine angle. Fom the gaph we can see that the sum of α and in quadant 2 =80º. Theefoe, = 80º α. = 80º 36.7º = 43.30º Check: Ente sin 43.3º into ou calculato to check. Thus, = 36.7º and 2 = 43.3º α = 36.7º = 43.3º It is sometimes desiable to find the angle in adians. Again ou calculato will do this opeation as long as ou change it to adian mode. DETERMINING GIVEN THE REFERENCE ANGLE. In step thee above, once ou have detemined the efeence angle ou will need to detemine angle using the efeence angle. The following diagam will help ou when detemining angle fom the efeence angle α in an quadant. CAST C is cosine A is all S is sine T is tangent Quadant II S = 80º α Quadant III T = 80º + α Quadant I A = α Quadant IV C = 360º α Calculato comment Most calculatos will not give moe than one answe fo an invese function. Be sue not to just cop the calculato answe. Thee usuall ae moe answes, so think it though. Module T2 Tigonometic Functions of An Angle 23
MORE EAMPLES Eample : Find the value given 0 < 2π cos = 0.9690 Step : Detemine the quadant(s) will be in. The cosine of is positive. Accoding to the CAST ule cosine is positive in quadants and 4. The condition 0 < 2π indicates that must be geate than o equal to 0 adians and less than 2π adians. Thee will be two solutions fo. One solution in quadant and one solution in quadant 4. Step 2: Detemine the efeence angle α. (Make sue ou calculato is in adian mode.) cos = 0.9690 cos α = 0.9690 α = cos 0.9690 α = 0.2496 Step 3: Use the efeence angle to detemine. Fom step we know we need solutions in quadants and 4. It will help to daw the angles to detemine. Answe in Quadant : = α = 0.2496 Check: cos 0.2496 = 0.9690 Answe in Quadant 4: = 2π α = 2π 0.2496 = 6.0336 Check: cos 6.0336 = 0.969 (off slightl due to ounding) α = 0. 2496 Thus = 0.2496 and 2 = 6.0336 24 Module T2 Tigonometic Functions of An Angle
Eample 2: Find the value given 0º < 360º sec = 5.2408 Step : Detemine the quadant(s) will be in. The secant of is positive. Accoding to the CAST ule secant is positive in quadants and 4 since it is the ecipocal of cosine. The condition 0º < 360º indicates that must be geate than o equal to 0º and less than 360º. Thee will be two solutions fo. One solution in quadant and one solution in quadant 4. Step 2: Detemine the efeence angle α. sec = 5.2408 sec α = 5.2408 Since calculatos do not have sec functions, it is necessa to fist convet this to the cosine function. cos α = = sec 5.2408 α = cos 5.2408 α = 79.0º Step 3: Use the efeence angle to detemine. Fom step we know we need solutions in quadants and 4. It will help to daw the angles to detemine. Answe in Quadant : = α = 79.0º Check: sec 79.0º = cos 79.0 Answe in Quadant 4: = 360º α = 360º 79.0º = 28.0º Check: sec 28.0º = cos 28.0 Thus = 79.0º and 2 = 28.0º = 5.2408 = 5.2408 α = 79.0º Module T2 Tigonometic Functions of An Angle 25
Eample 3: Find the value given 0º < 360º sin = 0.366 Step : Detemine the quadant(s) will be in. The sine of is negative. Accoding to the CAST ule sine is negative in quadants 3 and 4. The condition 0º < 360º indicates that must be geate than o equal to 0º and less than 360º. Thee will be two solutions fo. One solution in quadant 3 and one solution in quadant 4. Step 2: Detemine the efeence angle α. sin = 0.366 sin α = 0.366 α = sin 0.366 α = 2.2º Step 3: Use the efeence angle to detemine. The tigonometic atio of the efeence angle α is the absolute value of the tigonometic atio of. Answe in Quadant 3: = 80 + α = 80º + 2.2º = 20.2º Check: sin 20.2º = 0.366 α = 2.2º 26 Module T2 Tigonometic Functions of An Angle
Answe in Quadant 4: = 360º α = 360º 2.2º = 338.8º Check: sin 338.8º = 0.366 α = 2.2º Thus = 20.2º and 2 = 338.8º Eample 4: Find the value given 0º < 360º csc =.7965 Step : Detemine the quadant(s) will be in. The cosecant of is negative. Accoding to the CAST ule cosecant is negative in quadants 3 and 4 since it is the ecipocal of sine. The condition 0º < 360º indicates that must be geate than o equal to 0º and less than 360º. Thee will be two solutions fo. One solution in quadant 3 and one solution in quadant 4. Step 2: Detemine the efeence angle α. csc =.7965 csc α =.7965 convet to sine function sin α =.7965 α = sin.7965 α = 33.8º The tigonometic atio of the efeence angle α is the absolute value of the tigonometic atio of. α = 33.8º Module T2 Tigonometic Functions of An Angle 27
Step 3: Use the efeence angle to detemine. Answe in Quadant 3: = 80º + α = 80º +33.8º = 23.8º Check: csc 23.8º = sin 23.8 =.7965 Answe in Quadant 4: = 360º α = 360º 33.8º = 326.2º Check: csc 326.2º = sin 326.2 =.7965 α = 33.8º Thus 3 = 23.8º and 4 = 326.2.2º 28 Module T2 Tigonometic Functions of An Angle
ALGEBRAIC SIGNS OF TRIGONOMETRIC FUNCTIONS OF QUADRANTAL ANGLES The last tigonometic functions that we ll look at in this module ae tig functions of quadantal angles. Quadantal Angle Definition Angles with a teminal side on the o -ais ae called quadantal angles. Eample 90º, π, 270º, 2π, 450º, 90º, π, and 3π 2 Tigonometic Functions of Quadantal Angles The following demonstates how to detemine the tigonometic functions of quadantal angles less than 360º. The same ules appl fo angles 360º. ou calculato will also give the tigonometic functions of angles 360º b enteing the angles diectl. Module T2 Tigonometic Functions of An Angle 29
Tigonometic functions of 90º Let s assume = emembeing that is alwas positive. B using the cicle method, we notice that as appoaches 90º, appoaches zeo and appoaches. π When = 90º & = 2 = 0 =. = = Since falls on the positive -ais =. = 0 We can now use the si tig functions fom the cicle method and substitute in the values π fo,, and fo 90º. Remembe that 90º = and these concepts appl to adians as 2 well as degees. sin = csc = cos = sec = tan = cot = sin 90º = csc 90º = cos 90º = 0 sec 90º = 0 tan 90º = 0 cot 90º = 0 sin 90º = csc 90º = cos 90º = 0 sec 90º = undefined tan 90º = undefined cot 90º = 0 sin 2 π = csc 2 π = cos 2 π = 0 sec 2 π = 0 tan 2 π = 0 cot 2 π = 0 sin 2 π = csc 2 π = cos 2 π = 0 sec 2 π = tan 2 π = cot 2 π = 0 undefined undefined 30 Module T2 Tigonometic Functions of An Angle
TRIGONOMETRIC FUNCTIONS OF OTHER QUADRANTAL ANGLES. = 80º & = π = 0 =. Since falls on the negative - ais =. = 0 = = Using the si tig functions fom the cicle method and substituting in the values fo,, and fo 80º we get the following. sin = csc = cos = sec = tan = cot = sin 80º = 0 csc 80º = 0 cos π = sec 80º = tan 80º = 0 cot π = 0 sin 80º = 0 csc 80º = undefined cos π = sec 80º = tan 80º = 0 cot π = undefined = 270º & = 3π 2 = 0 =. Since falls on the negative -ais =. = = 0 = Using the si tig functions fom the cicle method and substituting in the values fo,, and fo 270º we get the following. sin = csc = cos = sec = tan = cot = 3π sin = 2 csc 3π = 2 cos 270º = 0 3π sec = 2 0 tan 270º = 0 cot 270º = 0 3π sin = 2 3π csc 2 = cos 270º = 0 3π sec = 2 undefined tan 270º = undefined cot 270º = 0 Module T2 Tigonometic Functions of An Angle 3
QUADRANTAL ANGLES CONTINUED = 0º o 360º & = 2π = 0 =. Since falls on the positive - ais =. = = = 0 Using the si tig functions fom the cicle method and substituting in the values fo,, and fo 360º we get the following. sin = csc = cos = sec = tan = cot = sin 360º = 0 csc 360º = 0 cos 2π = sec 360º = tan 2π = 0 cot 2π = 0 sin 360º = 0 csc 360º = undefined cos 2π = sec 360º = tan 2π = 0 cot 2π = undefined Tigonometic functions that ae undefined An tigonometic definition that esults in a denominato containing a zeo (0) value will poduce an undefined tigonometic function at that paticula angle. Eample: csc 360º = = 0 = undefined Tigonometic functions that esult in a zeo (0) value An tigonometic definition that esults in a zeo numeato (denominato not = 0) will poduce a zeo value. Eample: tan 2π = = 0 = 0 Note: In cases whee questions ma ask ou to indicate the algebaic sign of a tigonometic function, if the answe is 0, ente 0, not positive o negative. 32 Module T2 Tigonometic Functions of An Angle
SUMMAR OF QUADRANTAL ANGLES The following labeled diagam should be of assistance to ou when ou ae solving invese function poblems of quadantal angles 80º sin 80º = 0 csc 80º = undefined cos 80º = sec 80º = tan 80º = 0 cot 80º = undefined 90º sin 90º = csc 90º = cos 90º = 0 sec 90º = undefined tan 90º = undefined cot 90º = 0 270º sin 270º = csc 270º = cos 270º = 0 sec 270º = undefined tan 270º = undefined cot 270º = 0 0º and 360º sin 0º = 0 csc 0º = undefined cos 0º = sec 0º = tan 0º = 0 cot 0º = undefined ou could memoize the above elationships. The ae not as difficult as the ma appea, i.e., note the patten fo sine as ou go fom 0º to 90º to 80º to 270º: sine goes 0,, 0,. Simila pattens eist fo cosine and tangent. CALCULATOR USAGE AND QUADRANTAL ANGLES Use ou calculato to evaluate the following and then compae ou answes to the chat. tan ( 90º) csc π cot 270º ou should get an ERROR answe. Does this mean that eve ERROR answe is equivalent to an undefined tigonometic function? No, it means that cot 270º and cot 90º cannot be evaluated using ou calculato!!! Wh? Thee isn t a cot button on ou calculato so we use the following elationship to evaluate the cot 270º. cot 270º = tan 270 Using the calculato evaluates as = undefined. tan 270 undefined Since tan 270º = undefined, we cannot use to evaluate cot 270º. tan 270 Module T2 Tigonometic Functions of An Angle 33
Epeiential Activit Five. Find all the values of given 0º < 360º (neaest tenth of a degee) a) sin = 0.900 f) cos = 0.7070 b) cos = 0.8200 g) sin =.0000 c) tan = 0.400 h) tan = 0.2500 d) sin = 0.4300 i) cos =.0000 e) tan =.2000 2. Find all the values of given 0 < 2π (answe to 4 decimals) a) sin = 0.582 f) cos = 0.0279 b) cos = 0.4555 g) sin = 0.7804 c) tan = 3.505 h) cos = 0.9763 d) sin = 0.3778 e) tan =.076 Show Me. 3. Complete the table fo 0º < 360º. Note: Unless is given, ou will have two angles fo and two values fo an tigonometic functions not given. Find to the neaest tenth of a degee and othe answes to 3 significant digits Tigonometic atios to 3 significant digits. sin cos tan 50.0º 233.5º 345.0º 64.3º 360.0º -0.460 0.968-0.95 0.66 0.680 2.32 34 Module T2 Tigonometic Functions of An Angle
4. Solve each of the following fo A when 0º A < 360º. Round answes to one decimal place. (Each of these will have TWO answes; detemine the quadant in which the teminal am lies fo those angles fist.) a) csc A =.9084 Show Me. b) cot A = 4.5067 c) tan A = 0.8645 d) sec A = 2.4 e) sec A =.2352 f) sec A = 3.2633 g) cot A = 4.5294 h) csc A =.0067 i) cos A = 0.7654 Show Me. j) sin A = 0.8789 k) cos A = 0.4563 l) tan A = 5.4000 5. Without using a calculato, give the tigonometic atios of each of these quadantal angles. Angle sin A cos A tan A csc A sec A cot A 0º 90º 80º 270º Module T2 Tigonometic Functions of An Angle 35
Epeiential Activit Five Answes. a) 65.5º, 4.5º b) 34.9º, 325.º c) 22.3º, 202.3º d) 205.5º, 334.5º e) 50.2º, 230.2º f) 45.0º, 35.0º g) 90.0º h) 4.0º, 94.0º i) 80.0º 2. a) 0.623, 2.5203 b) 2.0437, 4.2395 c).8483, 4.9899 d) 3.5290, 5.8958 e) 2.396, 5.462 f).5987, 4.6845 g) 0.8953, 2.2463 h) 2.9234, 3.3597 3. sin cos tan 50.0º 0.500 0.866 0.577 233.5º 0.804 0.595.35 345.0º 0.259 0.966 0.268 207.4º 0.888 0.58 0.460 332.6º 0.888 0.58 62.0º 0.309 0.325 0.95 98.0º 0.309 0.325 34.2º 0.562 0.827 24.2º 0.562 0.827 0.680 64.3º 0.27 0.963 0.28 75.5º 0.25 3.86 0.968 04.5º 0.25 3.86 80.4º 0.986 5.9 0.66 279.6º 0.986 5.9 3.3º 0.98 0.396 293.3º 0.98 0.396 2.32 360.0º 0 0 4. a) A is in quad 3 and 4, A=2.6º, 328.4º, b) A is in quad and 3, A= 2.5º, 92.5º c) A is in quad 2 and 4, A= 39.2º, 39.2º d) A is in quad and 4, A= 6.7º, 298.3º e) A is in quad 2 and 3, A= 44.º, 25.9º f) A is in quad 2 and 3, A= 07.8º, 252.2º g) A is in quad 2 and 4, A= 67.5º, 347.5º h) A is in quad and 2, A= 83.4º, 96.6º i) A is in quad 2 and 3, A= 39.9º, 220.º j) A is in quad 3 and 4, A= 24.5º, 298.5º k) A is in quad and 4, A= 62.9º, 297.º l) A is in quad and 3, A= 79.5º, 259.5º 5. Angle sin A cos A tan A csc A sec A cot A 0º 0 0 undefined undefined 90º 0 undefined undefined 0 80º 0 0 undefined undefined 270º 0 undefined undefined 0 36 Module T2 Tigonometic Functions of An Angle
Pactical Application Activit Complete the Tigonometic Functions of An Angle assignment in TLM. Summa This module showed the student how to deal with the tigonometic functions of angles that ae geate than 90º. Module T2 Tigonometic Functions of An Angle 37