Trigonometry: Angles between 0 and 360

Transcription

1 Chapte 6 Tigonomet: Angles between 0 and 360 Leaning objectives B the end of this chapte, the students should be able to:. Detemine the sine, cosine and tangent of an angle between 0 and Given sin θ, cos θ and tan θ, detemine θ, whee 0 θ Use the unit cicle to develop gaphs fo sin θ, cos θ and tan θ, fo 0 θ Sketch the gaphs of the fom A ± B sin nθ fo simple numeical values of A, B and n. Teaching and leaning mateials Students: Tetbook, eecise book, witing mateials, gaph books o gaph pape and aids fo dawing cuves (wie o twine). Teache: Gaph boads, aids fo dawing cuves (wie o twine). If an ovehead pojecto is available: tanspaencies of gaph pape and tanspaenc pens. Teaching notes Make sue that the students undestand the fact that sin θ = _, cos θ = _ and tan θ = _ and that the size of θ detemines the values of the - and -coodinates and that the value of is alwas positive. (, ) The sin-atio contains a -coodinate and is, theefoe, positive in the second quadant. In the sketch, A and A' ae smmetical aound the -ais. So, all the angles in the second quadant can be witten as 80 θ, whee θ is an acute angle. Fo eample: sin 0 = sin (80 60 ) = sin 60 cos 0 = cos (80 60 ) = cos 60 tan 0 = tan (80 60 ) = tan 60 In the thid quadant, both the - and -coodinates ae negative; so all the tigonometic atios that contain eithe an -coodinate o a -coodinate, will be negative. The ae the sin and cos atios A(a, b) A'( a, b) Then eplain the following: In the second quadant, the -coodinate is negative; so all the tigonometic atios that contain an -coodinate will be negative. The ae the cos and tan atios. A ( a, b) B 80 A(a, b) (, ) The tangent-atio contains both an - and a -coodinate; and since a negative numbe divided b a negative numbe gives a positive answe, the tangent-atio is positive in the thid quadant. In the sketch, A and A', ae smmetical aound the oigin. So, the angles in the thid quadant can all be witten as 80 + θ, whee θ is an acute angle. Fo eample: sin 40 = sin ( ) = sin 60 cos 40 = cos ( ) = cos 60 tan 40 = tan ( ) = tan 60 4 Chapte 6: Tigonomet: Angles between 0 and 360

2 In the fouth quadant, the -coodinates ae negative; so all the tigonometic atios that contain a -coodinate will be negative. The ae the sin and tan atios. 360 A(a, b) D(a, b) The cos-atio contains an -coodinate and will be positive in the fouth quadant. In the sketch, A and A' ae smmetical aound the -ais. So, the angles in the fouth quadant can all be witten as 360 θ whee θ is an acute angle. Fo eample: sin 300 = sin ( ) = sin 60 cos 300 = cos ( ) = cos 60 tan 300 = tan ( ) = tan 60 So, eplain to the students that the sign of the tigonometic atio depends entiel on the quadant in which the adius of the cicle falls. When the sine and cosine gaphs ae seen as waves, students can also egad: Thei maimum deviation fom the -ais as the amplitude of the wave, which, in these cases, is. One epetition of the gaph as its peiod and fo = sin θ and = cos θ the peiod is 360. Tell the students that in = m cos θ o = m sin θ, the m epesents the amplitude of the gaph. So, if the have to daw a gaph like, fo eample, = sin θ o = _ cos θ, the maimum deviation (amplitude) fom the -ais is and _ espectivel: = cos θ and = _ cos θ (0, ) (, 0) 0 /360 (, 0) (0, ) 70 = sin θ and = sin θ When ou eplain to the students how to daw sketch gaphs of = sin θ and = cos θ, eplain that the onl have to show whee these gaphs intesect the - and -aes, and the coodinates of thei tuning points. So, to daw these gaphs students can use the unit cicle, and the points whee it intesects the - and -aes: At 0 : sin θ = _ = 0_ = 0 cos θ = _ = _ = At 90 : sin θ = _ = _ = cos θ = _ = 0_ = 0 At 80 : sin θ = _ = 0_ = 0 cos θ = _ = = At 70 : sin θ = _ = = cos θ = _ = 0_ = 0 At 360 : sin θ = _ = 0_ = 0 cos θ = _ = _ = Chapte 6: Tigonomet: Angles between 0 and 360 5

3 When students have to daw a gaph of = sin 3θ, fo eample, tell them that the can again use the unit cicle, but that the now have to find the values of θ, if the values of 3θ ae 0, 90, 80, 70 and 360 to find one peiod (o epetition) of the gaph as shown in this table: = cos() = cos θ θ = sin 3θ The emainde of the gaph then is just a epetition of the patten of one peiod, and can easil be dawn as shown below: When students have to daw a gaph of = cos _ θ, fo eample, tell them that the can again use the unit cicle, but that the now have to find the values of θ, if the values of _ θ ae 0, 90, 80, 70 and 360 to find one peiod (o epetition) of the gaph as shown in this table: θ _ θ = cos _ As befoe, students will find it eas to complete the emainde of the patten of the gaph. Eplain to the students that the can find the peiod of: = sin a and = cos a (the numbe of degees fo one epetition of the gaph) b dividing 360 b a. The can find the new peiod of = tan a b dividing 80 b a. If, fo eample, = tan _, then the new peiod is 80 _ = 80 = 360. When students have to daw gaphs of = a + sin θ o = a + cos θ, eplain to them that it simpl means that these gaphs ae tanslated a units upwads (if a > 0) o a units downwads (if a < 0). In tigonomet, the Geek lette θ (theta) is often used to indicate the size of an unknown angle. Othe Geek lettes often used to indicate sizes of angles in tigonomet ae α (alpha), β (beta) and φ (phi). You can eplain to the students that fom the unit cicle the should be able to see the following: When θ = 0, tan 0 = _ = 0_ = 0 When θ = 90, tan 90 = _ = _, which is not 0 defined. When θ = 80, tan 80 = _ = 0 = 0 When θ = 70, tan 70 =, which is not defined. When θ = 360, tan 360 = _ = 0_ = 0. You could also eplain to them that, if the t to detemine tan 90 o tan 70, b using scientific calculatos, thei calculatos would give something like Eo. = 0 6 Chapte 6: Tigonomet: Angles between 0 and 360

4 If scientific calculatos ae available, ou could let ou students investigate what happens to the tangent atio aound 90 and 70 if the answe these questions:. Detemine: a) tan 89.9 b) tan c) tan d) tan e) tan f) What do ou notice?. Detemine: a) tan 90. b) tan 90.0 c) tan d) tan e) tan f) What do ou notice? 3. Detemine: a) tan 69.9 b) tan c) tan d) tan e) tan f) What do ou notice? 4. Detemine: a) tan 70. b) tan 70.0 c) tan d) tan e) tan f) What do ou notice? 5. Wh can ou not find tan 90 o tan 70? You can then eplain the following to the class: If θ inceases fom 0 to 90, tan θ inceases fom 0 to an infinitel lage positive numbe. If θ inceases fom 90 to 80, tan θ inceases fom an infinitel big negative numbe to 0. If θ inceases fom 80 to 70, tan θ inceases fom 0 to an infinitel big numbe. If θ inceases fom 70 to 360, tan θ inceases fom an infinitel big negative numbe to 0. It is impossible to detemine tan 90 o tan 70, because at 90 o at 70 the -coodinate is equal to 0 and division b 0 is undefined. Since tan 90 and tan 70 ae not defined, we sa that the gaph has asmptotes thee. These asmptotes ae shown b a vetical boken line though 90 and 70. An asmptote is a line towads which the gaph comes neae and neae but can neve intesect. The gaph of the tan function, theefoe, consists of sepaate pats, and we sa that this function is discontinuous because it is not defined fo all values of the angle θ. If ou look at the gaph ou dew, ou will see that the gaph epeats itself eve 80. We, theefoe, sa that the tan function has a peiod of 80. Lastl ou can eplain to ou students that fo a gaph of the tan-function the need to show the following: The asmptotes with a vetical boken line at 90, 70,. Whee the gaph intesects the -ais (the hoizontal ais). Whee the gaph intesects the -ais (the vetical ais). The coodinates of the points whee tan θ =, fo eample, (45, ). The coodinates of the points whee tan θ =, fo eample, (35, ). When students daw the gaph of = tan θ, the have to find the values of θ, if the values of θ ae 0, 45, 90, 35, 80 to find one peiod (o epetition) of the gaph as shown in this table: θ θ = tan θ 0 Not defined = tan = tan Chapte 6: Tigonomet: Angles between 0 and 360 7

5 Eplain to the students that these gaphs ae eamples of gaphs of peiodic functions. A peiodic function is when its gaph epeats itself afte a cetain peiod. Eplain to the students that, when we solve an algebaic equation, we ae looking fo the values(s) of the vaiable that will satisf the When we solve a tigonometic equation, we must find the angle(s) that will satisf the If fo eample, the equation cos = has to be solved, we can see fom the gaph that = 0, if cos =. But if we look at the gaph of = cos, we see that 0 is onl one of the solutions of the The othe solution of is 40, because = intesects the gaph of = cos twice. You can eplain the othe tigonometic equations also with the help of gaphs Aeas of difficult and common mistakes If students use scientific calculatos to solve a tigonometic equation such as cos θ = 0.834, the tend to not fist take the positive value of the cos atio and fist wok out the angle in the fist quadant and then wok out the angles in the nd and 3 d quadants. Emphasise that the immediatel wite the following if, fo eample the cos-atio is negative: Step : θ is in the second quadant and the thid quadant Step : Wite down θ = 80 o θ = Step 3: Now find the acute angle, if cos θ = Step 4: Now complete Step b witing: θ = o θ = and complete ou solution b witing down the values of θ. Students ma find it difficult to daw sketch gaphs of combinations, fo eample, not onl = cos, but = + cos o not onl = sin, but = sin 3. Then teach them to fist daw, fo eample = cos. Then tanslate that gaph unit upwads, o fist daw the gaph of = sin 3 and then make the amplitude of the gaph double its pevious value. So, let them daw the gaphs in steps. - 8 Chapte 6: Tigonomet: Angles between 0 and 360