FURTHER TRIGONOMETRY B the end of this set of eercises, ou should be able to (a) recognise the graphs of sine, cosine and tangent functions sketch and identif other trigonometric functions solve simple trigonometric equations (in degrees) define the period of trigonometric functions, either from their graphs or from their equations (e) simplif trigonometric epressions using sin + cos = 1 sin and tan = cos Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 5
A. Sine, cosine and tangent graphs Eercise 1 You ma have drawn the sine, cosine and tangent graphs as part of the introduction to trigonometr in Maths Intermediate. If ou have retained the graphs, ou ma miss out questions 1 to 3 of Eercise 1 below. 1. The Sine Graph (a) Make a cop of this table and use our calculator to help fill it in, giving each answer correct to decimal places. 0 0 40 60 80 90 100 10 140 160 180 sin 0á00 0á34 0á64 0á87 0á98 1á00............... 00 0 40 60 70 80 300 30 340 360 sin.............................. Use a piece of mm graph paper to draw a set of aes as illustrated below. 1 0 90 180 70 360 Ð1 Plot as accuratel as possible the 1 points from our table. Join them up smoothl to create the graph of the function = sin.. Repeat question 1 (a) to for the function = cos 3. Repeat for the graph of = tan (a different scale will be required for the vertical ais). 4. Stud our three graphs carefull. You should now be able to sketch the sine, cosine (and tangent) graphs fairl quickl (about 30 seconds) indicating the main points where the graphs cut the and aes and the general shape of each graph. Tr them now. Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 6
Sketches of the three trigonometric graphs, for comparison: 1 = sin = tan 90 180 70 360 Ð1 1 = cos 90 180 90 180 70 360 Ð1 Check our graphs are similar to those shown above. B. Sketching and identifing trigonometric functions Eercise A 1. Write down the equations of the trigonometric functions represented b the following graphs: (a) 10 6 90 180 70 360 90 180 70 360 Ð10 Ð6 5 0á8 90 180 45 90 Ð5 Ð0á8 Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 7
(e) 0 (f) 80 180 360 180 360 Ð0 Ð80 (g) 5 (h) 4 360 70 180 360 Ð5 Ð4 (i) 50 (j) 60 10 Ð50 45 90 (k) 0á (l) 15 360 70 180 360 Ð0á Ð15. Make neat sketches of the following trigonometric functions, clearl indicating (i) the shape of the graph (draw one ÔccleÕ of it onl) (ii) the important values on the horizontal ais (iii) the maimum and minimum values of the function. contõd... Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 8
(a) = 3sin = 4cos = tan 3 = 10sin 3 (e) = 1cos (f) = 0á7sin 4 (g) = 1ácos 4 (h) = 30sin 6 (i) = 100cos 5 (j) = Ðsin (k) = Ð6sin 1 / (l) = Ð0cos 3 3. Make neat sketches of the following over the given range of values: (a) = 60sin 0 360 = á5cos 3 0 360 = 40sin 4 0 180 = Ðcos 6 0 180 (e) = Ð15sin 8 0 180 (f) = 1á8cos 30 0 1 Eercise B 1. Write down the equations of the trigonometric functions in the form = k sin( Ð a) or = k cos( Ð a) represented b the following graphs: (a) 5 8 Ð30 90 180 70 360 Ð0 90 180 70 360 Ð5 Ð8 5 0á6 10 90 180 70 360 40 90 180 70 360 Ð5 Ð0á6 (e) 4 (f) 15 35 90 180 70 360 Ð70 90 180 70 360 Ð4 Ð15. Make neat sketches of the following trigonometric functions, showing clearl: (i) where each graph cuts the - ais. (ii) the maimum and minimum values. (a) = 10sin( Ð 15) = 1á4 cos( Ð 0) = 15sin( + 10) = á4 cos( + 30) (e) = 300sin( Ð 80) (f) = tan( Ð 10) Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 9
C. Solving trigonometric equations Eercise 3 S (in) T (an) A (ll) C (os) 1. Find the two solutions for each of the following in the range 0 360: (Give each answer correct to the nearest whole degree). (a) sin = 0á500 cos = 0á707 tan = 0á869 cos = 0á940 (e) tan = 1á80 (f) sin = 0á574 (g) sin = 0á990 (h) tan = 6á314 (i) cos = 0á391 (j) cos = 0á985 (k) sin = 0á866 (l) tan = 1á73. Rearrange each of the following and solve them in the range 0 360. (Give our answers correct to 1 decimal place). (a) cos Ð 1 = 0 5sin Ð 4 = 0 10tan Ð 7 = 0 1 Ð 3sin = 0 (e) 5 Ð 6cos = 0 (f) 3tan Ð 5 = 0 3. Find the two solutions for each of the following in the range 0 360: (Give each answer correct to the nearest whole degree). (a) sin = Ð0á500 cos = Ð0á707 tan = Ð0á384 cos = Ð0á9 (e) tan = Ð1á000 (f) sin = Ð0á866 (g) tan = Ð 4 (h) sin = Ð0á174 (i) cos = Ð0á97 4. Rearrange each of the following and solve them in the range 0 360. (Give our answers correct to 1 decimal place). (a) 4sin + 1 = 0 5cos + 3 = 0 3tan + 1 = 0 7 + 8cos = 0 (e) 0á4sin + 0á3 = 0 (f) 5tan + 8 = 0 5. Solve the following miture of trigonometric equations in the range 0 360. (Give our answers correct to 1 decimal place). (a) sin = 0á33 cos = Ð0á9 tan = 0á678 cos = 1 / 4 (e) sin = Ð0á707 (f) tan = Ð (g) sin = 3 / 5 (h) cos = Ð0á111 (i) tan = 5 / 8 (j) 8sin + 5 = 0 (k) 6cos + 3 = 0 (l) 1 Ð 5tan = 0 (m) 0sin Ð 17 = 0 (n) 15 Ð 5cos = 0 (o) 8tan + 7 = 0 (p) 5sin + 3 = sin + 5 (q) 7cos Ð 1 = cos + 4 (r) 10tan + 8 = 3tan + 4 (s) 6sin + 11 = 3sin + 10 Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 30
D. The period of a trigonometric function Eercise 4 1. Determine the period and the maimum and minimum values of these trig functions. (a) 7 8 90 180 30 60 Ð7 Ð8 0á5 5 45 18 Ð0á5 Ð5 (e) 9 (f) 6 360 Ð9 30 Ð6 (g) 0á1 180 Ð0á1. Determine the period and the maimum and minimum values of these trig functions. (a) = 5sin = 3cos3 = 10tan4 = ácos (e) = 30sin 6 (f) = Ð5cos30 (g) = 50sin 90 (h) = Ð 4cos 1 / (i) = 18sin 1 / 4 (j) = 0á9cos 60 (k) = 1 / sin 5 (l) = 3 / 4 cos 9 (m) = 11sin 180 (n) = 8sin 1á5 (o) = 40cos á5 Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 31
E. Trigonometric identities Eercise 5 sin Remember :- sin + cos = 1; and tan = cos 1. Simplif the following using the above identities: (a) sin + cos 5cos + 5sin 3sin 5sin cos cos. Write down a simple epression, identical to: sin (a) 1 - sin 1 - cos tan cos cos 3. Simplif: (a) 1 - cos 1 - sin sin sin cos cos 1 - sin 1 - cos (e) (f) tan ( 1 - sin ) cos 5sin 4. Prove the following trigonometric identities: (a) 3-3sin = 3cos 5-5cos = 5sin 1 - cos = sin tan 1 - sin = sin 1 - cos (e) 1 - sin = tan 1 - sin 1 (f) 1 - cos = tan Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 3
Checkup for further trigonometr 1. Make a sketch of the sine, cosine and tangent graphs, indicating all their main features.. Write down the equations of the trigonometric functions associated with the following graphs: (a) 8 0á7 90 180 45 90 Ð8 Ð0á7 á7 180 360 Ðá7 30 60 3. Make neat sketches of the following, indicating all the main points and features: (a) = 0sin 4 0 90 = 1á6cos 0 360 = Ð8sin 8 0 90 = tan 0 90 4. Find the two solutions for each of the following in the range 0 360: (Give our answers correct to 1 decimal place). (a) sin = 0á911 cos = 0á444 tan = 3 cos = Ð0á605 (e) tan = Ð0á8 (f) sin = Ð 4 / 5 (g) sin Ð1 = 0 (h) 8cos + 6 = 0 (i) 4tan Ð 3 = 0 (j) 3cos Ð = 0 (k) 1 + 4sin = 0 (l) 5tan = 3tan Ð Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 33
5. What are the periods of the following trigonometric graphs and functions? (a) 6 7 60 180 360 Ð6 Ð7 15 180 Ð15 0 40 (e) = 10sin 10 (f) = á3cos 30 (g) = Ð 4sin 9 (h) = 5tan 4 6. (a) Simplif: (i) 6-6sin (ii) cos sin Prove these identities :- (i) 1 - cos = 1 sin (ii) ( 1- sin )( 1+ sin )= cos 7. Write down the equations of the following trigonometric graphs: (a) 8 3á6 Ð15 180 345 360 0 00 380 Ð8 Ð3á6 8. Sketch the following trigonometric graphs, indicating their main features: (a) = 18cos( + 30) 0 360 = sin( Ð 10) 0 360 Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 34
Further trigonometr Eercise 1 1. Check Graphs Ð see graphs at top of page 15 for comparison. Eercise A 1. (a) = 10sin = 6cos = 5sin = 0á8cos 4 (e) = 0sin (f) = 80cos (g) = 5sin 1 / (h) = Ð4sin (i) = Ð50cos 3 (j) = tan (k) = 0ácos 1 / (l) = 15sin 4. (a) 3 4 Ð3 360 Ð4 180 360 30 60 (e) (f) 1 10 0á7 Ð10 10 Ð1 90 180 90 Ð0á7 (g) (h) (i) 1á 30 100 45 90 60 36 7 Ð1á Ð30 Ð100 (j) (k) (l) 1 6 0 360 70 60 10 Ð1 Ð6 Ð0 Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 46
3. (a) 60 á5 40 360 360 180 Ð60 Ðá5 Ð40 (e) (f) 15 1á8 Ð 180 Ð15 180 Ð1á8 6 1 Eercise B 1. (a) = 5sin( + 30) = 8cos( + 0) = 5sin( Ð 10) = 0á6sin( Ð 40) (e) = 4cos( Ð 35) (f) = 15sin( + 70) or = 15cos( Ð 0). (a) 10 1á4 Ð10 15 375 Ð1á4 0 380 15 á4 Ð10 350 Ðá4 30 330 Ð15 (e) 300 Ð300 (f) 440 80 10 100 190 Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 47
Eercise 3 1. (a) 30, 150 45, 315 41, 1 0, 340 (e) 5, 3 (f) 35, 145 (g) 8, 98 (h) 81, 61 (i) 67, 93 (j) 10, 350 (k) 60, 10 (l) 60, 40.. (a) 60, 300 53á1 or 16á9 35á0, 15á0 19á5, 160á5 (e) 33á6, 36á4 (f) 59á0, 39á0. 3. (a) 10, 330 135, 5 159, 339 107, 53 (e) 135, 315 (f) 40, 300 (g) 104, 84 (h) 190, 350 (i) 158, 0. 4. (a) 194á5, 345á5 16á9, 33á1 161á6, 341á6 151á0, 09á0 (e) 8á6, 311á4 (f) 1á0, 30á0. 5. (a) 18á8, 161á 154á, 05á8 34á1, 14á1 75á5, 84á5 (e) 5, 315 (f) 116á6, 96á6 (g) 36á9, 143á1 (h) 96á4, 63á6 (i) 3á0, 1á0 (j) 18á7, 31á3 (k) 10, 40 (l) 11á3, 191á3 (m) 58á, 11á8 (n) 53á1, 306á9 (o) 138á8, 318á8 (p) 41á8, 138á (q) 33á6, 36á4 (r) 150á3, 330á3 (s) 199á5, 340á5. Eercise 4 1. Graph (a) (e) (f) (g) Period 180 60 90 36 180 10 360 Ma/Min ±7 ±8 ±0á5 ±5 ±9 ±6 ±0á1. Question (a) (e) (f) (g) (h) Period 180 10 45 180 60 1 4 70 Ma/Min ±5 ±3 ±10 ±á ±30 ±5 ±50 ±4 Question (i) (j) (k) (l) (m) (n) (o) Period 1440 6 7 40 40 144 Ma/Min ±18 ±0á9 ±0á5 ±0á75 ±11 ±8 ±40 Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 48
Eercise 5 1. (a) 5 3tan 5 / tan. (a) cos sin sin tan 3. (a) 1 1 / tan cos (e) 1 / 5 sin (f) sin 4. All proofs. Checkup for further trigonometr 1. See sketches on page 15.. (a) = 8sin = 0á7cos4 =Ðá7sin3 = tan3 3. (a) 0 90 1á6 Ð0 Ð1á6 360 Ð10 8 90 45 90 Ð8 4. (a) 65á6, 114á4 63á6, 96á4 71á6, 51á6 17á, 3á8 (e) 141á3, 31á3 (f) 33á1, 306á9 (g) 30, 150 (h) 138á6, 1á4 (i) 36á9, 16á9 (j) 48á, 311á8 (k) 194á5, 345á5 (l) 135, 315. 5. (a) 10 180 60 40 (e) 36 (f) 1 (g) 40 (h) 90. 6. (a) (i) 6cos (ii) 1 tan Proof. 7. (a) = 8sin( + 15) = 3á6cos( Ð 0) 8. (a) 18 Ð30 330 10 Ð 370 Ð18 Mathematics Support Materials: Mathematics 3 (Int ) Ð Student Materials 49