PracMaths. Trigonometry is Easy Grades 10 & 11. Seeliger ~ Mouton. Set by / Opgestel deur

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2 PracMaths Trigonometry is Easy Grades 0 & Set by / Ogestel deur Seeliger ~ Mouton Trigonometry is Easy Grades 0 & ~ PS JNM PULISHERS (Pty) Ltd 07 PO ox 955 WTERKLOOF 045 Tel: (0) Fax: (0) Fax: megarint@icon.co.za john@systemmaths.co.za OPYRIGHT RESERVED Unauthorised reroduction by hotocoying or any other method STRITLY PROHIITED. KOPIEREG VOOREHOU Ongemagtigde reroduksie deur fotokoiëring of enige ander metode STRENG VERODE.

3 Trigonometry / Trigonometrie Grade 0 and Trigonometry Section Toic Page Page Questions nswers Grade 0 Trigonometric ratios in a right-angled triangle alculations using a calculator 5 Solving right-angles triangles 9 D The function values of 0 o 45 o and 60 o E E Trigonometric ratios in terms of the co-ordinates of oints in the artesian lane 5 F ngles of elevation and deression 0 5 G/P G/P4 Grahs defined by y = a sin + q y = a cos + q and y = a tan + q 7 Grade H/ Function values of 80 o and 60 o 4 H/ Function values of secial angles 46 H/ Function values of 90 o and 90 o + 49 H/4 Function values of ( ) and ( + 60 o k) 5 I/ Trigonometric identities 54 I/ Restrictions 57 4 I/ Trigonometric equations 58 4 I/4 Trigonometric equations involving identities 59 4 J/ Solution of right-angled triangles 64 5 J/ The sine rule 67 6 J/ The area rule 70 6 J/4 The cosine rule 7 7 J/5 Mixed questions 74 7 K/P & K/P Grahs defined by y = a sin k(x + ) y = a cos k(x + ) 76 9 and y = a tan k(x + ) L Mixed questions 8 ongratulations you are on your way to success in Maths! Learners say The books work!!! oyright reserved

4 Trigonometry is easy Grade 0 Grade 0 Trigonometry. Trigonometric ratios in a right-angled triangle.. asic background. a) Trigonometry (from Greek trigonon a triangle and metron a measure) is the branch of Mathematics which deals with the relationshi between the sides and angles of a triangle. b) When denoting angles by single letters we use: i) aital letters: etc. ii) Greek letters: a (alha) β (beta) γ (gamma) θ (theta) etc. c) When caital letters are used to name the vertex angles of a triangle the sides oosite them are named by the corresonding small letters e.g. side b is oosite ˆ. d) In the adjacent right-angled triangle i) with resect to ˆ is the oosite side c b is the adjacent side and is the hyotenuse. a ii) with resect to  is the oosite side is the adjacent side and c a is the hyotenuse. b. a) In each of the triangles in the adjacent figure: and DE is the side oosite  and and E the corresonding hyotenuses. Thus for any angle  the ratio of the length of the side oosite  to the length of the hyotenuse is a constant. This constant ratio is called the sine of  or sin. E D b) In each of the triangles the ratio of the length of the side adjacent to  to the length of the hyotenuse is a constant. This constant ratio: D in right-angled triangle and E in right-angled ΔDE is called the cosine of  or cos. ED c) In right-angled triangles and DE the value of and is a constant called the D tangent of  or tan Â. N: The trigonometric ratios are indeendent of the lengths of the sides of a right-angled triangle and deend only on the angle size. Hence trigonometric ratios are considered to be functions of the angles. Grade 0 Trigonometry is easy

5 Trigonometry is easy Grade 0. Study: a) In Δ: b) In Δ: oosite side length sin = = hyotenuse length adjacent side length cos = = hyotenuse length oosite side length tan = = adjacent side length sin = cos = and tan = 4. omlete: D 4. i) sin E = ii) sin D = cos E = cos D = tan E = tan D = E F 4. P i) sin R = ii) sin P = cos R = cos P = tan R = tan P = 4. T R or i) sin = ii) sin = cos = cos = tan = tan = 4.4 D i) sin = ii) tan = sin = tan = cos = sin D = cos = sin D = Grade 0 Trigonometry is easy

6 Trigonometry is easy Grade In Δ: sin = In ΔD: sin = D In Δ: sin = In ΔD: sin = In ΔD: cos = In ΔD: cos = In ΔD: tan = In Δ: tan = 5. omlete: G H D E F 5. In ΔDEH: 5. In ΔDFG: 5. In ΔHEF: 5.4 In ΔGHF: HE HE DE = = and = DH DE DH GF DF GF = = and = DG DG DF HE EF HE = = and = HF HF EF HF HF HG = = and = GF HG GF 6. Study: 6. is the recirocal of t if = t or t =. 6. In the adjacent triangle: a c a b b c by definition: = sin = cos and = tan. b c a The recirocal ratios b b and c are resectively called the cosecant secant and cotangent of Â. a c a Grade 0 Trigonometry is easy

7 Trigonometry is easy Grade omlete: e.g. 7. sin = 5 means the recirocal cosec = 5 7. sin = 7. cos = tan = 5 4 means the recirocal = means the recirocal = means the recirocal = 7.5 cosec = 5 means the recirocal = 7.6 sec = cot = 7 means the recirocal = means the recirocal = 8. Write down the ratio value in terms of the sides of the given triangle: 8. cosec P sec P cot P 8. cosec R sec R cot R in triangle (b). a) R b) T P P T P R R T 9. Write as function values of the angles in the given figure: e.g. 9. D = cosec D or sec D ˆ in D D. 9. = or in D = = or or in in 9.5 D D = or in Grade 0 Trigonometry is easy

8 Trigonometry is easy Grade nswer each of the following questions without using a calculator. Leave the answer in the surd form if necessary. Examle: 0. Determine the value of: i) a ii) tan α iii) cosec α 5 a α i) a = 5 6 (Pyth) 4 = 9 a = ii) tan α = 4 iii) cosec α = 5 0. Determine the value of t sin θ and sec θ. t 5 θ 0. Determine the value of t cos and cot. 6 0 t alculations using a calculator. Study:. Use the calculator keys sin cos tan to determine the value correct to decimal laces of: e.g. i) sin 7 o ii) sin(46 o + o ) iii) sin 46 o + sin o iv) sin 7 o v) sin o nswers: i) sin 7 o = 096 : sin 7 ii) sin(46 o + o ) = sin 59 o = 086 : sin (46 + ) = iii) sin 46 + sin o = 095 : sin 46 + sin ) = iv) sin 7 o = 66 : sin 7 = v) sin o = (sin o ) = 08 : sin = x Grade 0 Trigonometry is easy

9 Trigonometry is easy Grade 0 6. omlete correct to decimal laces. a) cos o = b) sin(0 o + 0 o ) = sin 50 o = sin 7 o = cos(64 o 4 o ) = tan 6 o = tan(8 o + 4 o ) = c) sin 70 o = d) sin 40 o + sin o cos 8 o = = sin 6 o = = 4 tan( o + o ) = sin(40 o + o ) = o cos 6 = sin 6 o = tan o = cos8 o = = cos 80 o cos 0 o = = sin 6 o. cos 4 o = cos(80 o 0 o ) = cos 5 o. tan 8 o = = = e) cos 58 o = f) tan 95 o = cos 47 o = tan 95 o = cos 47 o = tan 7 o + tan 5 o = cos 49 o + cos 6 o = tan(7 o + 5 o ) = cos(49 o + 6 o ) = tan 6 o = g) sin (4 o + 8 o ) = cos (79 o 4 o ) = tan (4 o ) = cos 6 o + cos 6 o = Grade 0 Trigonometry is easy sin 96 o =. Without using a calculator mark whether each of the following statements is correct ( ) or incorrect ( ). a) i) sin(4 o + 0 o ) = sin 4 o + sin 0 o b) i) sin 60 = sin 60 ii) cos 76 o cos 0 o = cos(76 o 0 o ) ii) cos 46 o = (cos 46 o ) iii) tan o = tan 44 o iii) tan(0 o + 0 o ) = tan 50 o iv) cos(80 o 0 o ) = cos 60 o iv) cosec 70 o = sin 70 o v) tan 5 o = tan 50 o v) sec 40 o = sin 40 o vi) cos 40 o = cos 0 o vi) cos 60 o = sec60 o o o

10 Trigonometry is easy Grade alculate correct to decimal laces. a) i) cos 5 o cos o o = b) i) sin 46 = ii) tan 4 o + tan 4 o o = ii) cos 69 = iii) o 6 o cos cos 6 = iii) 8 tan 80 4 o = iv) sin 60o = iv) sin 0 o = 5. There is no cosec sec or cot key on a calculator. Thus we use the fact that cosec α = sin α sec α = cos α and cot α = tan α and the calculator key: x- to find the value of recirocal ratios. e.g. sin 6 o = means cosec 6 o = Key sequence : e.g. sin 6 o = or sin 6 = ans - x = 6. Determine the value of each trig. number correct to decimal laces. a) cosec 48 = b) sec 7 = c) cot 6 = cosec 6 = sec 648 = cot 78 = cosec 5 = sec 5 = cot 00 = 7. Use the calculator keys sin cos tan to calculate the size of a correct to decimal lace if the given angle is an acute angle. e.g. a) sin a = 0768 b) cos a = 0768 a = sin α = cos α = 50 α = 98 c) cos α = 056 d) sin α = 0498 = = = = e) tan α = 46 f) tan α = 0845 = = = = Grade 0 Trigonometry is easy

11 Trigonometry is easy Grade alculate the value of correct to decimal lace if 0 < < 90. e.g. a) sec = 64 b) cosec ( + 0 ) = 46 cos = cosec ( + 0 ) = 07 = cos sin ( + 0 ) = = = sin x - = cos - ns = ( + 0 ) = or 64 = cos - ns = = 487 c) cosec = 55 d) sec = 55 e) cot = 55 sin = cos = tan = = = = = = = f) cosec ( 0 ) = 08 g) sec ( + 0 ) = 8 h) cot = Question: If cos α = 08 and 0 < α < 90 evaluate cos α correct to decimal laces. nswer: cos α = 08 α = and α = cos α = 0 0. alculate each final answer correct to decimal laces. a) If sin α 0476 = 0 and 0 < α < 90 evaluate cos α. b) If tan α = 88 and 0 < α < 90 evaluate cos α. a) b) Grade 0 Trigonometry is easy

12 Trigonometry is easy Grade 0 9 c) alculate the value of tan a if cos α = 096 and 0 < α < 90. d) alculate the value of sin α if sin α = 46 and 0 < α < 90. c) d) o o e) If sec(α + 0 ) = 786 and 0 < α + 0 < 90 evaluate cosec (α 0 ). f) If cot α = 076 and 0 < α < 90 evaluate sec α. e) f) Solving right-angled triangles.. triangle has sides and angles. If the measures of of these of which one at least is a side are known it is ossible to determine the other. This rocess of finding the missing measurements is called solving the triangle.. alculate the value of correct to decimal lace. e.g. a) b) = sin 8 = cosec 4 or = sin 4 o = 6 sin 8 = cosec 4 or = sin 4 o = 99 = 76 or = 76 Grade 0 Trigonometry is easy

13 Trigonometry is easy Grade 0 0 c) d) e) f) g) h) i) j) k) l) m) Determine the size of β correct to decimal lace. e.g. a) b) β 9 β tan β = 4 9 sin β = 59 6 tan β = sin β = β = tan β = sin β = 05 β = 79 Grade 0 Trigonometry is easy

14 Trigonometry is easy Grade 0 c) d) e) β 8 β 4 β 0 f) g) h) β β β Solve triangle. e.g. a) In ˆ = 90º a = and c = 5. nswer: i) tan = 5 = 0 46 Ĉ = 6º ii) Â = 90º 6º = 674º (sum of s of = 80º) or tan = = 4 Â = 674º 5 b 5 iii) b² = 5² + ² (Pyth) b² = 69 and b = or 5 5 = sin 6º 5 = b sin 6º o b sin 6 or b 5 = cosec 6 o b = 5 cosec 6 o = = b and = b or b = cos 6º b = o cos 6 b = or b = sec 6 o b = sec 6 o = Grade 0 Trigonometry is easy

15 Trigonometry is easy Grade 0 b) c) D The function values of the secial angles 0 45 and 60.. Study each diagram and answer the questions that follow. a) b) 0 t α = a 60 If cos 60 = then: = (Pyth.) = = t = (Pyth.) t = Y c) O(0;0) is the centre of the circle with radius = unit. 0 X lies on the terminal arm of 0 and lies on the terminal arm of 90. The co-ordinates of oint are ( : ) and the co-ordinates of oint are ( ; ). Grade 0 Trigonometry is easy

16 Trigonometry is easy Grade 0. The following diagrams illustrate what we learnt in a b and c Y (0;) 0 X (;0) omlete: sin 45 = sin 60 = sin 0 = sin 0 = sin 90 = cosec 45 = cosec 60 = cosec 0 = cosec 0 = cosec 90 = cos 45 = cos 60 = cos 0 = cos 0 = cos 90 = sec 45 = sec 60 = sec 0 = sec 0 = sec 90 = tan 45 = tan 60 = tan 0 = tan 0 = tan 90 = cot 45 = cot 60 = cot 0 = cot 0 = cot 90 =. Draw the 4 secial angle diagrams and use them to do the given calculations. Remember that sin 60 = sin 60 and sin 60 = (sin60 ) Examles: a) i) sin 90 cos 0 = ()() = ii) sin 45 sin 0 = (sin 45 ) (sin 0 ) = ( ) ( ) = = b) tan 0. sin 0 = c) sin 90. cos 0 = d) tan 60. cosec 60 = e) 4 sin 0. cos 60 = = = f) cot 45 + tan 60 = g) cosec 0 + sec 60 = = = h) cosec 90 + cot 0 = i) sin 0 + cos 60 = = = = = Grade 0 Trigonometry is easy

17 Trigonometry is easy Grade 0 4 j) cos 0 sin 0 = k) sec 0 tan 0 = = = = = 4. Evaluate without using a calculator. a) cos 60 + cos 45 + tan 45 b) sin 90. cos 60 sin 0 cos 0 = ( ) + ( ) + ( ) = = = = = c) sin 45 cos 45 tan 45 d) (sin 0 sin 90 ) = = = = 5. Without using a calculator determine the value of β if 0 β 90. Examles: a) sin β = b) sin β = sin β = β = 90 β = 0 β = 45 c) tan β = d) secβ = e) o tan( β + 0 ) = 0 f) o sin(β 0 ) = Grade 0 Trigonometry is easy

18 Gr 0& Trigonometry Gr 0& Trigonometrie NSWERS NTWOORDE Set by / Ogestel deur Seeliger~Mouton Tel (0) PS oyright Reserved Koiereg Voorbehou

19 Trigonometry is easy / Trigonometrie is maklik Grade/Graad 0 &. 4. i) ii) 4. i) ii) 4. i) ii) 4.4 i) ii) 4.5 sin = sin = cos = D tan = D nswers / ntwoorde Grade 0 / Graad 0 DF EF DF DE DE EF EF DF EF DE DE DF PT TR PT PR PR TR TR PT TR PR PR PT D D D D D D D D D D sin = D sin = D cos = D tan = D 5. sin D tan D cos D 5. sin D cos D tan D 5. sin F cos F tan F 5.4 sin G tan G cos G 7. cosec = 7. sec = cot = sin = cos = 7.7 tan = a) PR PR PT RT PT RT b) PR PR RT PT RT PT 9. cosec or/of sec in Δ 9. tan or/of cot in Δ 9.4 sec or/of cosec in Δ 9.5 cos D or/of sin D in ΔD 0. t = = 69 t = 69 = sin = 5 & sec = 0. t = 00 6 = 64 t = 8 cos = 8 = cot = 8 = a) 09 b) c) 88 d) e) f) g) nswers / ntwoorde a) i) b) i) ii) ii) iii) iii) iv) iv) v) v) vi) vi) 4.a) i) -049 b) i) 078 ii) 49 ii) 76 iii) 06 iii) 9 iv) 047 iv) 4 6.a) 4 b) c) c) α = 75 d) α = 99 e) α = 556 f) α = 4097 α = 0 8. c) sin = = 476 d) cos = = 44 e) tan = = 64 f) sin( - 0 ) = = = 778 g) cos( + 0 ) = = 6907 = 59 h) tan = 7... = = 6

20 Trigonometry is easy / Trigonometrie is maklik Grade/Graad 0 & 0. a) sin α = 0476 α = sin α = cos α = 759 b) α = tan - 88 α = α = cos α = 075 c) α = cos α = α = tan α = 064 d) sin α = 06 α = sin - 06 α = α = sin α = 0487 e) cos(α + 0 ) = α + 0 = α = α 0 = 595 cosec(α 0 ) = 70 f) cot α = 058 tan α = 858 α = 679 α = 54 sec α = c) = cos 46 5 = 5 cos 46 = 04 d) = cosec 46 0 = 0 cosec 46 = 9 e) f) g) h) i) j) k) l) m) = tan = 68 tan 50 = 8 = cot 8 77 = 77 cot 0 = = sin 0 or/of cos 70 0 = 0 sin 0 = 68 = cosec 6 or/of sec 64 4 = 4 cos 6 = 8 = cos 40 or/of sin 50 8 = 8 cos 40 = 6 = sec or/of cosec 77 = sec = = tan 6 6 = 6 tan 6 = 4 = tan 48 8 = 8 tan 48 = = cot 57 or/of tan 84 = 84 cot 57 = 55 nswers / ntwoorde. c) tan β = 5 8 = 065 β = 0 d) tan β = 6 4 = 5 β = 544 e) cos β = 0 6 = 065 β = 5 f) cos β = 4 68 = 05 β = 578 g) sin β = 9 68 = 046 β = 5 h) sin β = 6 8 = 0768 β = b) Â = 4 = sin 48 6 = 6 sin 48 = 9 = cos 48 6 = 6 cos 48 = 07 c) tan = = 67 = 59 Â = 09 = cosec 59 or/of sec = 97 cosec 59 =

21 Trigonometry is easy / Trigonometrie is maklik Grade/Graad 0 & D.. a) α = 45 = & = b) = t = & t = c) ( ; 0) & (0 ; ). 0 und/ong 0 und/ong 0 und/ong und/ong 0. b) (0)(0) = 0 c) ()() = d). = e) 4( )( ) = f) + = 4 g) + = 4 h) + = 4 i) + = 4 4 j) = k) = a) ( ) + ( ) + () = = 4 b) ()( ) ( )() = 0 c) ( )( )() = d) ( ) = ( ) = 4 5. a) tan β = β = 0o d) sec β = β = 60 o e) tan(β + 0 o ) = β + 0 o = 60 o β = 50 o f) sin(β 0 o ) = β 0 o = 60 o β = 90 o β = 45 o _ E/.. a) 4 b) 4 _ 4. a) sec α & cot α b) sin tan cos & 5 6. a) r = 5 x = 5 x = 7 x = 4 b) r = 5 y = 4 r = x = c) r = r = x = 5 y = 5 8. a) sec α = 5 i) x = 4 ii) b) cos β or sec β i) y = 5 ii) c) tan or of cot i) r = ii) nswers / ntwoorde 9. a) sin α = b) cot β = 4 c) sin α = d) cos α = 4 0. a) sin α = 4 b) cos α = c) tan α = α 5 α α r α 4 (7;y) (x;) (;6). a) b) c) i) ( ) - 4 ( ) = = 7 5 ii) ( )( 4) = iii) = iv). 5 = _