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3 Spring 999 HIGHER STILL 0DWKHPDWLFV 0DWKHPDWLFV,QWHUPHGLDWH 6XSSRUW0DWHULDOV *+,-./

4 This publication ma be reproduced in whole or in part for educational purposes provided that no profit is derived from the reproduction and that, if reproduced in part, the source is acknowledged. First published 999 Higher Still Development Unit PO Bo 74 Ladwell House Ladwell Road Edinburgh EH 7YH Mathematics Support Materials: Mathematics (Int )

5 STAFF NOTES INTRODUCTION These support materials for Mathematics were developed as part of the Higher Still Development Programme in response to needs identified at needs analsis meetings and national seminars. Advice on learning and teaching ma be found in Achievement for All (SOEID 996), Effective Learning and Teaching in Mathematics (SOEID 99) and in the Mathematics Subject Guide. This support package provides student material to cover the content of Mathematics within the Intermediate course. The depth of treatment is therefore more than is required to demonstrate competence in the unit assessment; that is, it goes beond minimum grade C. The content of Mathematics (Int ) is set out in the landscape pages of content in the Arrangements document where the requirements of the unit Mathematics (Int ) are also stated. Students are unlikel to have met much of the materials of this unit before - Algebraic Operations, Quadratic Functions and non-right Angled Triangle work. The material is designed to be directed b the teacher/lecturer, who will decide on the was of introducing topics and in the use of eercises for consolidation and for formative assessment. The use of a scientific calculator will be necessar for Part C but students should be encouraged to set down all working and, where appropriate, use mental calculations. The use of computers or graphics calculators is obviousl highl desirable for some of the graphical aspects in Parts B and C of the course. An attempt has been made to have the ÔeasÕ questions at the start of each eercise, leading to more testing questions towards the end of the eercise. While students ma tackle most of the questions individuall, there are opportunities for collaborative working. Staff ma wish to discuss points raised with individuals, groups and the whole class. The specimen assessment questions at the end of the package are not intended to be onl at minimum grade C. The National Assessment Bank packages for Mathematics (Int ) contain questions that meet the requirements of this unit. This package gives opportunities to practise core skills. Information on the core skills embedded in the unit, Mathematics (Int ) and in the Intermediate course is given in the final version of the Arrangements document. General advice and details of the Core Skills Framework can be found in the Core Skills Manual (HSDU June 998). Brief notes of advice on the teaching of each topic are given. The introductor notes for teachers/lecturers comprise pages -6 inclusive. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes

6 Format of Student Material Eercises on Algebraic Operations Checkup for Algebraic Operations Eercises on Quadratic Functions Checkup for Quadratic Functions Eercises on Further Trigonometr Checkup for Further Trigonometr Specimen Assessment Questions Answers for all eercises Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes

7 ALGEBRAIC OPERATIONS A. Reducing algebraic fractions to their simplest form Students should be shown the method for Ôcancelling downõ a vulgar fraction, then fractions with letters can be introduced. * Before this eercise some revision of factorisation ma be necessar. Eample Simplif / 49 Eample Simplif 6a /9a Ans. / 49 = / 7 Ans. 6a /9a = / 6 Eample Simplif p Ð pq Eample 4 Simplif p Ans. 6 Ans. = p Ð pq p(p Ð q) p = p = p Ð q Note the factorisation in the numerator! Eercise Questions and ma now be attempted. The following two eamples could be used to illustrate the need for factorising the numerator and/or denominator before cancelling. a + a Ð a + a Ð a + a Ð a + a Ð Eample Simplif v Ð Eample 6 Simplif Ans. v Ð v Ð Ans. v Ð = (v Ð )(v + ) v Ð = = v + = (a Ð )(a + ) (a Ð )(a + ) (a + ) (a + ) Eercise Question ma now be attempted. B. Appling the four rules to algebraic fractions The following eight eamples can be used to illustrate the four rules. It ma be that additional eamples will be required to be shown to the class in order to reinforce the method. Eample Simplif a Eample Simplif b 8 b a Ans. a Ans. b 8 b a a = 4 = b ba = = a contd. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes 6

8 Eample Simplif 6 Eample 4 Simplif c c d d Ans. 6 Ans. c c d d = = c 6 d d c = / (b cancelling) = c (b cancelling) Eercise Questions and ma now be attempted. Eample Simplif + Eample 6 Simplif Ð Ans. + Ans. Ð = + = 0 Ð 9 = 8 / = / Eample 7 Simplif 4r + s Eample 8 Simplif Ans. 4r + s Ans. v Ð = 8r + s 0 0 = w Ð vw = 8r + s = w Ð v 0 vw v w v vw Ð w Eercise Questions and 4 ma now be attempted. Eample 9 Ans. Simplif = 4( + ) + = = Ð Ð 4 Ð 4 ( Ð ) Eercise Question ma now be attempted. This eample should be followed b the simplification of + Ð Ð 4 Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes 4

9 C. Changing the subject of a formula It should be emphasised that ALL WORKING and ALL STEPS should be shown. The following eamples could be used to introduce changing the subject of a formula. The eamples are a miture of Ôchange side, change signõ, cross multiplication, dividing and finding the square root. The eamples are a miture of Ôchange side, change signõ, cross multiplication, dividing and finding the square root. Eample Change + = m to. Eample Change a /b = p to b Ans. + = m Ans. a / b = p = m Ð bp = a (cross ) b = a /p Eample Change a Ð c = d to. Eample 4 Change A = pd to d Ans. a Ð c = d Ans. A = pd a = d + c pd = A = d = d + c A /p a d = A / p Question 6 of Eercise provides 6 harder eamples, this eample could be used as etension: Eample: Change = v Ð z z to z. v Ð z Ans. = z z = v Ð z z + z = v z( + ) = v z = v + Eercise ma now be attempted. Ö D. Simplifing surds Irrational numbers could be introduced b first revising sets of numbers. E.g. Real nos. - all the numbers which can be represented on a number line. Whole nos. 0,,,, 4,, 6,... Integers...-, -, -, 0,,,, 4,... Rational nos. 8, -, /, -/4 etc. numbers which can be epressed as a fraction. Then eplaining that numbers like Ö, Ö, p... cannot be epressed as a fraction, therefore these are irrational. contd. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes

10 A SURD is a special kind of irrational number. It is a square root, a cube root, etc. which cannot be epressed as a rational number. Ö, Ö, Ö0 are all surds, whereas Ö and Ö8 are not surds as Ö = and Ö8 =. The following eamples could be used to show students how to simplif surds: Eample Epress Ö8 in its simplest form. Ans. Ö8 = Ö(9 ) = Ö Eplain wh Ö8 = Ö(6 ) is not used. Largest square number which divides into 8 Eample Simplif Ö8 + Ö Ans. Ö8 + Ö = Ö(4 ) + Ö = Ö + Ö = 7Ö Eercise 4 ma now be attempted. Eample Simplif Ö6 Ö6 Eample 4 Simplif Ö Ö6 Ans. Ö6 Ö6 = Ö6 = 6 Ans. Ö Ö6 = 0Ö8 Eample Simplif Ö(4 Ð Ö) = 0Ö(9 ) Ans. Ö(4 Ð Ö) = 0 Ö = 4Ö Ð ÖÖ = 0Ö = 4Ö Ð () = 4Ö Ð Eercise ma now be attempted. E. Rationalising a surd denominator Students should be reminded of the difference between a numerator and a denominator and an eplanation given of what rationalising a denominator means. Eample Epress with a rational denominator. Ö6 Ans. can be multiplied b, without changing its value. Ö6 The number ÔÕ can be written here as Ö6. Ö6 So Ö6 will have the same value as but will be written differentl. Ö6 Ö6 Ö6 Ö6 = Ö6 = Ö6 Ö6 Ö6 6 Eample Epress 9 with a rational denominator. Ö 0 Ans. 90 = = Ö0 = Ö0 Ö Ö0 Ö0 Ö0 0 Eercise 6 Q and Q ma now be attempted. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes 6

11 Eercise 6 Q contains eight eamples appropriate to grades A/B. Eample Epress + Ö with a rational denominator. Ans. Here, multipl b Ð Ö to rationalise the denominator. Ð Ö So, Ð Ö = Ð Ö = Ð Ö + Ö Ð Ö 4 Ð Rule = a mn Notice - no Ö term in denominator Eercise 6 Q ma be attempted now (for etension to grades A/B). F. Simplif epressions using the laws of indices Basicall, there are 6 rules for the students to learn. The should be lead through them, doing eamples of each tpe, before attempting Eercise 0, containing miscellaneous eamples. The rules are: Rule a m a n = a m+n Rule a m a n = a mðn Rule (a m ) n = a mn Rule 4 a 0 = Rule a Ðm = /a m Rule 6 a m/n = n Öa m Eamples Rule 7 4 = = 4 = 0 + = 0 7 Rule Rule 7 4 = 7 Ð 4 = (6 ) = 6 = 6 6a 8 a 6 = a 8 Ð 6 = a ( ) 4 = 4 4 = 8 Eercise 7 ma now be attempted. Rule 4 Rule Ðve power = /+ve power (An number) 0 = Ð = / = /9 (046) 0 = 4 / Ð = 4 // = 4 Eercise 8 ma now be attempted. a (a Ð a Ð4 ) = a + Ð a Ð4 = a Ð a Ð = a Ð /a Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes 7

12 Rule 6 / = Ö Öw = w / Ð/ = / / = /Ö = / Eercise 9 ma now be attempted. Eercise 0, containing Miscellaneous Eamples, comes net, there are eamples in this eercise which are appropriate to grades A/B. Eample Eample (a Ð/ ) = a Ð = /a 8w /4 w Ð/4 = 4w /4 Ð(Ð/4) = 4w = 4w Eample If m = 64, find the value of m /. Ans. m / = (64 / ) = [ Ö(64) ] = (4 ) = 6 = Eample 4 Epress with positive indices: (p 4/ ) Ð/4 Ans. (p 4/ ) Ð/4 = p 4/ Ð/4 = p Ð = /p Eample Simplif :- Ð Ans. = = = = Ð Ð Ð + 0 Ð / / / Eercise 0 ma now be attempted. The idea in Eercise is to epress the numerator in inde form. Illustrate b two eamples on the board Eample 9 = 9 / = 9 Ð Eample 9 4Ö = 9 9 Ð/ 4 / = 4 Eercise ma now be attempted. *This eercise is appropriate to grades A/B The checkup eercise ma then also be attempted. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes 8

13 QUADRATIC FUNCTIONS A. Quadratic graphs = Students should be made aware of the importance of recognising (,4) the basic parabolas associated with (,Ð) = and = Ð. From this, develop with the students (,) (,Ð4) the graphs associated with :Ð =, =, = /, = Ð, = Ð, = Ð /, etc., = Ð b plotting or points on the graph and drawing a smooth parabola shape through them. Students should then be encouraged to develop further the idea of graphing the functions = ±, = ±, etc., b ÔmovingÕ the basic parabola = up or down. Similar development of = Ð ±, = Ð ±, etc. should then follow. A class set of graphic calculators could be used to allow students to investigate the transformation of quadratic graphs. Eercise ma now be attempted. B. Turning points - their nature Go over the idea of moving the turning point awa from the origin to a point (a,b). = = becomes Ð b = ( Ð a) (a,b) => = ( Ð a) + b Similarl, = Ð => Ð b = Ð( Ð a) => = Ð( Ð a) + b. Eample Eample (,) = Ð( Ð ) + (4,) = ( Ð 4) + Eercise ma now be attempted. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes 9

14 C. Solving quadratic equations graphicall The difference between a Ôquadratic equationõ and a Ôquadratic epressionõ should be eplained to students. The ÔrootsÕ of a quadratic equation should be defined graphicall. Again, graphic calculators provide an efficient means of reinforcing the teaching of this topic, as man will not onl draw the functions but also show a table of values. Eample Roots (from graph) are at = Ð, = Ð 4 Ð Ð Eample = + Ð 8 then draw parabola (or pick out roots from table) Ð Ð4 Ð Ð Ð Ð Ð8 Ð9 Ð8 Ð 0 7 Ð4 Eercise ma now be attempted. D. Solving quadratic equations algebraicall Factorisation should be revised: (a) Common factors Ð, 8 +, etc. (a) Difference of squares Ð 4, 4 Ð, etc. (a) Trinomials , Ð Ð, etc. Students should then be shown how to use factorisation to solve quadratic equations (up to simple trinomals with at most in the trinomial). Eamples: Ð 6 = 0 ( Ð ) = 0 => = 0 or Ð = 0 => = 0 or Ð 9 = 0 ( Ð )( + ) = 0 Ð = 0 or + = 0 = or Ð Ð Ð = 0 ( Ð )( + ) = 0 Ð = 0 or + = 0 = or Ð Eercise 4 Questions Ð ma now be attempted. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes 0

15 Factorisation of trinomial quadratic equation of the form: a + b + c = 0 (a ¹ ). The same technique as in (Maths Int ) should be used. Eamples + 9 Ð = 0 ( Ð )( + ) = 0 Ð = 0 or + = 0 = or + = 0 = / or Ð + Ð 6 = 0 ( Ð )(4 + ) = 0 Ð = 0 or 4 + = 0 = or 4 = Ð = / or Ð / 4 Eamples of quadratics not in the form a + b + c = 0. ( Ð ) = 4 Ð = 4 Ð Ð 4 = 0 ( Ð 7)( + ) = 0 = 7 or Ð ( Ð )( + ) = Ð + 6 Ð = + Ð 8 = 0 ( + 9)( Ð ) = 0 = Ð 9 / or Eercise 4 Questions, 4 ma now be attempted. E. Quadratic formula Teachers/lecturers ma wish to go through the proof of the quadratic formula. Proof a + b + c = 0 + b a + c a = 0 + b a = Ð c a + b a + b = b 4a 4a ( + b a ) = b - 4 ac 4a + b a = ± b ± 4 ac a = ±b b - 4ac ± a a = ± b b ± ± 4 ac a Ð c a Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes

16 Eample Solve Ð 8 + = 0 Method factorising Method quadratic formula Ñ> ( Ð 6)( + ) = 0 = Ð or 6 = ± b b ± ± 4 ac a 8± = = 8 ± 4 = 6 or Ð Eample Solve + Ð 6 = 0 Method Ð Does not work! Method ± ± = 6 ± ± 76 = 6 = á or Ðá79 Students should be made aware of the following possible errors when using the formula. (i) make certain to assign a, b, c the correct sign (ii) the Ðb (iii) the effect of a negative value for c on b Ð 4ac (iv) pressing Ô=Õ on calculator before doing final division or calculation. Eercise ma now be attempted. The checkup eercise ma also be attempted. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes

17 FURTHER TRIGONOMETRY A. Basic sine, cosine and tangent graphs Eercise provides step b step instructions for the graphs of = sin, = cos and = tan to be drawn b hand. Access to graphic calculators ma mean that staff wish to omit this eercise and use the graphic calculators to remind students of the graphs. B. Sketching trigonometric graphs Students should be encouraged to memorise sketches of these functions as shown on page 6. Students should then investigate the effect: (i) ÔaÕ has in = a sin, = a cos and = a tan (encourage them to sketch the sine, cosine or tangent graph as appropriate and then fill in the scales on the aes). (ii) ÔbÕ has in = sin(b), = cos(b), = tan(b) 60 i.e. period becomes b, 60 and 80 b b. A variet of eamples should be discussed, such as: = 0sin, = cos, = tan, = ásin, = Ð0á8cos4 (a) Students must be able to recognise the function from a given graph = 7sin Ð7 (b) Students must be able to sketch the graph given the function. á8 = á8cos Ðá8 Eercise A ma now be attempted. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes

18 Students should investigate graphs of the form = sin( Ð c), = cos( Ð c) and = tan( Ð c) and conclude that c ÔslidesÕ the basic curve b ÔcÕ horizontall. i.e. = sin( Ð 0) slides 0 right = 0cos( + 0) slides 0 left 0 Ð0 Students should also be shown how to sketch curves such as: = sin( + ) and = 0á6cos( Ð ) = sin( + ) Eercise B ma now be attempted. Solving trigonometric equations The graphs of = sin, = cos and = tan can be used to build up the mnenomics for solving equations. S(in) + A(ll) + T(an) + C(os) + 80 Ð Eamples with positive values sin = 0á407 = 4 or (80 Ð 4) = 4 or 6 cos = 4 cos = 4 / = 0á8 = 6á9 or (60 Ð 6á9) = 6á9 or á Ð tan Ð = 0 tan = tan = / = á = 6á or (80 + 6á) = 6á or 6á Eercise, questions and, ma now be attempted. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes 4

19 Eamples with negative values. Eample cos = Ð 0á60 Ñ> rule (forget negative sign; look up inv cos 0á60 => ) (use to find the two correct answers) ( = 80 Ð or 80 + ) ( = 7 or ) (score out the Ð it is not in answer) Eample Eample sin = Ð 4tan + = 0 sin = Ð / 4tan = Ð (If sin = / => 6á9) tan = Ð / 4 = á9 or 60 Ð 6á9 (If tan = / 4 => 6á9) = 6á9 or á = 80 Ð 6á9 or 60 Ð 6á9 = 4á or á Eercise, questions to, ma now be attempted. D. Period of a trig function = 0sin goes through three complete ccles between 0 and 60 => Period = 60 = 0 This could be illustrated with graphic calculators. Eample 0 Ð0 4 => Period of = 0cos4 is 60 4 = 90 Eercise 4 ma now be attempted. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes

20 E. Trigonometric identities The two standard identities could be developed using the triangle opposite. h a o A. sin + cos B. o a = æ ö è hø ö è hø = o + a h sin cos = o a h h = o h h a = o a = h h = tan = Alternativel, a graphical approach can be taken using graphic calculators. For eample, draw = (sin) + (cos) and notice that it is the same graph as =. Similarl, draw = sin cos and notice that it is the same graph as = tan. These identities should be learned. sin cos + = sin tan = cos Eample Simplif sin + cos Eample Prove that - cos sin = Eercise ma now be attempted. The checkup eercise ma also be attempted. Mathematics Support Materials: Mathematics (Int ) Ð Staff Notes 6

21 STUDENT MATERIALS CONTENTS Algebraic Operations A. Reducing Algebraic Fractions to their Simplest Form B. Adding, Subtracting, Multipling and Dividing Algebraic Fractions C. Changing the Subject of a Formula D. Simplifing Surds E. Rationalising a Surd Denominator F. Simplifing Epressions using Laws of Indices Checkup Quadratic Functions A. Recognising Quadratics from their Graphs B. The Nature and Coordinates of Turning Points C. Solving Quadratic Equations Graphicall D. Solving Quadratic Equations b Factorising E. Solving Quadratic Equations using the Formula Checkup Further Trigonometr A. Sine, Cosine and Tangent Graphs B. Sketching and Identifing Trigonometric Functions C. Solving Trigonometric Equations D. The Period of the Trigonometric Function E. Trigonometric Identities Checkup Specimen Assessment Questions Answers Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

22 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

23 ALGEBRAIC OPERATIONS B the end of this set of eercises, ou should be able to (a) (b) (c) (d) (e) (f) reduce an algebraic fraction to its simplest form appl the four rules to algebraic fractions change the subject of a formula simplif surds rationalise a surd denominator simplif epressions using the laws of indices. Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

24 ALGEBRAIC OPERATIONS A. Reducing algebraic fractions to their simplest form Eercise. Simplif these fractions: 6 (a) (b) 0 9 (c) 8 (d) 44 (e) (f) 4a a (g) p p (h) v w (i) rs rt (j) ab ac (k) a ad (l) m m (m) a 4a (n) 8z z (o) (p) d d (q) 6 (r) v 6w (s) 8 4 (t) a ab (u) pq 6p (v) b b (w) z z () ef 7ef () pq pq (z) 8 4 (aa) ( ) ( ) ( ) ( ) + + (ab) - -. Factorise either the numerator or the denominator, then simplif: (a) (b) (c) (e) (i) (f) (j) pq + p p (g) + (k) v + v v. Factorise the numerator and/or the denominator, then simplif: a + 6 c + 9 (a) (b) (c) d - a + c + d - 6 (d) (h) (l) (d) a - a a g + g g + (e) + + (f) p+ q 7p+ 7q (g) 4-4w - w (h) (i) - - (j) (k) a - a - (l) w + 0 w - 00 (m) (n) v -v- v - 4 (o) (p) Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 4

25 B. Multipling, dividing, adding and subtracting algebraic fractions Eercise. Simplif these fractions b multipling: (a) (b) (c) (d) (e) (f) 9 (g) 7 (h) (i) a c b d (j) v (k) m m (l) a b w n n b a (m) (n) (o) (p) a a n 7 (q) a 6 6 (r) d (s) a c (t) a c a 9 n 9 a. Change these divisions to multiplications and simplif: (a) 6 (b) 8 (c) 9 (d) (e) (f) a a d (g) d m (h) m (i) a a (j) (k) (l) b b 6 r 4 r 6 a a b b (m) a a (n) (o) (p) d d w w a a d (q) a b a b (r) a d a d. Do the following additions and subtractions: (a) + (b) + (c) Ð (d) (e) Ð (f) 4 Ð (g) + (h) Ð 7 (i) (j) (k) (l) 8 Ð a c + d 0 (m) e Ð h (n) m Ð n (o) + k (p) u w Ð (q) 4r + s (r) a d (s) (t) Ð + + u 4 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

26 4. B finding a common denominator with letters, work out these additions/subtractions: (a) + (b) Ð (c) 4 + (d) Ð a b c d p q (e) + (f) Ð (g) 7 + (h) Ð 8 v w g h k n. Add or subtract these fractions: (a) + + (b) (c) (d) + + Ð Ð + + (e) + + (f) (g) (h) Ð Ð Ð Ð Ð C. Changing the subject of a formula Eercise A This eercise has a mied selection of formulae. Change the subject of each formula to the letter shown in the brackets. ALL WORKING and ALL STEPS SHOULD BE SHOWN.. + = c (). Ð 4 = c (). + p = q () 4. Ð p = q (). / = a () 6. / 7 = a () 7. / = a () 8. / p = m () 9. / r = s () 0. 4 = 0 (). 4 = a (). g = h (). n = t () 4. + = (). + = b () 6. + c = b () 7. a + c = b () 8. p + q = r () 9. v Ð w = () 0. D = S T (S). C = pd (d). = 6 (). = () 4. A = pr (r). T = D / S (S) 6. A = () 7. P = pr (p) 8. P = pr (r) 9. h Ð p = q (h) 0. h Ð p = q (p). h Ð p = q (h). h Ð p = q (p). b Ð c = a () Eercise B. Change the subject of each formula to h. (a) g = hf (b) e = g + h (c) k = h /f(d) e = g Ð h. Change the subject of each formula to r. (a) Q = r (b) N = pr (c) M = pr (d) P = pr w Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 6

27 . Change the subject of each formula to m. (a) A = klm (b) B = Km (c) C = pmr (d) D = pm 4. Change the subject to. (a) p = q + (b) r = s Ð (c) r = s Ð (d) r = 7 Ð (e) m = ( + ) (f) m = / ( Ð ) (g) n = / ( + ) (h) p = / ( + q). Change the subject of the formula to the letter in brackets. (a) P /Q = R (P) (b) t = /s (s) (c) M = P /Q (Q) (d) v (e) d e = (f) (f) f K mn = T (n) (g) R w = (w) z = 7 (s) (h) a + b = c (a) 9s 6. Harder eamples. Change the subject of the formula to the letter in brackets. (a) A + d = V /T (T) (b) p + q = r () (c) a = b + c () (d) m = r Ð s (s) (e) = v Ð w (w) (f) p = Ör Ð (r) s v + w D. Simplifing surds Eercise 4. Epress each of the following in its simplest form: (a) Ö8 (b) Ö (c) Ö7 (d) Ö0 (e) Ö0 (f) Ö8 (g) Ö8 (h) Ö4 (i) Ö00 (j) Ö7 (k) Ö4 (l) Ö7 (m) Ö00 (n) Ö47 (o) Ö4 (p) 7Ö8 (q) Ö (r) 6Ö40. Add or subtract the following: (a) Ö + Ö (b) 6Ö Ð Ö (c) 8Ö0 + Ö0 (d) 9Ö0 Ð 9Ö0 (e) Ö6 Ð Ö6 (f) Ö + Ö Ð Ö (g) Ö7 Ð 8Ö7 + Ö7 (h) 0Ö + 0Ö. Simplif: (a) Ö8 Ð Ö (b) Ö8 Ð Ö (c) Ö + Ö (d) Ö48 + Ö (e) Ö4 + Ö0 (f) Ö6 Ð Ö8 (g) Ö0 + Ö8 (h) Ö7 Ð Ö Eercise. Simplif: (a) Ö Ö (b) Ö Ö (c) Ö6 Ö6 (d) Ö Ö (e) Ö Ö (f) Ö Ö (g) Ö4 Ö (h) Ö6 Öa (i) Ö Öc (j) Ö Ö (k) Ö Ö8 (l) Ö Ö (m) Ö6 Ö (n) Ö0 Ö0 (o) Ö Ö (p) Ö Ö contd. Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 7

28 . Multipl out the brackets: (a) Ö( + Ö) (b) Ö(Ö + ) (c) Ö(Ö + ) (d) Ö7(+ Ö7) (e) Ö( Ð Ö) (f) Ö( Ð 4Ö) (g) (Ö + )(Ö Ð ) (h) (Ö Ð )(Ö + ) (i) (Ö + Ö)(Ö Ð Ö) (j) (Ö Ð Ö)(Ö + Ö) (k) (Ö + Ö) (l) (Ö Ð Ö). If a = + Ö and b = Ð Ö, simplif: (a) a + b (b) ab (c) a + b 4. If r = Ö + Ö and s = Ö Ð Ö, simplif: (a) r Ð s (b) rs (c) r Ð s. A rectangle has sides of length (Ö + ) cm and (Ö Ð ) cm. Calculate: (a) its area (b) the length of a diagonal Ö + Ö Ð E. Rationalising a surd denominator Eercise 6. Rationalise the denominators in the following and simplif where possible: (a) (b) (c) (d) (e) (i) (m) (f) (j) (n) (g) (k) (o) 6 0 (h) (l) (p) Epress each of the following in its simplest form with a rational denominator: (a) 4 (b) (c) 4 0. Simplif the following b rationalising the denominator: (d) (e) (f) a b (a) - (b) 7 - (c) + (d) 4 + (e) - (f) - (g) + (h) 9 - Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 8

29 F. Simplifing epressions using the laws of indices Eercise 7 Rule a m a n = a m+n. Use Rule to write down the simplest form of the products in the following: (a) 4 (b) (c) 8 8 (d) (e) a a 4 (f) b b (g) c c 6 (h) d d (i) v v 8 (j) 0 (k) w w (l) z z (m) f 4 f 7 (n) g 0 g 0 (o) k k (p) m 00 m Rule a m a n = a mðn. Use Rule to write down the simplest form of the quotients in the following: (a) 4 (b) (c) 8 8 (d) (e) a 7 a 6 (f) b b (g) c 9 c 4 (h) d d (i) v v (j) 0 (k) w w (l) z z (m) f 7 f 4 (n) g 0 g 0 (o) k k (p) m 00 m (q) 6 (r) m (s) a (t) r 9 m a r 8 Rule (a m ) n = a mn ( ) (8 ) u 7 u. Simplif these. For eample, (a ) 4 = a. (a) ( ) (b) ( ) (c) (z ) (d) (g ) 8 (e) (a ) 7 (f) (b 4 ) 4 (g) (c ) 6 (h) (d ) 7 4. Epress the following without brackets, writing answers in inde form: (a) (b) (7 ) (c) (6 ) 4 (d) (e) ( 7 ) 7 (f) ( 4 ) (g) (9 ) (h) ( ). Note (ab) m = a m b m Use this result to simplif: (a) (ab) (b) (cd) 6 (c) ( ) 0 (d) (pq) 6. Use the rules learned so far to simplif the following: (a) (b) t 4 t (c) (d) v 7 v (e) (f) 4 (g) 6 (h) 8 (i) 6 (j) ( + 4 ) (k) ( Ð 4) (l) (uv) 7 (m) () (n) (mn ) 8 (o) 4 (p) u Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 9

30 7. Find m. For eample: if m = 8, m = since = 8. (a) m = 6 (b) m = (c) 4 m = 64 (d) m = 6 Eercise 8 Rule 4 a 0 =. Write down the values of: (a) 0 (b) 0 (c) 0 (d) (46) 0 Rule a Ðm = /a m. Write the following with positive indices. For eample: Ð = /. (a) Ð (b) Ð7 (c) a Ð4 (d) b Ð9 (e) Ð (f) Ð (g) Ð (h) - (i) t (j) 6 - c (k) - (l) - -. Epress these in a form without indices: (a) 0 (b) 7 Ð (c) Ð (d) 8 Ð (e) (f) - 4. Simplif the following. For eample: Ð = Ð+ =. 7 æ ö è ø (a) a Ð a 4 (b) b 6 b Ð4 (c) c Ð c Ð (d) d Ð6 d 6 (e) e 6 e Ð (f) g Ð4 g 4 (g) w Ð w Ð (h) ( Ð ) (i) ( ) Ð (j) (z Ð4 ) Ð4 (k) (klm) 0.. Epress with positive indices: (a) 4 4 Ð (b) 6 Ð 6 Ð (c) 7 Ð (d) 4 Ð ( ) (e) 4 Ð (f) (9 0 ) Ð8 (g) (w Ð ) Ð (h) - - (i) (j) (k) 6h Ð (l) 9s Ð - - (m) k - (n) m Multipl out the brackets: (a) ( + Ð ) (b) ( Ð Ð ) (c) Ð ( 4 + ) (d) Ð ( + ) (e) ( Ð Ð7 ) (f) Ð ( Ð ) (g) Ð ( Ð ) (h) ( Ð Ð ) 7. (a) Find the value of:, 0, Ð,, Ð. - (b) Write in the form 4 Ð p : / 4, / 6, / 64. Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 0

31 Eercise 9 Rule 6 a m/n = n Öa m 4. Write these in root form. For eample: = 4 4 (a) (b) m (c) r (d) w (e) n (f) r - 4. Write these in inde form. For eample: Ö 4 = 4/ (a) (b) b 4 (c) z 4 (d) w (e). Evaluate the following. For eample: 8 = 8 = () = 4 (a) 9 (b) 64 (c) 8 (d) 64 (e) 00 (f) 7 4 (g) 9 (h) 6 (i) 49 (j) 6 4 (m) 8 (n) 8 4. Simplif: (a) 4-4 (o) æ ö è 4ø - (p) - æ ö è 8ø (k) 7 - (f) (l) ( ) (b) m 9 ( ) (c) c 6 ( ) (d) ( n - ) - u ( ) (e) n - æ 4 (f) g - ö ç è ø - (g) æ ö çb è ø - æ (h) z - ö ç è ø - (i) æ ç è 4 ö æ (j) ç ø è 4 ö (k) + ø ( ) (l) + + ( ) Eercise 0 Miscellaneous eamples. Simplif: (a) a (e) e a (b) b b e - (f) æ ö ç z è ø (c) c c (g) æ çw- è - ö ø (d) d d (h) æ w - ç è ö ø 0. Simplif: (a) a / a Ð/ (b) b / 4b / (c) 6c / c / (d) d / d Ð/ (e) 4e / e / (f) 8v /4 v Ð/4 (g) z Ð/ z Ð/ Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

32 . Multipl out the brackets: (a) / ( / Ð Ð / ) (b) / ( 4/ + / ) (c) Ð/ ( 6/ + Ð/ ) 4. Evaluate the following for = 6 and = 7 : (a) / (b) 4 /4 (c) / (d) 8 Ð/ (e) Ð/4 /. Simplif the following, epressing our answers with positive indices: (a) p 6 p / (b) p p Ð/ (c) (p / ) Ð/ (d) (p Ð/4 ) (e) p /4 p Ð/4 (f) 7p / 7p Ð/ 6. Simplif: (a) 4 - (b) (c) - - (d) (e) - (f) Multipl out the brackets: (a) ( Ð )( Ð Ð ) (b) ( Ð + )( Ð Ð ) (c) ( / + 4)( Ð/ Ð 4) Eercise Epress the following with in the numerator in inde form: For eample :- - - = = = a a a m m ( 6 ). ( ) 4 Ð Mathematics Support Materials: Mathematics (Int ) Ð Student Materials Ð/4 /4

33 Checkup for algebraic operations Ð/4 /4 Ð/. Factorise the numerator and/or denominator if possible, then simplif: (a) 7v 9a (b) b 6 4v (c) ab 6 Ð 8 (d) (e) + a Ð 8 (f) (g) v + v a Ð 64 v Ð 4 v Ð vw v. Simplif: (a) b (b) a (c) z (d) z a a 4 b a 6 b b (e) c (f) (g) (h) c u v Ð + Ð k m. Change the subject of each formula to. (a) + = 6 (b) Ð a = w (c) a + m = p (d) / z = p /w (e) N = p (f) T = 4 + (g) w = / ( + ) (h) M = 8 4. Epress each of these in its simplest form: (a) Ö (b) Ö000 (c) Ö4 (d) Ö4 Ð Ö0 9 (e) Ö0 Ð Ö8 (f) Ö7 + Ö0 (g) (h) Ö8 Ð Ö a. If = + Ö and = Ð Ö, simplif: (a) + (b) (c) + 6. Rationalise the denominators in the following and simplif where possible: (a) /Ö (b) 8 / Ö (c) /Ö (d) Ö/ Ö6 7. Write in their simplest form: (a) 6 8 (b) 8 6 (c) (d) (a ) (e) (4a b) (f) (g) (p ) 0 (h) ( Ð ) 8. Write with positive indices: (a) Ð (b) ab Ð (c) ( Ð ) Ð (d) 9. Write these in root form: 0. Write these in inde form: (a) b / (b) c Ð/ (a) Ö 4 (b) / Öa. Evaluate: (a) 6 / (b) Ð/ (c) ( 8 ) / (d) ( 7/ ) Ð (e) a Ð/ (a / Ð a Ð/ ) (f) s / 4s Ð/ (g) Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

34 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 4

35 QUADRATIC FUNCTIONS B the end of this unit, ou should be able to: (a) (b) (c) (d) (e) recognise quadratics of the form = k and = ( + a) + b ; a, b Î Z, from their graphs identif the nature and coordinates of the turning points and the equation of the ais of smmetr of a quadratic of the form = k( + a) + b; a, b Î Z, k = ± know the meaning of Ôroot of an quadratic equationõ and solve a quadratic equation graphicall solve quadratic equations b factorisation solve quadratic equations b using the quadratic formula Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

36 A. Recognising quadratics from their graphs Eercise. Each of the following graphs represents a simple parabola of the form = k. Find k each time and hence write down the equation representing each parabola. (a) (b) (c) (,8) (,4) (4,8) (,) (,) (,) (d) (e) (f) (,-) (,- /) (,-) (,-8) (4,-8) (,-0). Each of the following parabolas can be represented b the equation = + b or = Ð + b, where b is an integer. Write down their equations. (a) (b) (c) (0,) (0,) (0,-) Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 6

37 (d) (e) (f) (0,4) (0,) (0,-). Each of the following parabolas can be represented b an equation of the form = ( + a) + b, (where a and b are integers). Write down the equation of each one. (a) (b) (c) (,) (,) (,0) (d) (e) (f) (-,) (-4,0) (-,-) (g) (h) (i) Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 7

38 B. The nature and coordinates of turning points Eercise. (a) Write down the coordinates of the minimum turning point of this parabola. (b) Write down the equation of the ais of smmetr. (the dotted line). (c) The equation of the parabola is of the form = ( + a) + b. Find the values of a and b and hence write down the equation of the parabola.. For each of the following parabolas: (i) write down the coordinates of its minimum turning point. (ii) write down the equation of its ais of smmetr. (iii) write down the equation of the function represented b the parabola. (a) (b) (c) (d) (e) (f). Without making a sketch, write down the coordinates of the minimum turning points and the equation of the aes of smmetr of these parabolas. (a) = ( Ð 4) + (b) = ( Ð ) + 7 (c) = ( Ð 8) + (d) = ( + ) + (e) = ( Ð ) Ð (f) = ( + ) Ð 7 (g) = ( Ð ) (h) = ( + ) (i) = + Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 8

39 4. This parabola is of the form = Ð( + a) + b (a) Write down the coordinates of the maimum turning point. (b) Write down the equation of the aes of smmetr. (c) Find the values of a and b and hence write down the equation of the function represented b the parabola.. For each of the following parabolas: (i) write down the coordinates of its maimum turning point. (ii) write down the equation of its ais of smmetr. (iii) write down the equation of the function represented b the parabola. (a) (b) (c) (d) (e) (f) 6. Without a sketch, write down the coordinates of the maimum turning points and the equation of the aes of smmetr of these parabolas. (a) = Ð( Ð ) + 6 (b) = Ð( Ð ) + (c) = Ð( Ð 6) Ð (d) = Ð( + ) + 7 (e) = Ð( + 4) Ð (f) = Ð( + ) (g) = 7 Ð ( Ð ) (h) = Ð ( Ð 8) (i) = Ð Ð ( + ) Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 9

40 C. Solve quadratic equations graphicall Eercise. In each of the following, the graph has been sketched for ou. Solve the quadratic equation associated with it. (a) (b) (c) = Ð Ð = Ð 4 + = Ð Solve Ð Ð = 0 Solve Ð 4 + = 0 Solve Ð 4 = 0. For the quadratic function = Ð Ð (a) Cop and complete this table. Ð Ð 0 = Ð Ð 4... Ð (b) Draw a set of aes, plot the 6 points above and draw a smooth (parabolic) curve through them. (c) Use our graph to solve the quadratic equation, Ð Ð = 0 [i.e. find the roots]. Solve each of the following quadratic equations b: (i) completing the table (ii) plotting the points and drawing the smooth parabola (iii) reading off the roots from the graph. (a) Ð 4 = 0 (b) + Ð = 0 (c) Ð = 0 (d) 6 Ð Ð = 0 Ð 0 4 = Ð 4... Ð Ð Ð Ð 0 = + Ð 4... Ð = Ð Ð4 Ð Ð Ð 0 = 6 Ð Ð Ð Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 0

41 (e) Ð 4 = 0 (f) + 4 Ð = 0 Ð Ð Ð 0 = Ð Ð Ð Ð = + 4 Ð D. Solve quadratic equations b factorising Eercise 4 In this eercise ou are going to solve quadratic equations b factorisation.. B considering a common factor, factorise and solve the following: (a) Ð 4 = 0 (b) Ð 0 = 0 (c) 8 Ð = 0 (d) + 6 = 0 (e) + = 0 (f) Ð = 0 (g) Ð 6 = 0 (h) + = 0 (i) Ð 8 = 0. B considering the difference of squares, factorise and solve the following quadratic equations: (a) Ð 4 = 0 (b) Ð 9 = 0 (c) Ð = 0 (d) 6 Ð = 0 (e) Ð 00 = 0 (f) = 49 (g) = 8 (h) Ð 9 = 0 (i) Ð 6 = 0. Factorise the following trinomials and solve the quadratic equations: (a) + + = 0 (b) Ð + 6 = 0 (c) = 0 (d) Ð = 0 (e) = 0 (f) Ð = 0 (g) Ð 7 + = 0 (h) Ð = 0 (i) Ð + 4 = 0 (j) + Ð 0 = 0 (k) Ð Ð 4 = 0 (l) + Ð 8 = 0 (m) Ð Ð 0 = 0 (n) + Ð = 0 (o) + Ð = 0 (p) + 4 Ð = 0 (q) + Ð 8 = 0 (r) = 0 (s) Ð = 0 (t) Ð 0 Ð 4 = 0 (u) + Ð 4 = 0 (v) Ð Ð 4 = 0 (w) Ð Ð 4 = 0 () Ð + 4 = 0 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

42 4. The following are harder and take a little longer to do. Solve: (a) = 0 (b) + + = 0 (c) = 0 (d) Ð = 0 (e) = 0 (f) + + = 0 (g) Ð Ð 8 = 0 (h) Ð Ð = 0 (i) + Ð = 0 (j) Ð 7 Ð 4 = 0 (k) + Ð 6 = 0 (l) = 0. Rearrange the following into quadratic equations of the form a + b + c = 0 and solve them: (a) ( + ) = (b) ( Ð ) = 0 (c) ( Ð ) = 0 (d) ( Ð ) = 6 (e) ( + ) = 70 (f) ( + ) = 6 (g) ( + )( + ) = (h) ( Ð )( + ) = 8 (i) ( + )( Ð ) = (j) ( Ð )( + ) = 8 (k) ( Ð )( + ) = (l) ( Ð )( Ð ) = (m) + Ð = Ð Ð 4 (n) + Ð 8 = + + (o) ( + ) = 6 (p) ( Ð ) = 0 (q) ( Ð ) = 0 (r) ( + ) = 7 (s) ( Ð )( + ) = 0 (t) ( + )( Ð ) = E. Solve quadratic equations using the formula Eercise b b ac = - ± - 4 a. Solve the following quadratic equations using the formula, correct to decimal places: (a) = 0 (b) + + = 0 (c) = 0 (d) = 0 (e) Ð 4 + = 0 (f) Ð + = 0 (g) = 0 (h) = 0 (i) 4 Ð 7 + = 0. Solve the following quadratic equations using the formula, correct to decimal places: (a) + Ð = 0 (b) + Ð = 0 (c) + Ð 6 = 0 (d) Ð Ð = 0 (e) Ð Ð = 0 (f) + 8 Ð = 0 (g) + Ð 4 = 0 (h) Ð Ð = 0 (i) 8 Ð Ð = 0. Rearrange the following and solve each equation giving the roots correct to significant figures ( figures of accurac): (a) ( + ) = (b) ( Ð ) = (c) ( + ) = (d) ( Ð ) = 0 (e) ( + )( + 4) = 7 (f) ( Ð ) = Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

43 Checkup for quadratic equations. Write down the equations of the quadratics corresponding to these parabolas: (each one is of the form = k or = ( + a) + b ) (a) (b) (c) (,) (,4) (,) (,) (,Ð) (,Ð 4) (d) (e) (f) (,) (Ð,) (Ð,Ð 4). (i) Write down the coordinates of the turning point for each of the following, stating whether it is a maimum or a minimum. (ii) Write down the equation of the aes of smmetr for each one. (a) (b) (c) Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

44 . Write down the coordinates of the turning point of each of the following. State whether each is a maimum or a minimum. Give also the equation of the aes of smmetr. (a) = ( Ð ) + (b) = ( + ) Ð (c) = Ð( Ð ) + 4. Solve each of the following quadratic equations b (i) completing the table. (ii) plotting the points and drawing the smooth parabola. (iii) reading off the roots from the graph. (a) Ð Ð 8 = 0 (b) + Ð = 0 Ð Ð Ð 0 4 = Ð Ð Ð Ð4 Ð Ð Ð 0 = + Ð Ð Solve the following quadratic equations b factorising: (a) Ð 7 = 0 (b) Ð 9 = 0 (c) = 0 (d) = 0 (e) Ð = 0 (f) Ð Ð 0 = 0 (g) 4 Ð 9 = 0 (h) Ð = 0 (i) + 7 Ð = 0 (j) ( + ) = 4 (k) ( Ð )( + ) = 8 (l) ( Ð ) = 6 6. Solve the following quadratic equations using the formula. (Give the answers correct to decimal places): b b ac = - ± - 4 a (a) = 0 (b) Ð 4 + = 0 (c) = 0 (d) + Ð = 0 (e) Ð 6 Ð 4 = 0 (f) Ð Ð = 0 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 4

45 FURTHER TRIGONOMETRY B the end of this set of eercises, ou should be able to (a) (b) (c) (d) 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 = sin and tan = cos Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

46 A. Sine, cosine and tangent graphs Eercise 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 to of Eercise below.. 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 sin 0á00 0á4 0á64 0á87 0á98 á sin (b) Use a piece of mm graph paper to draw a set of aes as illustrated below Ð (c) (d) Plot as accuratel as possible the points from our table. Join them up smoothl to create the graph of the function = sin.. Repeat question (a) to (d) for the function = cos. 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 0 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 (Int ) Ð Student Materials 6

47 Sketches of the three trigonometric graphs, for comparison: = sin = tan Ð = cos Ð Check our graphs are similar to those shown above. B. Sketching and identifing trigonometric functions Eercise A. Write down the equations of the trigonometric functions represented b the following graphs: (a) 0 (b) Ð0 Ð6 (c) (d) 0á Ð Ð0á8 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 7

48 (e) 0 (f) Ð0 Ð80 (g) (h) Ð Ð4 (i) 0 (j) 60 0 Ð (k) 0á (l) Ð0á Ð. 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 (Int ) Ð Student Materials 8

49 (a) = sin (b) = 4cos (c) = tan (d) = 0sin (e) = cos (f) = 0á7sin 4 (g) = ácos 4 (h) = 0sin 6 (i) = 00cos (j) = Ðsin (k) = Ð6sin / (l) = Ð0cos. Make neat sketches of the following over the given range of values: (a) = 60sin 0 60 (b) = ácos 0 60 (c) = 40sin (d) = Ðcos (e) = Ðsin (f) = á8cos 0 0 Eercise B. 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) (b) 8 Ð Ð Ð Ð8 (c) (d) 0á Ð Ð0á6 (e) 4 (f) Ð Ð4 Ð. Make neat sketches of the following trigonometric functions, showing clearl: (i) where each graph cuts the - ais. (ii) the maimum and minimum values. (a) = 0sin( Ð ) (b) = á4 cos( Ð 0) (c) = sin( + 0) (d) = á4 cos( + 0) (e) = 00sin( Ð 80) (f) = tan( Ð 0) Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 9

50 C. Solving trigonometric equations Eercise S (in) T (an) A (ll) C (os). Find the two solutions for each of the following in the range 0 60: (Give each answer correct to the nearest whole degree). (a) sin = 0á00 (b) cos = 0á707 (c) tan = 0á869 (d) cos = 0á940 (e) tan = á80 (f) sin = 0á74 (g) sin = 0á990 (h) tan = 6á4 (i) cos = 0á9 (j) cos = 0á98 (k) sin = 0á866 (l) tan = á7. Rearrange each of the following and solve them in the range (Give our answers correct to decimal place). (a) cos Ð = 0 (b) sin Ð 4 = 0 (c) 0tan Ð 7 = 0 (d) Ð sin = 0 (e) Ð 6cos = 0 (f) tan Ð = 0. Find the two solutions for each of the following in the range 0 60: (Give each answer correct to the nearest whole degree). (a) sin = Ð0á00 (b) cos = Ð0á707 (c) tan = Ð0á84 (d) cos = Ð0á9 (e) tan = Ðá000 (f) sin = Ð0á866 (g) tan = Ð 4 (h) sin = Ð0á74 (i) cos = Ð0á97 4. Rearrange each of the following and solve them in the range (Give our answers correct to decimal place). (a) 4sin + = 0 (b) cos + = 0 (c) tan + = 0 (d) 7 + 8cos = 0 (e) 0á4sin + 0á = 0 (f) tan + 8 = 0. Solve the following miture of trigonometric equations in the range (Give our answers correct to decimal place). (a) sin = 0á (b) cos = Ð0á9 (c) tan = 0á678 (d) cos = / 4 (e) sin = Ð0á707 (f) tan = Ð (g) sin = / (h) cos = Ð0á (i) tan = / 8 (j) 8sin + = 0 (k) 6cos + = 0 (l) Ð tan = 0 (m) 0sin Ð 7 = 0 (n) Ð cos = 0 (o) 8tan + 7 = 0 (p) sin + = sin + (q) 7cos Ð = cos + 4 (r) 0tan + 8 = tan + 4 (s) 6sin + = sin + 0 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 0

51 D. The period of a trigonometric function Eercise 4. Determine the period and the maimum and minimum values of these trig functions. (a) (b) Ð7 Ð8 (c) 0á (d) 4 8 Ð0á Ð (e) 9 (f) 6 60 Ð9 0 Ð6 (g) 0á 80 Ð0á. Determine the period and the maimum and minimum values of these trig functions. (a) = sin (b) = cos (c) = 0tan4 (d) = ácos (e) = 0sin 6 (f) = Ðcos0 (g) = 0sin 90 (h) = Ð 4cos / (i) = 8sin / 4 (j) = 0á9cos 60 (k) = / sin (l) = / 4 cos 9 (m) = sin 80 (n) = 8sin á (o) = 40cos á Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

52 E. Trigonometric identities Eercise sin Remember :- sin + cos = ; and tan = cos. Simplif the following using the above identities: (a) sin + cos (b) cos + sin sin sin (c) (d) cos cos. Write down a simple epression, identical to: sin (a) - sin (b) - cos (c) tan cos (d) cos. Simplif: (a) (d) - cos - sin sin (b) (c) sin cos cos - sin - cos (e) (f) tan ( - sin ) cos sin 4. Prove the following trigonometric identities: (a) - sin = cos (b) - cos = sin (c) - cos = sin (d) tan - sin = sin - cos (e) - sin = tan - sin (f) - cos = tan Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

53 Checkup for further trigonometr. 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 (b) 0á Ð8 Ð0á7 (c) á7 (d) Ðá Make neat sketches of the following, indicating all the main points and features: (a) = 0sin (b) = á6cos 0 60 (c) = Ð8sin (d) = tan Find the two solutions for each of the following in the range 0 60: (Give our answers correct to decimal place). (a) sin = 0á9 (b) cos = 0á444 (c) tan = (d) cos = Ð0á60 (e) tan = Ð0á8 (f) sin = Ð 4 / (g) sin Ð = 0 (h) 8cos + 6 = 0 (i) 4tan Ð = 0 (j) cos Ð = 0 (k) + 4sin = 0 (l) tan = tan Ð Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

54 . What are the periods of the following trigonometric graphs and functions? (a) (b) Ð6 Ð7 (c) (d) 80 Ð 0 40 (e) = 0sin 0 (f) = ácos 0 (g) = Ð 4sin 9 (h) = tan 4 6. (a) Simplif: (i) 6-6sin (ii) cos sin (b) Prove these identities :- (i) - cos = sin (ii) ( - sin )( + sin )= cos 7. Write down the equations of the following trigonometric graphs: (a) (b) 8 á6 Ð Ð8 Ðá6 8. Sketch the following trigonometric graphs, indicating their main features: (a) = 8cos( + 0) 0 60 (b) = sin( Ð 0) 0 60 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 4

55 SPECIMEN ASSESSMENT QUESTIONS. Factorise full: (a) (b) 4a Ð 8 (c) p Ð pq (d) w Ð (e) 8 p w Ð. Simplif: (a) v 6 c (b) (c) (d) (e) 9 v c a a p + t Ð t q z z a Ð a Ð a + a Change the subject of each formula to the letter in the bracket: a (a) p Ð q = r [p] (b) aw + g = h [w] (c) = 9 [] (d) / 4 (m + n) = V [m] 4. Epress each of these in its simplest form: (a) Ö8 (b) Ö (c) 9 (d) Ö8 Ð Ö Ö a 4. Rationalise the denominators and simplif: (a) 0 /Ö (b) Ö p /Ö (c) Öq 6. Simplif, giving answers with positive indices: (a) b b (b) (w ) Ð (c) a a 4 (d) p / p Ð/ (e) v (v Ð v Ð ) (f) 8 / 4 / a v 4 7. Evaluate the following for v = 4 and w = 8: (a) 0v / (b) w / (c) v Ð/ w Ð/ 8. Write down the equation of the quadratic functions corresponding to each of the following parabolas: (each is of the form = k or = ±( + a) + b). (a) (b) (Ð,6) (,4) (,-) (,-0) contõd... Mathematics Support Materials: Mathematics (Int ) Ð Student Materials

56 (c) (d) (Ð,) (,Ð) 9. Write down the coordinates of the turning point of each of the following, state whether each is a maimum or minimum and give the equation of the ais of smmetr. (a) = ( Ð ) + 7 (b) = Ð( + ) Ð 0. (a) Complete the following table for the function, = Ð Ð. Ð4 Ð Ð Ð = Ð Ð Ð Ð (b) Plot the points and sketch the parabola. (c) Write down the roots of the quadratic equation Ð Ð = 0. Solve the following quadratic equations b factorising: (a) Ð 6 = 0 (b) Ð 6 = 0 (c) = 0 (d) Ð Ð = 0 (e) ( Ð ) = 4 (f) ( Ð )( + ) = 8. Solve the following, correct to decimal places, using the quadratic formula: (a) + + = 0 (b) Ð Ð 7 = 0 (c) + Ð = 0 b b ac = - ± - 4 a. Write down the equations of the trigonometric functions associated with these graphs: (a) (b) 0á Ð Ð0á contõd... Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 6

57 (c) (d) Ð 0 4. Sketch the following trigonometric graphs, indicating all the main features and points: (a) = sin 0 60 (b) = Ð cos Find two solutions for each of the following trig equations in the range (Give answers correct to decimal place). (a) sin = 0á4 (b) 8cos = (c) 0tan Ð 7 = 0 (d) cos = Ð0á4 (e) tan = Ð (f) sin + = 0 6. What are the periods of the following trigonometric functions: (a) = 0 sin (b) = Ð 4 cos 0 (c) = tan 7. (a) Simplif: sin Ð (b) Prove the identit: - cos - sin = tan 8. Write down the equation of the function given b this trigonometric graph: 8 Ð Sketch the graph of the function, = 0á7 cos( Ð 0) showing all the main points and features. Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 7

58 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 8

59 ANSWERS Algebraic operations Eercise. (a) / (b) / (c) / 4 (d) / 4 (e) / 4 (f) 4 / (g) / (h) v /w (i) s /t (j) b /c (k) /d (l) /m (m) a / 4 (n) 8 /z (o) (p) /d (q) / 6 (r) v / w (s) (t) a / b (u) q / (v) /b (w) () / 7 () / q (z) / (aa) /( + ) (ab) /( Ð ). (a) + 4 (b) + (c) Ð 4 (d) Ð (e) + (f) (g) (h) 4 (i) Ð (j) q + Ð + Ð (k) v + (l) Ð a. (a) (b) (c) / (d) g (e) / (f) / 7 (g) 4 (h) /8 (i) + (j) Ð (k) a + (l) Ð v + + /(w Ð 0) (m) + (n) v + (o) + (p) Ð + 4 Eercise. (a) / (b) / 9 (c) / (d) (e) (f) (g) 7 / (h) 6 / (i) ac /bd (j) v /w (k) m (l) (m) (n) (o) a (p) (q) a n 0 7 /d (r) (s) a (t) a. (a) / (b) / 4 (c) / 6 (d) / 4 (e) / (f) / 4 (g) (h) (i) a (j) b / (k) r / (l) (m) ad (n) / w (o) /a (p) d / (q) a (r) /. (a) 7 / (b) / (c) / 0 (d) 8 / (e) / (f) / 4 (g) 7 / 0 (h) / 0 (i) / 8 (j) 9 / 0 (k) + a (l) c + d (m) 4e Ð h (n) m Ð n (o) 4 + k (p) u Ð 4w (q) 8r + s (r) a Ð 6d (s) (t) + 8u (a) + (b) b Ð a (c) 4d + c (d) q Ð p (e) ab cd pq w + v vw Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 9

60 (f) h Ð g gh (g) 7n + k kn (h) Ð 8. (a) + 6 (b) (c) (d) Ð 4 (e) (f) (g) + 4 (h) Eercise A. = c Ð. = c + 4. = q Ð p 4. = q + p. = a 6. = 7a 7. = a 8. = mp 9. = rs 0. =. = a /4. = h /g. = t /n 4. =. = b Ð 6. = b Ð c 7. = b a Ð c 8. = r Ð p q 9. = + v w 0. S = D /T. d = C /p. = 4 or Ð4. = Ö or ÐÖ 4. r = Ö( A /p). S = D /T 6. = ÖA (or ÐÖA) 7. p = P /r 8. r = Ö( P /p) 9. h = q + p 0. p = h Ð q. h = q + p. p = h Ð q. = b a Ð c Eercise B. (a) h = g /f (b) h = e Ð g (c) h = kf (d) h = g Ð e. (a) r = ÖQ (b) r = Ö( N /p) (c) r = Ö( M /p) (d) r = Ö( P /pw). (a) M = A /kl (b) m = B /K (c) m = c /pr (d) m = /pd 4. (a) = p Ð q (b) = s Ð r (c) = (s Ð r) / (d) = (r + ) /7 (e) = (m Ð ) / (f) = m + (g) = n Ð (h) = p Ð q. (a) P = QR (b) s = /t (c) Q = Ö( P /M) (d) w = v z (e) f = e /d (f) n = K /mt (g) s = Ö( 7 /9r) (h) a = Ö(c Ð b ) 6. (a) T = V / (a+d) (b) = r /(p + q) (c) = c /(a Ð b) (d) = r /(m + ) (e) w = (v Ð v) / ( + ) (f) r = (p + ) 4 Eercise 4. (a) Ö (b) Ö (c) Ö (d) Ö (e) Ö (f) Ö7 (g) Ö (h) Ö6 (i) 0Ö (j) Ö (k) Ö (l) 6Ö (m) 0Ö (n) 7Ö (o) Ö6 (p) 4Ö (q) 0Ö (r) Ö0 Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 40

61 . (a) 8Ö (b) Ö (c) Ö0 (d) 0 (e) ÐÖ6 (f) ÐÖ (g) 0 (h) 0Ö + 0Ö!!. (a) Ö (b) Ö (c) 0Ö (d) 6Ö (e) Ö (f) Ö7 (g) 8Ö (h) Ö Eercise. (a) (b) (c) 6 (d) (e) (f) Ö6 (g) Ö (h) 4Öa (i) Ö(c) (j) Ö() (k) 4 (l) 8 (m) Ö (n) 0Ö (o) 6 (p) 6Ö6. (a) Ö + (b) Ö6 + Ö (c) + Ö (d) Ö7 + 7 (e) Ö Ð (f) Ö Ð 8 (g) + Ö (h) ÐÖ (i) (j) (k) + Ö6 (l) 8 Ð Ö. (a) 6 (b) Ð (c) 6 4. (a) 4Ö (b) 0 (c) 4Ö. (a) 4 cm (b) Ö6 cm Eercise 6. (a) Ö / (b) Ö / (c) Ö / (d) Ö6 /6 (e) Ö7 /7 (f) Ö (g) Ö / (h) Ö / (i) 0Ö (j) Ö (k) Ö6 (l) Ö /0 (m) Ö / (n) Ö /0 (o) Ö /0 (p) Ö. (a) Ö / (b) Ö0 / (c) Ö0 / (d) Ö / (e) Ö / (f). (a) Ö + (b) (Ö7 + ) /6 (c) ( Ð Ö) / (d) Ö Ð (e) 6 + Ö (f) Ö + Ö (g) Ö Ð Ö (h) Ö + Ö Eercise 7. (a) 7 (b) 8 (c) 8 7 (d) 0 0 (e) a 7 (f) b 8 (g) c 8 (h) d 0 (i) v (j) (k) w 4 (l) z 6 (m) f (n) g 0 (o) k (p) m 0. (a) (b) (c) 8 (d) 0 0 (e) a (f) b (g) c (h) (i) v (j) 8 (k) w (l) z (m) f (n) (o) k 0 (p) m 99 (q) 4 (r) m (s) a 4 (t) r. (a) 6 (b) (c) z 0 (d) g 6 (e) a (f) b 6 (g) c 0 (h) d 4. (a) 0 (b) 7 (c) 6 0 (d) 8 (e) 49 (f) (g) 9 0 (h). (a) a b (b) c 6 d 6 (c) 0 0 (d) 4p q 6. (a) (b) t (c) (d) v 6 (e) (f) 4 7 (g) 4 (h) 4 (i) (j) + 6 (k) 6 Ð 4 (l) u 7 v 7 (m) (n) m 8 n 6 (o) (p) u 6 7. (a) m = 4 (b) m = (c) m = (d) m = 4 Ö(ab) b Mathematics Support Materials: Mathematics (Int ) Ð Student Materials 4