Maths

No TOPIC PAGE 1 GEOMETRY 1 - 24 2 TRIGONOMETRY 25 - 36 GROWING KWAZULU-NATAL TOGETHER

EUCLIDEAN GEOMETRY Term 1 Duration 3 Days Sub-topics Grade 11 Geometry ATP  Accept results established in earlier grades as axioms and also that a tangent to a circle is perpendicular to the radius, drawn to the point of contact. REFERENCE  Investigate and prove the following theorems of the geometry of circles, assuming results from earlier grades:  The line drawn from the centre of a circle perpendicular to a chord bisects the chord.  The perpendicular bisector of a chord passes through the centre of the circle.  The angle at the centre of a circle is double the size of the angle at the circle.  Angles subtended by a chord of the circle, on the same side of the chord, are equal.  Accept results established in earlier grades as axioms and that a tangent to a circle is perpendicular to the radius, drawn to the point of contact. Then investigate and prove the following theorems of the geometry of circles:  The opposite angles of a cyclic quadrilateral are supplementary.  Two tangents drawn to a circle from the same point outside the circle are equal in length.  The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment  Solve circle geometry problems, providing reasons for statements. The proofs of the FIVE theorems printed in bold above can be asked in examinations. (See Gr. 11 Examination Guidelines. Class Work Activities QUESTION 1 1.1 In the diagram below, O is the centre of the circle and PTSR is a cyclic quadrilateral. ˆ  180 . Prove the theorem that states that P̂  S (6) QUESTION 2 In the figure, QS and PR are diameters of the circle with centre O such that PQ // SR. PS is ˆ Q produced to T. N is a point on the circle such that Q ˆ . SN is drawn. RS intersect QN at 1 2 M. Ŝ'1  48 . 1

2.1 Determine with reasons the size of: (a) Q̂1 (3 ) (b) R̂ (2) (c) M̂1 (2) 2.2 Prove that ST is a tangent to a circle passing through M, N and S. (2) [15] QUESTION 3 In the diagram, O is the centre of the circle. Chord AC is perpendicular to the radius OD at B. OB  2x units and AC  8x units. Show that length of BD is 2x  5  1 units. [5] 2

QUESTION 4 Complete the folloeing statement: 4.1 The angle between the tangent and the chord at the point of contact is equal to …………………………………... (1) 4.2 In the diagram, A, B, C, D and E are points on the circumference of the circle such that AE // BC. BE and CD produced meet in F. GBH is a tangent to the circle at B. B̂1  68 and F̂  20 . Determine the size of each of the following: 4.2.1 Ê1 (2) 4.2.2 B̂3 (1) 4.2.3 D̂1 (2) 4.2.4 Ê2 (1) 4.2.5 Ĉ (2) [09] 3

Home QUESTION 5 Work In the diagram below, CD is the common cord of the two circles. Straight line ADE and BCF Activities are drawn. Chords AB and EF are drawn. Prove that: EF // AB [5] QUESTION 6 In the diagram, O is the centre of the circle. A, B and C are points on the circumference of the circle. Chords AC and BC and radii AO, BO and CO are drawn. Â  x and B̂  y . 4

6.1 Determine the size of Ô1 in terms of x. (3) 6.2 Hence, prove the theorem that states that the angle subtended by an arc at the centre is equal to twice the angle subtended by the same arc at the ˆ  2 ACB circumference, that is AOB ˆ . (3) 06] QUESTION 7 In the diagram, PQ is a common chord of the two circles. The centre, M, of the larger circle lies on the circumference of the smaller circle. PMNQ is a cyclic quadrilateral in the smaller circle. QN is produced to R, a point on larger circle. NM produced meets the chord PR at S. P̂  x . 7.1 Give a reason why N̂ 2  x . (1) 7.2 Write down another angle equal in size to x. Give a reason. (2) 7.3 Determine the size of R̂ ïn terms of x. (3) 7.4 Prove that PS  SR . (3) [09] 5

QUESTION 8 In the figure Q, P, T, S and R lie on the circle. M lies on QS such that PM // SR. ˆ  145. PTS Calculate the size of: 8.1 Q̂ (2) 8.2 R̂ (2) 8.3 P̂2 (1) 8.4 Why is PR a tangent to the circle passing through P, M and Q? (1) [06] Class QUESTION 9 Work In the diagram, O is the centre of the circle. Points A, B, C and D lie on the circumference of Activities the circle. BOD is a diameter. AC and BD intersect at E. Aˆ1  51 and Bˆ1  29. Determine: 6

9.1 the size of Oˆ1. (2) 9.2 the size of Aˆ2 . (2) 9.3 the size of Dˆ . (2) 9.4 the size of Bˆ2 . (2) [14] QUESTION 10 O is the centre of the circle. P, Q and R are points on the circle. PR, QR, PO, and QO are drawn. 10.1 Complete the following statement: The angle at centre of a circle is…….. (1) 10.2 ˆ  2 PRQ Hence prove the theorem which states that POQ ˆ . (6) [07] QUESTION 11 In the sketch below, O is the centre of the circle. Points P, Q, R and S lie on the circumference of the circle. Points T, O, and Q form a straight line such that T lies on PS. It is given that PQ = QR and Qˆ1  x. 7

11.1 Calculate P̂1 in terms of x. (7) 11.2 ˆ . Prove that line TQ bisects PQR (7) [14] Home QUESTION 12 Work Activities 12.1 Calculate, with reasons, the size of: 12.2 Prove that: [18] 8

TOPIC: GR. 12 GEOMETRY From Term: 1 Duration: 2 days Sub-topic Proportionality 1. Revise earlier work on the necessary and sufficient conditions for polygons to be similar. 2. Prove (accepting results established in earlier grades): ATP A line drawn parallel to one side of a triangle divides the other two sides proportionally Reference: (and the Midpoint Theorem as a special case of this theorem). 3. Solve proportionality problems and prove riders. Classwork 1. In ABC , PQ || BC and each side has 5. In the diagram below, prove that Activities the dimensions shown on the diagram. AB BC  . Determine the value of x. AD BF 2. In ABC , DE || BC , AB = 16mm, AD = 12 mm and EC = 6 mm. Determine the length of AE and AC. 6. In the diagram below, BF || CE , BC 3  and AE : ED = 4 : 3. AB 5 Determine the ratio DG : GB. 3. Determine the value of x: 7. In ABC , AD = 3 m, DB = 12 m, AE = 6 m and EC = 24 m. Prove that DE || BC . 4. Determine the value of x: 9

Classwork National June 2018: Activities (continued) Limpopo September 2019: 9.2 In the diagram QR of PQR is produced such that XS || YR . PX 2 PT 3  and  . XQ 5 TR 2 Calculate, giving reasons, QR the value of . (6) RS Free State September 2019: 10

Homework 1. Determine the value of x: 5. In the diagram below, prove that Activities PS RU  . SQ WU 2. Determine the value of x: 6. In the diagram below, PS || VT and PQ || SV . If 11 PV = 12 VR, determine the ratio QS : QT. 3. Determine the value of x: 7. In the diagram below, AD || EF , 4. In ΔADF , AE = EF and CG = 30 cm. BD : DC = 2 : 3 and AE : EC = 2 : 7. Calculate the length of FG if Determine BE : BG. AB = BC = CD. 11

Homework National June 2019: Activities (continued) AN CL Determine, with the reasons, the ratio of: 8.2.1 (2); 8.2.2 (3) AM LK Homework KZN June 2019: Activities (continued) Area of FEC 10.1 the length of CD (3); 10.2 the value of (4) Area of FAD 12

Limpopo September 2020: Calculate, giving reasons, EM CM Area of FDC 9.2.1 (3); 9.2.2 (3); 9.2.3 (4) AM ME Area of BDC National November 2018: 8.2.1 Explain why FC || GH . (1) 8.2.2 Calculate, with reasons, the length of DM. (5) 13

Homework National November 2020: Activities (continued) North West September 2019: Determine, with reasons, the following ratios: AS RT Area of RAS Area of TPC 10.2.1 ; 10.2.2 ; 10.2.3 ; 10.2.4 SC TC Area of RSC Area of RSC 14

TOPIC: GR. 12 GEOMETRY From Term: 1 Duration: 2 days Sub-topics Similarity 1. Prove (accepting results established in earlier grades): 1.1 that equiangular triangles are similar; ATP 1.2 that triangles with sides in proportion are similar; and Reference: 1.3 the Pythagorean Theorem by similar triangles. 2. Solve similarity problems and prove riders. Classwork 1. In the figure below, Ĉ1  B̂ . 3. Given SPR and SQP , prove that Activities 1.1 Prove that ΔABC ||| ΔACD SP̂R  Q̂ . 1.2 Hence, prove that AC 2  AB  AD . 4. ABC is right-angled at B. D is a point on AC such that BD  AC and E is a point on AB such that DE  AB . 2. In the diagram below, CD  AB and AD : DC = 3 : 2 and AD = 30 m. AF  BC . E is the midpoint of AF. 4.1 Prove that ΔBDA ||| ΔCDB . Prove that: 4.2 Calculate BD, leaving your answer in BD EF simplest surd form. 2.1  ; 2.2 EF 2  DE  CE . CD CF 4.3 Calculate AB, leaving your answer in simplest surd form. 15

Classwork 5. Activities (continued) 6. Homework 1. Activities 2. 16

Homework 3. Activities (continued) 4. 17

TOPIC: GR. 12 GEOMETRY From Term: 1 Duration: 1 day Sub-topics Proportionality and Similarity: More practice questions from Past Papers ATP 1. Solve proportionality problems and prove riders. Reference: 2. Solve similarity problems and prove riders. Classwork Western Cape September 2018: Activities 11. 11.1 Give, with reasons, three more angles each equal to x. (6) 11.2 Prove that ΔDEA ||| ΔDBC . (4) 11.3 Prove that BC.ED  AE 2 . (5) 11.4 Calculate the value of x if it is given that Â1  75 . (3) Gauteng September 2018: 11. Prove that 11.1 ΔNSR ||| ΔNPQ 11.2 ΔNQR ||| ΔNPS 11.3 NR.PQ  NP.QR  NQ.PR 18

Classwork National March 2018: Activities 9. (continued) 9.1 Give a reason why ΔDEG||| ΔEGF . (1) 9.2 Calculate the length of GE. (3) 9.3 Prove that ΔDEH||| ΔFGH . (3) 9.4 Hence, calculate the length of GH. (3) Homework Gauteng September 2019: Activities 11. Prove that 11.1 ΔNSR ||| ΔNPQ 11.2 ΔNQR ||| ΔNPS 11.3 NR.PQ  NP.QR  NQ.PR 19

Homework Limpopo September 2019: Activities 10.2 (continued) Prove that: 10.2.1 SB || RP (5) 10.2.2 APS | | | RWS (4) 10.2.3 RS 2  WS.AS (4) RW 2 10.2.4 AS   WS (4) WS Gauteng September 2020: 9.2.1 State, with reasons, three angles equal to M̂ 2 . (3) 9.2.2 Prove that ΔWMV ||| ΔQMN (3) MV MN 9.2.3 Prove that  . (3) WV PW 20

Homework Limpopo September 2020: Activities Two circles with centres A and B, intersect at X and Y. The radius of circle A is R and the (continued) radius of circle B is r. CX is a tangent to circle B at X and DX is a tangent to circle A at X. Prove that 10.1 ΔXYC||| ΔDYX , and hence deduce that XY 2  DY.YC . (5) 10.2 Â1  B̂1 (4) 10.3 If ΔCAY ||| ΔYBX , r 2 DY then  . (5) R 2 CY . National June 2019: 10.2.1 Prove that: (a) DGFO is a cyclic quadrilateral (3) (b) GC = CF (5) 10.2.2 If it is further given that CO = 11 units and DE = 14 units, calculate (a) the length of BC (3) (b) the length of CG (5) (c) the size of Ê . (4) 21

ANSWERS: GR. 11 GEOMETRY Question 1 QUESTION43 Bookwork 4.1 ….an angle in an alternate segment. Question 2.1 4.2 (a) Q̂1  24 4.2.1 Ê  68 1 (b) R̂=24 4.2.2 B̂3  68 (c) M̂1  48 4.2.3 D̂1  68 12.2 M̂1  Sˆ 1  48 4.2.4 Ê2  88 4.2.5 Ĉ  92 Question 3 4.3 Proof Proof Question 5 Proof QUESTION 6 QUESTION 4 5.1 6.1 Ĉ1  x 5.1.1 Qˆ  145  180 Ô1  2x  Qˆ  35 6.2 Proof 5.1.2 Rˆ  Qˆ  35 QUESTION 7 5.1.3 Pˆ2  Rˆ  35 7.1 Ext < of a cyclic quad 5.2 Pˆ  Qˆ  35 2 7.2 Q̂1  x 5.3 7.3 R̂  90  x 5.3.1. Oˆ1  2  51  Oˆ  102 1 7.4 Proof 5.3.2 Aˆ1  Aˆ2  90 Aˆ  90  51 2  Aˆ 2  39 QUESTION 6 QUESTION 8 6.1 6.1.1. The angle at centre of a circle is 2 times the angle at the circumference. 6.1.2. Proof 6.2 6.2.1 Rˆ  Qˆ  x (OQ  OR, radii ) 1 1 Oˆ1  180  2 x ( sum of  in a )  Pˆ1  90  x ( at centre.th.) 6.2.2 Proof 22

QUESTION 8 QUESTION 9 8.1 Q  35 opp  s of cycl quad supp 9.1 Q  102  at the centre = 2  at circ 1 8.2 R=35 opp  s of cycl quad supp 9.2 A 2  39  in a semi-circle 8.3 P2 =35 alt  s, PM SR 9.3 D  61 sum of  s of  8.4 P  Q  35 conv tan chord thm 9.4 B2  39  s opp = sides QUESTION 10 QUESTION 11 Proof 11.1 P1  90  x  at the centre = 2  at circ 11.2 Proof QUESTION 12 12.1 (a) A1  35 (b) O3  70  at the centre = 2  at circ (c) P3  55  s opp = sides (d) M1  35 sum of  s of  12.2 (a) Proof (b) Proof 23

ANSWERS: GR. 12 GEOMETRY Sub-topic: Proportionality Sub-topic: Similarity More practice questions Classwork Activities: Classwork Activities: Classwork Activities: 1. 4,8 mm 4.2 10 6 units Western Cape September 2018: 2. AE = 18 mm 11.4 35  AC = 24 mm 4.3 10 15 units National March 2018: 3. 40 units 5 b) 5 cm 9.2 60 4. 20 units 6 c) 2 41 cm 9.4 38,4 6. 2 : 1 6 d) 10,25 cm Homework Activities: National June 2018: Homework Activities: National June 2020: 9.2.1 8 3 d) 2,67 cm 10.2.2 a) 4 units 4 4 d) 2 cm 10.2.2 b) 8,49 units 9.2.2 25 10.2.3 c) 19,76 Limpopo September 2019: 4 9.2 25 Free State September 2019: 11.2 7,5 Homework Activities: 1. 42 units 2. 8 units 3. 10 cm 4. 90 cm 6. 23 : 34 7. 4 : 3 National June 2019: 3 8.2.1 5 23 8.2.2 4 KZN June 2019: 10.1 15 units 6 10.2 25 Limpopo September 2020: 3 9.2.1 7 7 9.2.2 3 3 9.2.3 7 National November 2018: 8.2.2 14 National November 2020: 8.2.2 31 North West September 2019: 3 2 10.2.1 ; 10.2.2 7 5 3 25 10.2.3 ; 10.2.4 7 49 24

TRIGONOMETRY PART 1 Sub-topics Solving Trig expressions using sketches, Reduction formula and Trig Identities using Compound and Double angles ATP Compound angle identities: REFERENCE 1. cos      cos  cos  sin  sin  2. sin      sin  cos   cos  sin  3. sin 2  2sin  cos  4. cos 2  cos 2   sin 2  5. cos 2  2cos 2   1 6. cos 2  1  2sin 2  Class Work Activities Feb/Mar 2018 5 1. If cos 2   , where 2  180; 270 , calculate, without using a calculator, the values in 6 simplest form of: 1.1 sin 2 1.2 sin 2  May/Jun 2021 2. Given: cos 35° = m Without using a calculator, determine the value of EACH of the following in terms of m: 2.1 cos 215° 2.2 sin 20° May/Jun 2021 3. Simplify the expression to a single trigonometric term: tan  x . cos x. sin  x  180   1 sin 4 x. cos 2 x  2 cos 4 x. sin x. cos x 4. Prove the identity:  cos 2 x  sin 2 x tan 2 x 25

Home Work Activities QUESTION 1 1.1 May/Jun (2019) Without using a calculator, write the following expressions in terms of sin 11 : 1.1.1 sin 191 1.1.2 cos 22 1.2 If sin 25  m , express each of the following in terms of m. 1.2.1 cos 205 1.2.2 cos50 1.2.3 sin 55 1.3 If cos 61  p , express the following in terms of p 1.3.1 sin 209 1.3.2 tan  421  1.3.3 cos1 1.4 May/Jun 2019 7 Given: sin P  sin Q  and P̂  Q̂  90 Without using a calculator, determine the value 5 of sin 2P . 1.5 3 If sin x  cos x  , calculate the value of sin 2x WITHOUT using a calculator. 4 QUESTION 2 2.1 May/Jun 2019 Simplify cos( x  180)  2 sin( x  45) to a single trigonometric ratio. 2.2 Feb/Mar 2018 Simplify sin 180  x  .cos   x   cos  90  x  .cos  180  to a single trigonometric ratio. 2.3 cos  90     sin 180    Simplify fully: sin 2 2.4 sin 290.tan  45  .sin160.sin  90  x  Evaluate without using a calculator: cos  x  360  .cos 300.sin140 2.5 1 Determine the value of:  tan 2 180    cos  360    .sin  90    2.6 Simplify without the use of a calculator:  sin104 2 cos 2 15  1  tan 38.sin 412 2 QUESTION 3: 3.1 sin 2 x  3cos 2 x  2 Prove the identity: 1 cos 2 x  sin 2 x 3.2 cos  2 Prove that tan    sin  sin 2 26

3.3 2  cos2   2sin  1  sin  Prove the following identity:  cos  2 1  sin  3.4 cos 2  90    1 Prove the following identity:  1 cos     sin  90    cos  cos  3.5 Prove the identity: cos 180  x   tan  x  180  sin  720  x  cos x  cos 2 x 2 TRIGONOMETRY PART 1: ANSWERS CLASSWORK ACTIVITIES 1.1 11 sin 2   6 1.2 11 sin 2   12 2.1 cos 215  m 2.2 sin 20  2m2  1 3  cos 2 x HOMEWORK ACTIVITIES QUESTION 1 1.1 1.3 1.1.1  sin11 1.3.1 p 1.1.2 1  2sin 11 2 1.3.2 1  p2  p 1.2 1.3.3 p  3  3 p2 2 1.2.1  1 p 2 1.4 24 sin 2P  25 1.2.2 1  2 p2 1.5 7 sin 2 x  16 1.2.3 3 p  1  p2 2 QUESTION 2 2.1 sin x 2.2 sin 2x 2.3 1 cos  2.4 1 2.5 1 2.6 3 27

TRIGONOMETRY PART 2 A] TRIG EQUATIONS Some notes to help you: 1) GENERAL SOLUTION  Period for trig graphs (period for sine & cosine is 360°,period for tan is 180°)  In General solution it is where there is a point of intersection of graphs or with axes or with certain value. It has its own way of solving.  Try to simplify trig ratios (have 1 or same trig ratios), try co-functions if not, try method of factorization, quadratic formula can be used (Trig Equation).  Divide with variable only if the answer will be tan of an angle. Once you divide with variable and get answer that is not tan, then one solution will be lost.  Specific solution (It is when domain is given) 2) FOR SPECIFIC SOLUTION: Where by the is a given interval e.g [𝟎°; 𝟑𝟔𝟎°]  Now substitute integers to general solution to get specific values.  Integers usually 𝑘 = −1 𝑘=0 𝑘=1 QUESTIONS: 1. Determine the general solution of the following equations: 1.1 sin  3 x  15   0, 4 1.2 sin 2x  cos x  0 1.3 6sin 2 x  7 cos x  3  0 1.4 cos 2 x  cos x  2  0 1.5 cos  x  30   sin 2 x 1.6 cos  x  45   2sin x 2. 1  cos 2 x  sin x Consider the expression: sin 2 x  cos x For which value(s) x   0;360 , will this expression be undefined? 3. Calculate the solutions to the equation: sin2𝑥 = cos(𝑥 − 30°) for −90° ≤ 𝑥 ≤ 90° 4. cos 2 x.tan x Given the expression: sin 2 x For which value(s) x   0;180 , will this expression be undefined? 5. cos  cos 2 Consider the identity:   tan  sin  sin  cos  For which value(s) of 𝜃 in the interval 0° < 𝜃 < 180° will the identity be undefined? 6. 4 Determine the general solution of: 6cos x  5  , cos x  0 cos x 7. Determine the general solution of: sin 2 x  cos 2 x  cos x  0 28

8. Given: sin x  cos 2x 1 8.1 Show that 2sin 2 x  sin x  0 8.2 Determine the general solution of the equation: sin x  cos 2x 1 9. 3 Determine the general solution of: cos 3 .cos   sin 3 .sin    2 10. Given: cos  A  B   cos A cos B+ sin A sin B 10.1 Use the formula for cos(A –B) to derive a formula for sin(A−B) 10.2 Without using a calculator, show that 1 sin  x  64  cos  x  379   sin  x  19  cos  x  244   for all values of 𝑥. 2 B] TRIG GRAPHS QUESTIONS: 1. In the diagram below, the graph of f ( x)  sin x  1 is drawn for 90  x  270 1.1 Write down the range of f . 1.2 Show that sin x  1  cos 2 x can be written as (2 sin x  1) sin x  0. 1.3 Hence, or otherwise, determine the general solution of sin x  1  cos 2 x. 1.4 Use the above grid to draw the graph of g ( x)  cos 2 x for 90  x  270 . 1.5 Determine the value(s) of x for which f ( x  30)  g ( x  30) in the interval 90  x  270 . 1.6 Consider the following geometric expression: 1  2 cos 2 x  4 cos 2 2 x  ... 1.7 Use the graph of g to determine the value(s) of x in the interval 0  x  90 for which this series will converge. 2. In the diagram below, the graphs of f ( x)  cos x  q and g ( x)  sin( x  p) on the same system of axes for 240  x  240 .  1 The graphs intersect at  0;  ,  120; 1 and  240; 1 . 2   29

2.1 Determine the values of p and q. 2.2 Determine the values of x in the interval 240  x  240 for which f ( x)  g ( x). 2.3 Describe a transformation that the graph of g has to undergo to form the graph of h, where h( x)   cos x. 3 In the diagram, the graphs of the functions f ( x)  a sin x and g ( x)  tan bx are drawn on the same system of axes for the interval 0  x  225 . 3.1 Write down the values of a and b. 3.2 Write down the period of f (3x). 3.3 Determine the values of x in the interval 90  x  225 for which f ( x).g ( x)  0. 30

TRIGONOMETRY PART 2 ANSWERS 1.1 x  12,86  k .120 k  or x  57, 41  k.120 k  1.2 x  90  k.360 k   or x  270  k.360 k   1.3 x  109, 47  k .360 k   or x  250,53  k .360 k   1.4 x  0  k.360 k   or x  360  k.360 k   1.5 x  20  k.120 k   or x  60  k.180 k   1.6 x  165, 43  k .180 k   or x  345, 43  k .180 k  2 x  90 or x  270 or x  30 or x  150 3 x  80; 40;60 4 x  0;90;180 5   90 6 x  120  k.360 k   or x  240  k.360 k   7 x  90  k.360 k   or x  270  k.360 k   x  0  k.360 k   or x  360  k.360 k   8.2 x  0  k.360 k   or x  180  k.360 k   x  210  k.360 k   or x  330  k.360 k   9. x  75  k.180 k   or x  105  k.180 k   Trig Graphs 1.1 0 y2 1.3 x  210  k.360 k  or x  330  k.360 k  1.4 1.5 x  60; 30;150;180 1.6 30  x  60 2.1 q 1 2 p  30 2.2 120  x  0 2.3 Shift g 60 to left and reflect about the x-axis or Shift g 120 to the right or Shift g 240 to the left 3.1 a  1 b2 3.2 120 3.3 0  x  45 or 90  x  135 or x  180 31

TOPIC: TRIGONOMETRY PART 3 -2D and 3D problems Term 1 Day(s) w.r.t. ATP 5 allocation Duration 1 lesson Weighting 8±3 Sub-topics 2D problems- Revision ATP REFERENCE  Solve problems in two and three dimensions Revision Activities Question 1 Question 2 32

4`Question 3 Question 4 Question 5 Question 6 33

TOPIC: TRIGONOMETRY PART 3 2D and 3D problems Term 1 Day(s) w.r.t. ATP 3 allocation Duration 1 lesson Weighting Sub-topics 3D problems ATP REF.  Solve problems in two and three dimensions Revision Activities Question 1 T, B and G are points on horizontal ground, TG is a vertical tower. The angles of elevation of T from B and A are x and y respectively, and ˆ = w . BGA GBA ˆ = 130 Prove that tan x . sin(50  w) sin w = tan y Question 2 In the diagram, DBCˆ  90, BCD ˆ  CAE ˆ  , BCA ˆ  , AB||EC and DB  x . 2.1 Find ABCˆ in term of α and β.  cos   2.2 Prove that AC  x   sin   .  tan   Question 3 A, B and C are three points in the same horizontal plane with vertical tower RC. AB  y and the angle of elevation of R from B is  . ˆ   and RBA RAB ˆ  . 34

y sin  .sin  3.1 Prove that RC  sin     3.2 Hence determine the height of the tower if y  40 metres,   38,   74 and   52. Question 4 E is the top of a mountain EF. A and B are two points 800m from each other and in the same horizontal plane as the foot of the  mountain. EA = 4273m, E A B  59,4 and the angle of elevation of E from B is 15,6°. Calculate the height of the mountain. Question 5 A mountain climber wants to determine the height, MN, of a mountain. He uses a theodolite (instrument used to measure angles). The mountain climber is standing on flat ground. P is in the same horizontal plane as the foot of the mountain N. He measures the following angles from P:  The angle of elevation of the top of the mountain M, is x  MP̂Q is y He then walks d metres to point Q , which is also in the same horizontal plane as P and N, and measures PQ̂M which is z . d sin x sin z 5.1 Show that: MN  sin  y  z  5.2 Calculate MN, without using a calculator if d  1 000 m, x  90  y and y  z . 35

Question 6 Principal Mr Anders wants to provide tarred paths between the different buildings on his school grounds. He draws a plan of the five paths as shown in the diagram alongside. The paths connect the buildings which are positioned at A, B, C and D. B1 = 𝜃 and ̂ ̂ A = 𝛼. AB = k and BD = w. AD  CD Show that the length of the path AB is given by 𝑤 sin(𝜃 + 𝛼) 𝑘= sin 𝛼 TRIGONOMETRY PART 3A Answers to 2D and 3D Question Answer to 2D problems 1 Area =19.39 2 Proof 3 AB=12.11 4a AB=17.56 𝜃 = 25.33 4b QS=253.48 PQ=169.37 𝜃 = 83.71 5a ℎ 𝐷𝐸 = 𝑠𝑖𝑛𝛼 5b ̂2 = 𝛼 − 𝛽 𝐷 5c Proof 5d ℎ =197.51 6 𝑥 = 7.15 TRIGONOMETRY PART 3B Answer to 3D problems 1. Proof 2.1 90 − 𝛽 + 𝛼 2.2 Proof 3.1 Proof 3.2 RC=23.67 4 EF=1096.33m 5.1 Proof 5.2 MN=500 6 Proof 36