JIT Gr 11 T2 2018

JUST IN TIME MATERIAL GRADE 11 CURRICULUM GRADES 10 – 12 DIRECTORATE TERM 2 – 2018 This document has been compiled by KZN Mathematics Advisors. It seeks to unpack the content and give more guidance to the teachers. Please note that this document is intended to supplement the Text book and not replace it!

TABLE OF CONTENTS TOPIC PAGE FUNCTIONS & GRAPHS 3 - 16 TRIGONOMETRY 17 - 28 KZN FET Mathematics 2017 Page 2 of 28

FUNCTIONS AND GRAPHS % COM- DATES CURRICULUM STATEMENT PLETED Revise the effect of a and q and investigate the effect of p on 10/4-13/4 the graphs of the functions defined by: 31% (4 days) y =f(x) =a (x + p)2 + q Revise the effect of the parameters 𝑎 and 𝑞 and investigate the 16/4-20/4 effect of 𝑝 on the graphs of the functions defined by: 35% 𝑎 (5 days) • 𝑦 = 𝑓(𝑥) = 𝑥+𝑝 + 𝑞; Revise the effect of the parameters 𝑎 and 𝑞 and investigate the 23/4-26/4 effect of 𝑝 on the graphs of the functions defined by: 38% (4 days) • 𝑦 = 𝑓(𝑥) = 𝑎. 𝑏 𝑥+𝑝 + 𝑞, where 𝑏 > 0 𝑎𝑛𝑑 𝑏 ≠ 1. • Investigate numerically the average gradient between two 02/5-04/5 points on a curve. (3 days) • Develop an intuitive understanding of the concept of the 40% gradient of a curve at a point. • Interpretations, applications and Practical problems. NB: Apply nature of roots with functions JUNE COMMON TEST WEIGHTING Functions & Graphs 40 ±3 marks out of 100 marks in P1 CAPS EXAM GUIDELINE WEIGHTING FOR FINAL EXAMINATION Functions & Graphs 45 ± 3 marks out of 150marks in P 1 KZN FET Mathematics 2017 Page 3 of 28

A. STRAIGHT LINE General representation or equation y  ax  q or y  mx  x . a or m is the gradient and q or c is the y  intercept Also note the shape of the following linear functions a<0 a=0 a>0 a is undefined q<0 y=q q<0 there is no q-value Domain and range is x   and y   respectively B. HYPERBOLA General representation or equation a a a a y or xy  a y q or y q or y q x x x p x p a0 a0 Dotted lines are asymptotes Dotted lines are asymptotes • q is the vertical translation • p is the horizontal translation a • For y  , p  0 and q  0 . The vertical asymptote is x  0 and the horizontal x asymptote is y  0 . The axis of symmetry are y  x (Positive) and y  x (Negative) Domain is x  0, x   and Range is y  0, y   KZN FET Mathematics 2017 Page 4 of 28

a • For y   q , p  0 . The vertical asymptote is x  0 and the horizontal asymptote is x y  q . The axis of symmetry are y  x  q (Positive) and y   x  q (Negative). Domain is x  0, x   and Range, y  q, y   a • For y   q  ( y  q)( x  p)  a , the vertical asymptote is x  p and the x p horizontal asymptote is y  q . The axis of symmetry is y  x  p   q . Domain is x   p, x   and Range, y  q, y    q   y  q x  p   a , the vertical asymptote is x   p and the a • For y  x p horizontal asymptote is y  q . The axis of symmetry is y  x  p   q . Domain is x   p, x   and range is y  q, y   Example 1 3 Given : f ( x)  1 x2 1.1. Write down the equations of the asymptotes of f . 1.2. Determine coordinates of B, the x-intercept of f. 1.3. Determine the coordinates of D, the y-intercept of f 1.4. Determine the domain and the range of f 1.5. Determine the decreasing and increasing functions of the axes of symmetry of f 1.6. Draw the sketch graph of f KZN FET Mathematics 2017 Page 5 of 28

Solutions: 1.1) Vertical asymptote is x  2  0 1.2) x  intercept  y  0 x2 3 0 1 Horizontal asymptote is y  1 x2  1( x  2)  3  x  2  3  x  1  x  1 1.3) y  intercept  x  0 1.4) Domain is x  2; x   3 3 2 1 Range is y  1; y   y 1   2 2 2 1.5) Axes of symmetry are: 1.6) y y   x  2   1 f 1 1  x  2  1 or y   x  2  1  12 2 x y  x  1 Increasing (positive gradient) or y  x  3 Decreasing (Negative gradient) C. PARABOLA General representation or Equation y  ax2 y  ax2  q y  a x  p   q y  ax2  bx  c 2 or or or Important Deductions for a  0 for a  0 • For y  ax , p  0 and q  0 , the turning point is (0 ; 0) and y-intercept is 2 y=0 KZN FET Mathematics 2017 Page 6 of 28

The domain is x  ℝ and the range is y  0; y  ℝ if a  0 or y  0; y  ℝ if a  0 • For y  ax  q , p  0 , the turning point is (0 ; q) and y-intercept is y = q 2 The domain is x  ℝ and the range is y  q ; y  ℝ if a  0 or y  0; y  ℝ if a  0 • For y  ax  p   q , the turning point is ( p ; q) and y-intercept is y  a p   q 2 2 The domain is x  ℝ and the range is y  q ; y  ℝ if a  0 or y  0; y  ℝ if a  0   b 4ac  b 2  • For y  ax  bx  c , the turning point is   and y-intercept is 2 ;  2a 4a  y=c 4ac  b 2 The domain is x  ℝ and the range is y  ; y  ℝ if a  0 or 4a 4ac  b 2 y ; y  ℝ if a  0 4a • The roots or x-intercepts are determined by equating y to zero and solve for x. Example 2: Sketched below are the graphs of: g x    2 x  8 ; f x  x 2  k and 6 h(x) = 1 x2 A and B are the x- and y - intercepts of h respectively, C (−6 ; 20) and E are the points of intersection of f and g . y C(−6 ; D 20) f h O E A x g B KZN FET Mathematics 2017 Page 7 of 28

2.1 Calculate the coordinates of A, B and E. 2.2 Show that the value of k   16 2.3 Determine the domain and the range of f 2.4 Write down the values of x for which g x   f x   0 2.5 Determine the equation of the symmetry axis of h if the gradient is negative. 2.6 Write down the range of s, if s(x) = f(x) + 2. 2.7 Write down the range of t, if t(x) = h(x) + 2 Solutions: To answer the above questions you need to identify all the functions in order to apply the deductions indicated above. 2.1 A and B are x and y intercepts of g respectively. 6 6 at A, y  0   1 0 at B, x  0  y  1 x2 2 6  x  2 y  3  1 4  x  y  2  x  4 Thus A  4;0 Thus B 0;2 E is the x- intercept of the straight line and the parabola. It is easy and straight forward to use the equation of the straight line to get the coordinates of E. At E, y  0; 0  2 x  8 2x  8 Thus E 4;0  x4 2.2) C(-6; 20) is on f and g , 2.3) Domain is x  ℝ substituting the into Range is y  16; y ℝ y  x 2  k  20   6  k 2  20  36  k k  16 KZN FET Mathematics 2017 Page 8 of 28

2.4) These are values of x for 2.5) For negative gradient, y  ( x  2)  1 which the graph of g and f intersect or f is below g . y  x  2 1 It is from C(-6 ; 20) and E(4 ; 0) y  x  3 That is  6  x  4 2.6) + 2 implies the value of p is 2.7) + 2 implies the value of p is increased by 2 increased by 2 The range of t is y  1  2; y  ℝ The range of s is y  16  2 y  3; y  ℝ y  14 D. EXPONENTIAL General representation or Equation: y  abx or y  abx  q or y  abx p  q The restriction is b  0; b  1 Important Deductions for for for for a  0 and 0  b  1 a  0 and b  1 a  0 and 0  b  1 a  0 and b  1 • For y  ab , the asymptote is y = 0 and the y-intercept is y  a x • For y  ab  q , the asymptote is y = q and y-intercept is y = a + q x x p • For y  ab  q , the asymptote is y = q and y-intercept is y  ab p  q Example 3 Given: 𝑓(𝑥) = 3−𝑥+1 − 3 3.1 Write f (x) in the form y  ab  q x 3.2 Draw the graph of 𝑓 showing all the intercepts with the axes and the asymptote. 3.3 What is the domain and the range of f ? KZN FET Mathematics 2017 Page 9 of 28

Solutions: x  x 1 x 1 3.1 y 3  3  3 .3  3  3.3  3  3   3 x  3 x 1 3.2 The asymptote is y  3 , x -intercept, y = 0, i.e. 3   3  0  3 x x 0 1 1 1 3   3     1     3 3  3 y x  0 0 f x 3 3.3. Domain is x  ℝ and range is y  3; y ℝ. KZN FET Mathematics 2017 Page 10 of 28

E. QUESTIONS FROM PAST EXAMINATION PAPERS. QUESTION 1 12 Given: ℎ(𝑥) = + 6 for 𝑥 > 0 𝑥−4 1.1 Draw a neat sketch graph of h in your work book . Show all intercepts with the axes and asymptotes. (4) 1.2 Write down the equation of k if k is the reflection of h about the x-axis. (3) QUESTION 2 2.1 Sketched below are the functions: 𝑓(𝑥) = 2𝑥 2 − 6𝑥 − 20 and 𝑔(𝑥) = −2𝑥 + 𝑘. y f 0 x A B C D g Determine: 2.1.1 the coordinates of turning point D. (2) 2.1.2 the coordinates of A and B. (3) 2.1.3 the value of k . (2) 2.1.4 the values of p if 2𝑥 2 − 6𝑥 + 𝑝 = 0 has no real roots. (2) 2.1.5 for which values of x is 𝑓(𝑥). 𝑔(𝑥) ≤ 0. (2) 2.1.6 the value of t if 𝑦 = −2𝑥 + 𝑡 is a tangent to f . (4) 2.2 Consider the following two functions: 𝑝(𝑥) = 𝑥 2 + 1 and 𝑟(𝑥) = 𝑥 2 + 2𝑥. 2.2.1 How will you shift p to become the function r ? (3) 2.2.2 Write down the range of p . (1) KZN FET Mathematics 2017 Page 11 of 28

QUESTION 3 a The diagram above shows the graph of f ( x)   q . The lines x = –1 and y = 1 x p are the asymptotes of f . P(2 ; 4) is a point on f and T is the x–intercept of f. y P(─2 ; 4) 1 O x ─1 T 3.1 Determine the values of a, p, and q. (4) 3.2 Calculate the coordinates of T, the x – intercept of f. (3) 3.3 If the graph of f is symmetrical with respect to the line y  x  c , determine the value of c. (2) [9] KZN FET Mathematics 2017 Page 12 of 28

QUESTION 4 The sketch below shows the graphs of g ( x)   x 2  2 x  3 and h( x)  ax  q . The graphs intersect at B and E. The graph of g intersects the x – axis at A and B and has a turning point at C. The graph of h intersects the y – axis at D. The length of CD is 6 units. y C h A O B x g D E 4.1 Determine the coordinates of B and C. (4) 4.2 Write down the coordinates of D. (2) 4.3 Write down the values of a and q. (2) 4.4 Determine the coordinates of E. (5) QUESTION 5 The function 𝑓 is given by the equation y = − 𝑥 2 − 2𝑥 + 8 5.1 Determine all the intercepts of 𝑓(𝑥) with the axes. (4) 5.2 Determine the turning point of 𝑓(𝑥). (3) 5.3 Determine the value of 𝑐 if the graph 𝑔(𝑥) = − 4𝑥 + 𝑐 is a tangent to the graph of 𝑓(𝑥). (4) KZN FET Mathematics 2017 Page 13 of 28

QUESTION 6 The graphs of the functions 𝑔(𝑥) = −1. 2(𝑥+𝑝) + 𝑞 and 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 are drawn below. Y 4 𝑓 𝑔 1 𝑥 0 6.1 Show that 𝑝 = 1 and 𝑞 = 4 (3) 6.2 Determine the 𝑦 − intercept of𝑔(𝑥). (2) 6.3 Determine the equation of𝑓(𝑥). (5) 6.4 Write down the range of 𝑔(𝑥). (1) 6.5 Explain how increasing the value of 𝑞 will change the graph of 𝑔(𝑥). (1) KZN FET Mathematics 2017 Page 14 of 28

QUESTION 7 2 Below is the graph of 𝑔 ∶ 𝑥 → 𝑥−3 + 𝑞, with a point (2; − 6) on the graph. 𝑦 𝑥 0 𝑔 ( 2 ; -6) 7.1 Write down the equations of the asymptotes of 𝑔. (3) 7.2 Determine the domain of the graph of 𝑔 (2) 7.3 Determine the equation of 𝑓(𝑥)=−2 𝑔( x+ 2) (3) QUESTION 8 3 The equation of a function f is given by f(x) = 1. x2 8.1 Determine the asymptotes of f(x). (2) 8.2 Determine the x and y intercepts. (3) 8.3 Draw a neat sketch of f on the given diagram sheet. (3) 8.4 Write down the asymptotes of k(x) if k(x) is a function where f(x) is moved 5 units to the left and 1 unit down. (2) KZN FET Mathematics 2017 Page 15 of 28

QUESTION 9 In the diagram is the graphs of: h(x) = px + q and g(x) = ax2 + bx + c f touches the x axis at (-2 ; 0). h and g intersects at (0;4). T(-1;6) lies on the graph of h y g T(-1 ; 6) • 4 h x (-2:0) O 9.1 Determine the equation of g. (4) 9.2 Determine the value of p and q. (4) 9.3 For what values of x is h(x) < g(x)? (1) 9.4 Describe the transformation from g to p if p(x) = x 2 (2) QUESTION 10 a The diagram below shows the graphy of: f ( x)   q T(5;3) is a point on f, x p x 10.1 Determine the values of p, q and a. (4) 10.2 Write down the equation of p(x) if p(x) = f(x + 1)  2. (2) 10.3 If the new graph p(x) is reflected across the line with equation y =  x + c, then the new graph will be exactly y = f(x). Determine the value of c. (3) KZN FET Mathematics 2017 Page 16 of 28

TRIGONOMETRY % COM- DATES CURRICULUM STATEMENT PLETED 07/5-11/5 • Basic graphs defined by y = a sin x , y = a cos x and y = 14/5-15/5 tan x for (7 days)   [360 0 ;360 0 ] • Investigate the effect of the parameter k on the graphs of the functions defined by: y = sin (kx), y = cos (kx) and y = tan (kx). • Investigate the effect of the parameter p on the graphs of the 45% functions defined by: y = sin 9x+p), y = cos (x+p) and y = tan (x+p). • Draw sketch graphs defined by: y = a sin k(x+p), y = a cos k(x+p) and y = a tan k(x+p) at most two parameters at a time. • Derivation and use of the identities 16/5-18/5 sin and sin 2   cos 2  1 tan   47% (3 days) cos  21/5-25/5 • Determine for which values of a variable an identity holds. (5 days) • Derivation and use of reduction formulae for sin( 900   ) , • cos ( 900   ) , sin (1800   ) , cos (1800   ) , tan (1800   ) , 51% sin (3600   ) , cos (3600   ) , tan (3600  ) sin ( ) , cos ( ) , tan ( ) 28/5-31/5 Determine the general solution and / or specific solutions (given 01/06 intervals) of trigonometric equations. 55% (5 days) JUNE EXAMINATION WEIGHTING Trigonometry 55 ±3 marks out of 100 marks in Paper 2 CAPS EXAM GUIDELINE WEIGHTING FOR FINAL EXAMINATION Trigonometry 50 ± 3 marks out of 150 marks in P2 KZN FET Mathematics 2017 Page 17 of 28

BASIC DEFINITIONS: 1. If 𝑃(𝑥; 𝑦) and the 𝑟𝑎𝑑𝑖𝑢𝑠 = 𝑟, then: 𝑦 sin 𝜃 = 𝑟 𝑥 cos 𝜃 = 𝑟 𝑦 tan 𝜃 = 𝑥 𝑦 sin 𝜃 𝑦 A = 𝑥𝑟 = 𝑥 = tan 𝜃 cos 𝜃 𝑟 𝑦2 𝑥2 𝑦 2 +𝑥 2 𝑟2 B 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 𝑟 2 + 𝑟 2 = = 𝑟2 = 1 𝑟2 sin  One of the methods in proving identities • tan   cos  is to change all to sin and/or cos. sin2   1  cos2  • sin2   cos2   1 or (variations) cos   1  sin2  2 *A and B above are called identities, holds true for any angle. 1.2 Because the signs of the coordinates of P change in the different quadrants the sine, cosine and tan functions vary from positive to negative. The sketch shows the quadrant in which a function is positive. KZN FET Mathematics 2017 Page 18 of 28

2. REDUCTION FORMULAE Second Quadrant First Quadrant sin(180° − 𝜃) = sin 𝜃 sin(360° + 𝜃) = sin 𝜃 cos(180° − 𝜃) = − cos 𝜃 cos(360° + 𝜃) = cos 𝜃 tan(180° − 𝜃) = − tan 𝜃 tan(360° + 𝜃) = tan 𝜃 Third Quadrant Fourth Quadrant sin(180° + 𝜃) = − sin 𝜃 sin(360° − 𝜃) = −sin 𝜃 cos(180° + 𝜃) = − cos 𝜃 cos(360° − 𝜃) = cos 𝜃 tan(180° + 𝜃) = tan 𝜃 tan(360° − 𝜃) = −tan 𝜃 Function values of negative angles If OP rotates, from the positive x-axis, clockwise around O, then 𝜃 is a negative angle. Reduction formulae for negative angles (for any value of 𝜃): sin(−𝜃) = −sin 𝜃 cos(−𝜃) = cos 𝜃 tan(−𝜃) = − tan 𝜃 KZN FET Mathematics 2017 Page 19 of 28

Co-Functions: Function values of (𝟗𝟎° ± 𝜽) a) sin(90° − 𝜃) = cos 𝜃 cos(90° − 𝜃) = sin 𝜃 b) cos(90° + 𝜃) = −sin 𝜃 ; because the cosine function is negative in the Second Quadrant sin(90° + 𝜃) = cos 𝜃 FUNCTION VALUES OF SPECIAL ANGLES Use the following sketches to find the function values of 𝟎° , 𝟑𝟎° , 𝟒𝟓° , 𝟔𝟎° and 𝟗𝟎° whe n problems must be solved without a calculator. The function values of 180° and 360° are the same as that of 0° and 270° is the same as 90° , just add the appropriate sign. EXERCISES GENERAL TRIGONOMETRY 1. If 𝑝 sin 𝜃 = −3 and 𝑝 cos 𝜃 = 3, 𝑝 > 0, determine the value of the following: 1.1 𝜃 for 𝜃 ∈ [0°; 360°] 1.2 p (Leave the answer in surd form if necessary.) 2. In the diagram, A is the point (−√3; −1). Calculate, without a calculator, the value of: 2.1.1 tan 𝜃 2.1.2 cos 𝜃 2.1.3 𝜃 KZN FET Mathematics 2017 Page 20 of 28

2.2 If B is the reflection of A in the y-axis, calculate the size of 𝐴𝑂̂𝐵. 3. In the figure, 𝑃(−3; 3√3) is a point in a Cartesian plane with 𝑃𝑂̂𝑋 = 𝜃. Q is a point on the x-axis such that 𝑂𝑃̂ 𝑄 = 90°. 3.1 Determine 𝜃 𝑎𝑛𝑑 𝛼 3.2 Determine the length of OP 3.3 Determine the coordinates of Q. 4. In the figure, P is the point (4; 7). 𝑂𝑃̂𝑄 = 90° and 𝑋𝑂̂𝑃 = 𝑥. Calculate: 4.1 OP 4.2 𝑥 4.3 the coordinates of Q 1 1 In the figure, 𝑃(√3; 1) and 𝑄 (− 2 ; − 2) are points in a Cartesian plane. 𝑋𝑂̂𝑃 = 5. 𝛼 and reflex 𝑋𝑂̂ 𝑄 = 𝛽. Without using a calculator, find the size of (𝛽 − 𝛼). KZN FET Mathematics 2017 Page 21 of 28

6. In the diagram, P(-4; m) is a point in the Cartesian plane. It is also given that 15 𝑅𝑂̂𝑋 = 𝜃 and sin 𝜃 = 17. Now determine, without using a calculator, the value of m. 7. If tan 23° = 𝑡, use a sketch to find the value of cos 473° in terms of 𝑡. 8. If cos 35° = 𝑡, express each of the following in terms of t without using a calculator: 8.1 sin 325° 8.2 𝑠𝑖𝑛2 215° + cos 180° . cos 240° + cos 755° REDUCTION FORMULAE (INCLUDING SPECIAL ANGLES) 1. Simplify: sin(90° − 𝑥). sin(90° + 𝑥) (sin 90° − sin 𝑥). (sin 90° + sin 𝑥) 2. Simplify: tan(180° − 𝜃). cos(90° − 𝜃). cos(360° − 𝜃) sin(−𝜃). (𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃) 3. Simplify without using a calculator: tan 234° sin(𝑥 − 90°) cos 𝑥 . cos(90° + 𝑥) + . tan 54° sin(𝑥 − 360°) cos(−180°) 4. Simplify: cos(−57°). cos 33° sin 237° . cos 147° − tan 123° 5. sin(−120°).tan(−330°) Determine the value(s) of 𝑥 if 𝑡𝑎𝑛2 𝑥 = and cos 240° 𝑥 ∈ [−90°; 90°], without using a calculator. KZN FET Mathematics 2017 Page 22 of 28

6. If 𝜃, 2𝜃 𝑎𝑛𝑑 3𝜃 are the angles of a triangle, evaluate without a calculator: 𝑐𝑜𝑠 2 𝜃 + 𝑐𝑜𝑠 2 2𝜃 + 𝑐𝑜𝑠 2 3𝜃 7. If 𝐴̂ + 𝐵̂ = 180°, calculate sin 𝐴 − sin 𝐵. 8. Simplify the following, without using a calculator: sin 337° . tan 157° . sin 67° . cos(−23°) cos 383° . sin 157° . 𝑐𝑜𝑠 2 113° 9. Simplify the following, without using a calculator: cos(𝐴 − 360°). sin(−𝐴) tan 𝐴 . cos 180° 10. Simplify the following, without using a calculator: 2 tan(180° − 𝛼) − 2 tan(𝛼 − 180°) sin(𝛼 − 90°) . sin(360° − 𝛼) 11 Simplify the following: 11.1 2𝐶𝑜𝑠150°. 𝑡𝑎𝑛240°. 𝑐𝑜𝑠 2 315° 𝑠𝑖𝑛2 30 + 𝑐𝑜𝑠 2 330° 11.2 𝑠𝑖𝑛120° − 𝑐𝑜𝑠180° 𝑐𝑜𝑠240° + 𝑠𝑖𝑛2 315°. 𝑐𝑜𝑠150° 11.3 𝑠𝑖𝑛210° + 𝑐𝑜𝑠225°. 𝑠𝑖𝑛135° − tan 315 ° 11.4 cos(−45°) . 𝑠𝑖𝑛45° − 𝑠𝑖𝑛210° 𝑡𝑎𝑛135° 11.5 𝑠𝑖𝑛225°. 𝑡𝑎𝑛150° √𝑐𝑜𝑠300°. sin(−120°) TRIGONOMETRY EQUATIONS & GENERAL SOLUTIONS 1. If cos 𝜃 = 0,56 , use the identity 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 to calculate the value(s) of sin 𝜃 rounded off to two decimal digits. 2. Solve for 𝜃 given that 𝜃 ∈ [0°; 360°], without a calculator: 2.1 1 𝑠𝑖𝑛2 𝜃 = 2 KZN FET Mathematics 2017 Page 23 of 28

2.2 1 1 sin 𝜃 = 2 2 2.3 1 tan 𝜃 = − √3 3. Find, giving answers rounded off to one decimal digit where necessary, the general solution to the equations: 3.1 4 cos(2𝑥 + 20°) = 2,178 3.2 𝑥 3 cos − 1 = 0 2 4. Determine, without using a calculator, the general solution of the following equations. 4.1 4𝑠𝑖𝑛2 𝑥 − 3 = 0 4.2 tan(𝑥 + 30°) + √3 = 0 4.3 1 sin(2𝑥 + 15°) = √2 4.4 √3 sin(𝜃 − 20°) = 2 5. Give the general solution for the following equations: 5.1 cos 2𝑥 = cos(𝑥 − 24°) 5.2 cos 2𝑥 = sin(𝑥 − 24°) 5.3 sin(2𝑥 + 30°) = cos 𝑥 5.4 cos(2𝜃 − 70°) = sin(𝜃 + 40°) 5.5 𝜃 𝜃 3 sin = 2 cos 2 2 5.6 1 cos 𝑥 = sin(𝑥 − 30°) 2 6. Give the general solution for the following equations: 6.1 2 2 sin 𝑥 + = 5; sin 𝑥 ≠ 0 sin 𝑥 2 6.2 5𝑐𝑜𝑠 𝑦 − 7 cos 𝑦 = −2 6.3 5 3 tan 𝑥 − = 2; tan 𝑥 ≠ 0 tan 𝑥 6.4 3𝑐𝑜𝑠 2 𝑥 + 10 sin 𝑥 + 5 = 0 7. Solve the following equations, answers rounded to one decimal digit: 7.1 cos 𝑥 = 0,37 ; 𝑥 ∈ [180°; 360°] 7.2 2 tan 𝑥 = −2,84 ; 𝑥 ∈ [0°; 360°] 7.3 sin 2𝑥 = −0,75 ; 0° ≤ 𝑥 ≤ 180° 7.4 2 tan 𝜃 = 1,5 ; 𝜃 ∈ [0°; 360°] 7.5 tan 𝑥 = 2 cos 306,1° ; 0° ≤ 𝑥 ≤ 270° KZN FET Mathematics 2017 Page 24 of 28

7.6 tan(2𝜃 − 10°) = sin 72° ; 2𝜃 ∈ [0°; 360°] 7.7 5𝑠𝑖𝑛2 𝑥 − 3 = 0 ; 90° ≤ 𝑥 ≤ 270° 7.8 3 + 7 = 0 ; 𝑥 ∈ [0°; 270°] cos 2𝑥 TRIGONOMETRIC IDENTITIES 1. Prove the identity: 1 + 𝑡𝑎𝑛2 𝑥 1 2 = 1 − 𝑡𝑎𝑛 𝑥 1 − 2𝑠𝑖𝑛2 𝑥 2.1 Prove the identity: 𝑠𝑖𝑛4 𝑥 + 𝑠𝑖𝑛2 𝑥. 𝑐𝑜𝑠 2 𝑥 = 1 − cos 𝑥 1 + cos 𝑥 2.2 If 𝑥 ∈ [0°; 360°], for what value(s) of x is the above identity not defined? 2.3 Hence, write down the range of the function which is defined by: 𝑠𝑖𝑛4 𝑥 + 𝑠𝑖𝑛2 𝑥. 𝑐𝑜𝑠 2 𝑥 𝑦= 1 + cos 𝑥 3.1 Use the fundamental identities to prove that: cos 𝑥 𝑠𝑖𝑛2 𝑥 (tan 𝑥 + ) = tan 𝑥 sin 𝑥 cos 𝑥 3.2 Hence, solve for x if 𝑠𝑖𝑛2 𝑥 (tan 𝑥 + ) = 2 where 𝑥 ∈ [0°; 180°] sin 𝑥 4. Prove the identities: 4.1 sin 𝜃 − sin 𝜃 . cos 𝜃 = tan 𝜃 cos 𝜃 − (1 − 𝑠𝑖𝑛2 𝜃) 4.2 sin 𝑥 . cos 𝑥 1 = tan 𝑥 1 + 𝑐𝑜𝑠 2 𝑥 − 𝑠𝑖𝑛2 𝑥 2 4.3 (sin 𝑃 + cos 𝑃)2 + (sin 𝑃 − cos 𝑃)2 = 2 4.4 𝑐𝑜𝑠 2 𝑥(2 + 𝑡𝑎𝑛2 𝑥) = 2 − 𝑠𝑖𝑛2 𝑥 4.5 cos 𝑥(tan 𝑥 − sin 𝑥) 1 = 𝑠𝑖𝑛3 𝑥 1 + cos 𝑥 4.6 1 − cos 𝑥 = tan 𝑥 sin 𝑥 cos 𝑥 4.7 cos 𝑥 1 1 sin 𝑥 (1 + tan 𝑥) + cos 𝑥 (1 + )= + sin 𝑥 sin 𝑥 cos 𝑥 4.8 2 tan 𝐴 − 2 sin 𝐴 cos 𝐴 = tan 𝐴 2𝑠𝑖𝑛2 𝐴 4.9 cos 𝛼 1 tan 𝛼 + = 1 + sin 𝛼 cos 𝛼 4.10 𝑐𝑜𝑠 4 𝑥 − 𝑠𝑖𝑛4 𝑥 = cos 𝑥 (cos 𝑥 − sin 𝑥)(1 + tan 𝑥) KZN FET Mathematics 2017 Page 25 of 28

5. Prove the identity: 𝑠𝑖𝑛2 𝜃 = cos 𝜃 + 1 1 − cos 𝜃 6. Prove the identity: sin 𝜃 cos 𝜃 1 + = 1 + cos 𝜃 sin 𝜃 sin 𝜃 TRIGONOMETRIC FUNCTIONS 1 The diagram shows the sketch graphs of 𝑓(𝑥) = 𝑎 sin 𝑏𝑥 𝑎𝑛𝑑 . 𝑔(𝑥) = 𝑑 cos 𝑡𝑥. 1.1 Determine the values of a, b, c, d and t. 1.2 What is the period of g? 1.3 For which values of x is f an increasing function? 1.4 Give the range of f. 2 Consider the sketch graphs of the following functions and determine the Value(s) of p, q and c. 2.1.1 𝑓(𝑥) = 𝑝 sin 𝑞𝑥 2.1.2 𝑔(𝑥) = tan(𝑥 + 𝑐) 2 Write down the equations of the two asymptotes of g. . KZN FET Mathematics 2017 Page 26 of 28

3 Consider the sketch graph of the function 𝑔(𝑥) = 𝑎 cos 𝑥 + 𝑏 below. . 3.1 Determine the value of a and b. 3.2 Write down the range of g. 4 In the diagram below: . 𝑓(𝑥) = tan(𝑥 − 45°) and 𝑔(𝑥) = 𝑎 cos 𝑥. 4.1 Write down the value of a 4.2 The graph of g is translated 300 horizontally to the left to form the graph in h. Write down the new equation of h. 4.3 Use your graph to find one value of x so that 𝑔(𝑥) − 𝑓(𝑥) = 3 4.4 Use your graph to find one value of x so that 𝑓(𝑥) − 𝑔(𝑥) = 1 KZN FET Mathematics 2017 Page 27 of 28

5 Consider the functions f ( x)  cos3x and g ( x)  sin x for x  [90 ; 180] . 5.1 Solve for x if f (x) = g(x). 5.2 Sketch the graphs of f and g on the system of axes on your workbook for x  [90 ; 180] . 5.3 Solve for x if f (x)  g (x) where x  [90 ; 0] . 1 6 Consider the functions f(x) = sin 2x and g(x) = tan x for x  [– 90° ; 180°]. 2 6.1 Sketch the graphs of f and g on the same system of axes 6.2 Calculate the x-coordinates of the points of intersection of f and g. 6.3 Determine the values of x for which g(x) > f(x). 7 Given: g ( x)  2 cos(x  30) 7.1 Sketch the graph of g for x  [90 ; 270] on your workbook 7.2 Use the symbols A and B to plot the two points on the graph of g for which cos(x  30)  0,5 7.3 Calculate the x-coordinates of the points A and B. 7.4 Write down the values of x, where x  [90 ; 270] and g ( x)  0 . 7.5 Use the graph to solve for x, x  [90 ; 270] , where g ( x)  0 8 Given: f ( x)  1  sin x and g ( x)  cos 2 x 8.1 Calculate the points of intersection of the graphs f and g for x  [ 180 ; 360] . 8.2 Draw sketch graphs of f and g for x  [ 180 ; 360] in your workbook. 8.3 For which values of x will f ( x)  g ( x) for x  [ 180 ; 360] ? REFERENCES 1. Department of Basic Education NSC Examination Question Papers 2009 - 2016 2. KZN Trial P1&P2 2015, Limpopo Trial P1& P2 2014, 2013, Mpumalanga Trial 2013, Western Cape Trial 2014 P1 & P2 3. DBE Second Chance Revision Booklet 2017 KZN FET Mathematics 2017 Page 28 of 28