Summer&Assignments& Ms.&Yoder& CCGPSPre5Cal … · Teachers’ Resource NSAM SHINGLEE PUBLISHERS PTE LTD 23. Prove the following identities: (a) 1 − cot 2 x 1 + cot2 x = 2 sin 2

Teachers’ Resource NSAM SHINGLEE PUBLISHERS PTE LTD

Summer Assignments

Ms. Yoder

Students (rising sophomores) planning to take CCGPS Pre-‐Calculus in Fall 2013 and rising juniors intending to take IB Mathematics SL have online assignments at Study Island. These are assignments to practice with prerequisite skills of things that you have learned already in previous courses. If you discover that you have areas of weakness that might need to be addressed, there are many resources available there. The courses are called CCGPS Pre-‐Calculus Summer Assignment and IB Mathematics SL Rising Juniors Summer Work and you have been pre-‐enrolled. All work is due by August 16 at the end of the day but don’t wait until then. You will regret that! It will count for a grade at the beginning of the semester. Home Page http://www.studyisland.com/web/index/ Login

User name FirstnameLastname@CHS Password Lunch Number Please contact Ms. Yoder at [email protected] if you have questions or concerns about the work. I will not be checking every my work e-‐mail every day but will at least once a week. I will also be working on a webpage over the summer but for now there are lots of useful links at www.the-‐y-‐axis.weebly.com to use as resources. Have a great summer!


Trigonometrical Ratios and Equations

1. Sketch on the same diagram, the graphs of y = |cos x| and y = 1 + cos 2x for 0 ≤ x ≤ 2π . Hence state the number of solutions in the range 0 ≤ x ≤ 2 π for (a) cos x = 1 + cos 2x (b) |cos x| = 1 + cos 2x [6] 2. Find all the angles between 0º and 360º which satisfy the equation (a) 4 sin θ + cosec θ = 4 , (b) 2 sin2 y + 3 cos y = 0. [7] 3. Prove the identity cot2 θ − cos2 θ = cot2 θ cos2 θ [3] 4. Find the angles between 0º and 360º inclusive which satisfy the equation (a) 2 tan x = sin x (b) 3 tan (x − 40º) = 1 [7] 5. Find all the angles between 0º and 360º inclusive which satisfy the equation (a) cos x + cos 30º = 0 (b) tan 2 θ = −1 (c) sin2 y + 2 cos2 y = 74

[9] 6. On 2 separate diagrams, sketch, for the values of 0 ≤ x ≤ 2π, the graph of (a) y = 1 − |2 sin x| (b) y = 4 sin 2x − 3 [4] 7. Prove that 1 − sin x cos x + 2 cos2x

sin2x = 3 cot2 x − cot x + 1. Solve the equation 1 − sin x cos x + 2 cos2 x = 3 sin2 x giving all the values of x between 0º and 360º. [7] 8. Find all possible values of x from 0º to 360º when (a) cos2 x = 0.25 (b) 7 sin x = 4 cos x (c) cosec x = 3 (d) 3 cos2 x + 2 sin x = 2 . [12] 9. Solve for −180º ≤ x ≤ 180º, for the following equations (a) 2 cot x − 3 = 0 (b) 3 cot x − 2 sin x = 0 [6] 10. Sketch the graph of y = sin 3x for 0º ≤ x ≤ 180º. Calculate all the values of x in the interval 0º ≤ x ≤ 180º for which (a) sin 3x = 12 (b) sin 3x = − 3

2

Hence write down the complete solution set of − 3 2 ≤ sin 3x ≤ 12 for 0º ≤ x ≤ 180º.

[7]


11. Solve the following equations, giving all the solutions between 0º and 360º: (a) tan x = −0.6 (b) sec2 θ = 8.549 (c) 6 cos2 y – 7 cos y + 2 = 0 [8] 12. Prove the identity

sec x – cos x = sin x tan x [3] 13. Sketch the graph of the function f(x) = |cos x| + 2 for the domain 0 ≤ x ≤ 2π State the range of the function for the given domain. [3] 14. Find all the angles between 0º and 360º inclusive which satisfy the equation (a) 2 sin (2x – 10º) = − 3 (b) 5 cosec2 y – 1 = 9 cot y [8]

15. Given that x

x2cos41

2sin+

= 149

5 where π2 ≤ x ≤ π, find the value of x

xcos41

sin+

.

[4] 16. Prove the identity (1 – sin2 θ) (1 – tan2 θ) = 2 cos2 θ − 1. [3] 17. (a) Given that 3 cot A = 4 and A is acute, find the value of

sin A - 2 cos A2 sin A + cos A

(b) If -π ≤ θ ≤ π, find the values of θ such that 2 cos2 θ − cos θ = 0 [3]

18. Prove the identity θ=θ+

+θ−

2sec2sin11

sin11 [3]

19. Prove the identity cot2 y – cos2 y = cot2 y cos2 y. [3] 20. Sketch, on the same diagram, the graph of y = |3 cos 2x| and y = 1 + 2

π x

for the interval -π2 ≤ x ≤ π2

State the number of solutions, in this interval, of the equation |3π cos 2x| = π + 2x . [4] 21. Prove that sin x

1 − tan x + cos x1 − cot x = 0.

Hence solve for 0º < x < 360º, the equation sin x

1 − tan x + cos x

1 − cot x = cos x (8 cos x + 3) [6]

22. Find all angles between 0º and 360º which satisfy the equation (a) sec 12 t = 2

3 [3] (b) tan (3x + 80º) = −1 [3]

(c) 2 cos y = cot y [3] (d) 2 sin2 z + cos z + 1 = 0 [4]


23. Prove the following identities: (a) 1 − cot2 x

1 + cot2 x = 2 sin2 x − 1 (b) 1 −cos A + sin A

1 − cos A = 1 + sin A + cos Asin A [7]

24. Find the values of the angle n between 0 and π radians such that cos 2x = cos 74 π − cos 32 π [4] 25. Solve the following equations for 0 ≤ x ≤ 360º. (a) tan x (1 + 2 cot x) = 3 cot x (b) cot x = 2 cos x (c) sin (x + 30º) = cos 120º [12]

26. Prove the identity 2 − 4 sin2 xsin x + cos x ≡ 2(cos x − sin x). [4]

27. Given that 2 ≤ a sin2 θ + 3b ≤ 6 for all values of θ where a > 0, find the values of a and b. [3] 28. Sketch the graph of y = 1 − cos x for 0º ≤ x ≤ 360º. Explain how you can obtain the graphical solution of cosec 2x − cos x cosec 2x = 2 using the graph you have sketched. State the number of solutions there are for the equation cosec 2x − cos x cosec 2x = 2 in this interval. [5] 29. Solve the following equations for 0º ≤ x ≤ 360º. (a) 3 tan(x - 150) = 5 [3]

(b) 5 tan x2 = 14 sin

x2 [5]

(c) 3 tan2(x + 15º) = 5 + 2

cos(x + 150) [4]

30. Prove the identity (1 – sin x)(1 + cosec x) ≡ cos x cot x. [3] 31. Prove the following identities

(a) (1 + cos x1 + sin x )(

1 + cosec x1 + sec x ) ≡ cot x. [4]

(b) tan 2x − tan xtan x ≡

1 2cos2 x − 1 . [4]

(c) (tan x + sec x)2 ≡ 1 + sin x1 − sin x . [4]


(d) cosec2 x − sec2 xcosec2 x + sec2 x ≡ 1 – 2 sin2 x. [4]

(e) cosec x – cot x ≡ sin x

1 + cos x . [4]

(f) cot x1 + cosec x −

cot x1 − cosec x ≡ 2 sec x. [4]

32. Find the maximum and minimum values of the following functions (a) y = 2 − 12 sin x (b) y = 14 cos(x + 45 º ) (c) y = 2 cos x −2 [6] 33. Solve the following equations for 0º ≤ x ≤ 360º (a) 5 sin2 x = 2 sin x cos x [4] (b) 5 sin(2x − 10º) = 2 [3] (c) 5 sin2 x(1 − cot x) = 2 [5]

34. If sin θ = −725 and cos θ < 0, find the values of sec θ and cot θ. [4]

35. If cos θ = −35 and sin θ > 0, find the values of tan θ − sin θ. [4]

36. If 2 tan2 θ + 3 sec2 θ = 18 and 90º < θ < 180º, find the value of cos θ. [4] 37. Draw, on the same axes for 0º ≤ x ≤ 360º, graphs of y = 3 cos 2x and of y = 2 + sin x. State the number of solutions of the equation 3 cos 2x = 2 + sin x in the range 0º ≤ x ≤ 360º. [4] 38. Sketch the following curves for 0° ≤ x ≤ 360°.

(a) xy 3sin= (b) 2

cos2 xy = (c) xy 4cos3= [9]

39. Sketch the graphs of y = x2sin and y = πx

+1 for the domain 0 ≤ x ≤ π. Hence state

the number of solutions in the domain of the equation xx 2sinππ =+ . [5] 40. Sketch on the same diagram the graphs of y = 1 + |sin x| and y = 3 cos x for values of

x between 0° and 360°. Hence state the number of solutions for the equation 1 + |sin x| - 3 cos x = 0. [5] 41. It is given that the graph of the function f:x → 2 sinnx –1, π20 ≤≤ x where n is a

positive integer, intersects the x-axis at 4 points. (a) State the value of n. [1]


(b) Hence, sketch the graph of f and state its range. [3] (c) Deduce the range of values of k, where k is positive, such that there are exactly 8

values of x between 0 and 2π which satisfy the equation knx =1-sin 2 . [2]


Answers 1. (a) 4 (b) 6 2. (a) 30º, 150º (b) 120º, 240º 4. (a) 0º, 180º, 360º (b) 70º, 250º 5. (a) 150º, 210º (b) 67.5º, 157.5º, 247.5º, 337.5º (c) 30º, 150º, 210º, 330º 7. 45º, 123.7º, 225º, 303.7º 8. (a) 60º, 120º, 240º, 300º (b) 29.7º, 209.7º (c) 9.7º, 80.3º, 189.7º, 260.3º (d) 90º, 199.5º, 340.5º 9. (a) -146.3º, 33.7º (b) -60º, 60º 10. (a) 15º, 50º, 130º, 170º (b) 80º, 100º; 0º ≤ x ≤ 15º, 50º ≤ x ≤ 80º, 100º ≤ x ≤ 130º or 170º ≤ x ≤ 180º 11. (a) 149.0º, 329.0º (b) 70º, 110º, 250º, 290º (c) 48.2º, 60º, 300º, 311.8º 13. 2 ≤ f(x) ≤ 3 14. (a) 120º, 150º, 300º, 330º (b) 45º, 51.3º, 225º, 231.3º 15. − 17

17. (a) −5

2 (b) −π2 , − π3 ,

π3 ,

π2

20. 4 21. 90º, 112.0º, 248.0º, 270º 22. (a) 120º (b) 18.3º, 78.3º, 138.3º, 198.3º, 258.3º, 318.3º (c) 30º, 90º, 150º, 270º (d) 180º 24. π8 ,

78 π ,

98 π ,

158 π

25. (a) 45º, 225º, 108.4º, 288.4º (b) 30º, 90º, 150º, 270º (c) 180º, 310º 27. a = 4, b = 23

28. Draw y = 2 sin 2x; 5


29. (a) 85.2º, 265.2º (b) 0º, 138.2º, 360º (c) 45º, 123.6º, 206.4º, 285º

32. (a) 52 , 32 (b) 14 , − 14 (c) 0, −4

33. (a) 0º, 41.8º, 180º, 221.8º, 360º (b) 13.2º, 86.8º, 193.2º, 266.8º

(c) 63.4º, 243.4º, 161.6º, 341.6º

34. sec θ = − 2524 , cot θ = 24

7

35. − 3215

36. – 12

37. 4

39. 2 40. 2 41. (a) 2 (b) 1)(f3 ≤≤− x (c) 10 << k

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Summer&Assignments& Ms.&Yoder& CCGPSPre5Cal … · Teachers’ Resource NSAM SHINGLEE PUBLISHERS PTE LTD 23. Prove the following identities: (a) 1 − cot 2 x 1 + cot2 x = 2 sin 2