XYZ SECONDARY SCHOOL
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.
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Please contact Ms. Yoder at Sandra.Yoder@ 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 the-y-axis. to use as resources.
Have a great summer!
Trigonometrical Ratios and Equations
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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(cos x ( sin x). [4]
27. Given that 2 ( a sin2 ( + 3b ( 6 for all values of ( 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 = 5 [3]
(b) 5 tan = 14 sin [5]
(c) 3 tan2(x + 15º) = 5 + [4]
30. Prove the identity
(1 – sin x)(1 + cosec x) ( cos x cot x. [3]
31. Prove the following identities
(a) ()() ( cot x. [4]
(b) ( . [4]
(c) (tan x + sec x)2 ( . [4]
(d) ( 1 – 2 sin2 x. [4]
(e) cosec x – cot x ( . [4]
(f) ( ( 2 sec x. [4]
32. Find the maximum and minimum values of the following functions
(a) y = 2 ( sin x (b) y = cos(x + 45 º ) (c) y = 2 cos x (2 [6]
33. Solve the following equations for 0º ( x ( 360º
(a) sin2 x = 2 sin x cos x [4]
(b) 5 sin(2x ( 10º) = [3]
(c) 5 sin2 x(1 ( cot x) = 2 [5]
34. If sin ( = ( and cos ( < 0, find the values of sec ( and cot (. [4]
35. If cos ( = ( and sin ( > 0, find the values of tan ( ( sin (. [4]
36. If 2 tan2 ( + 3 sec2 ( = 18 and 90º < ( < 180º, find the value of cos (. [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) [pic] (b) [pic] (c) [pic] [9]
39. Sketch the graphs of y =[pic] and y = [pic]for the domain 0 ( x ( (. Hence state the number of solutions in the domain of the equation [pic]. [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, [pic] 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 2( which satisfy the equation [pic]. [2]
Answers
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25. (a) 45º, 225º, 108.4º, 288.4º (b) 30º, 90º, 150º, 270º (c) 180º, 310º
27. a = 4, b =
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) , (b) , ( (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 ( = ( , cot ( =
35. (
36. –
37. 4
39. 2
40. 2
41. (a) 2 (b) [pic] (c) [pic]
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