Graph of Trigonometric and Inverse Trigonometric Functions and Solutions of Trigonometric Equations

Maths, HSSC-KPK First Year. This chapter is split into 85 topics and has 2,051 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Verifying identities of inverse trigonometric functions 24 questions
2. Even-odd identities of the six trigonometric functions 25 questions
3. Restricting the domain: principal trigonometric functions 22 questions
4. Maximum and minimum values of a + b sin theta 26 questions
5. Trigonometric equations reduced to one function using identities 24 questions
6. Graphs of sin A theta and cos A theta: number of periods 26 questions
7. Table of values for y = sin x and y = cos x 25 questions
8. Zeros and extreme points of sin kx and cos kx 25 questions
9. Addition and subtraction formulae for inverse sine 25 questions
10. Domains and ranges of principal and inverse trigonometric functions 28 questions
11. Equations solved by squaring and extraneous roots 22 questions
12. Periodic, even and translation properties of the cosine graph 24 questions
13. Periodic functions: definition and values 25 questions
14. Trigonometric equations: meaning, identities and the simplest forms 21 questions
15. Vertical shift, reflection and scaling of trigonometric graphs 24 questions
16. The inverse sine function: definition and graph 24 questions
17. Guessing the graph without drawing: period, frequency and amplitude 24 questions
18. The reduction identity a cos x + b sin x = c 22 questions
19. Graph of y = tan x 24 questions
20. Amplitude, period, frequency and wavelength 25 questions
21. Range of composite trigonometric functions 25 questions
22. Domain and range of secant and cosecant functions 25 questions
23. Graph of y = cot x 24 questions
24. Inverse cotangent, secant and cosecant: definitions and values 24 questions
25. Domain, range and period of a trigonometric function 25 questions
26. Graphical solution of trigonometric equations 24 questions
27. Graph of y = sin x from -2pi to 2pi 24 questions
28. Periods of the six basic functions and angle reduction 25 questions
29. Parity of sums, products and shifted trigonometric functions 26 questions
30. Composite values of inverse functions with rational arguments 25 questions
31. Expressing trigonometric functions of inverse functions algebraically 25 questions
32. Periodic, odd and translation properties of the tangent graph 24 questions
33. Equations of the form fn squared x = c and sec, cosec equations 22 questions
34. The inverse cosine function: definition and graph 24 questions
35. Graph of a function and the first shapes of sine, cosine and tangent 24 questions
36. Solving cos theta = k in an interval 25 questions
37. Exact values of inverse sine (Examples 15 to 17) 24 questions
38. Where the extreme values are attained 22 questions
39. Periods of sin, cos and tan: the theorems 20 questions
40. Trigonometric functions of a real number 25 questions
41. Maximum and minimum values of a + b cos theta 26 questions
42. Classifying trigonometric functions as even, odd or neither 26 questions
43. Maximum and minimum values of reciprocal functions 25 questions
44. Even and odd functions: definitions and symmetry 25 questions
45. Addition and subtraction formulae for inverse tangent 24 questions
46. Solving tan theta = k in an interval 25 questions
47. Graph of y = cosec x 24 questions
48. Identities for inverse cosecant, secant and cotangent 24 questions
49. Exact values of inverse cosine (Examples 18 and 19) 24 questions
50. Addition and subtraction formulae for inverse cosine 24 questions
51. General solution of tan theta = k 20 questions
52. Establishing identities with symmetric and translation properties 24 questions
53. Period of sec kx and cosec kx 25 questions
54. Maximum and minimum values of a + b cos(ctheta + d) 26 questions
55. Reading features of y = a sin bx and y = a cos bx without plotting 24 questions
56. Inverse of a function: one-one and onto 24 questions
57. Period of a f(kx + d) + b and mixed functions 25 questions
58. Solutions of general trigonometric equations: the unknown inside the argument 24 questions
59. Solving trigonometric equations by factorization 24 questions
60. Domain of composite trigonometric functions 25 questions
61. Deducing sin(theta + 2k pi) = sin theta and similar identities 24 questions
62. Graphs of inverse trigonometric functions: reflection and key points 21 questions
63. Period of tan kx and cot kx 25 questions
64. Trigonometric functions on the unit circle 26 questions
65. Domain and range of tangent and cotangent functions 23 questions
66. Maximum and minimum values of a + b sin(ctheta + d) 26 questions
67. Periodic, odd and translation properties of the sine graph 24 questions
68. Domain and range of sine and cosine functions 25 questions
69. General solution of sin theta = k 24 questions
70. Double and multiple angle relations of inverse functions 24 questions
71. Solving sin theta = k in an interval 25 questions
72. Inverse of a trigonometric function and the principal range 24 questions
73. Graph of y = sec x 24 questions
74. General solution of cos theta = k 23 questions
75. Solving equations involving inverse trigonometric functions 24 questions
76. The inverse tangent function: definition and values 24 questions
77. Graph of y = cos x from -2pi to 2pi 24 questions
78. Tables of values for tan, cot, sec and cosec 24 questions
79. Trigonometric equations that are quadratic in sin x, cos x or tan x 24 questions
80. Review Exercise 12 style questions 21 questions
81. Exact values of composites such as sin of inverse cosine 24 questions
82. Number of solutions of trigonometric equations in an interval 24 questions
83. Period of sin kx and cos kx 25 questions
84. Complementary relation of inverse sine and cosine 22 questions
85. Inverse functions of negative arguments 22 questions

Sample questions: Graph of Trigonometric and Inverse Trigonometric Functions and Solutions of Trigonometric Equations

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. How many solutions does sin θ = (4/5)θ have for −π/2 ≤ θ ≤ π/2?

• A. 2
• B. 4
• C. 3(correct answer)
• D. 1

Answer: C.The line y = 4/5θ and the curve y = sin θ meet at the origin; since the slope is greater than 2/π (the chord to (π/2, 1)), the line also meets the curve once on each side, so 3 intersections.

Q2. How many solutions does sin θ = (1/4)θ have for −π/2 ≤ θ ≤ π/2?

• A. 2
• B. 4
• C. 3
• D. 1(correct answer)

Answer: D.The line y = 1/4θ and the curve y = sin θ meet at the origin; since the slope is less than 2/π, the line stays below the curve for 0 < θ ≤ π/2 (and symmetrically), so only the origin.

Q3. How many solutions does sin θ = (1/5)θ have for −π/2 ≤ θ ≤ π/2?

• A. 1(correct answer)
• B. 2
• C. 3
• D. 4

Answer: A.The line y = 1/5θ and the curve y = sin θ meet at the origin; since the slope is less than 2/π, the line stays below the curve for 0 < θ ≤ π/2 (and symmetrically), so only the origin.

Q4. How many solutions does sin θ = (5/6)θ have for −π/2 ≤ θ ≤ π/2?

• A. 3(correct answer)
• B. 1
• C. 4
• D. 2

Answer: A.The line y = 5/6θ and the curve y = sin θ meet at the origin; since the slope is greater than 2/π (the chord to (π/2, 1)), the line also meets the curve once on each side, so 3 intersections.

Q5. How many solutions does sin θ = (2/3)θ have for −π/2 ≤ θ ≤ π/2?

• A. 2
• B. 4
• C. 1
• D. 3(correct answer)

Answer: D.The line y = 2/3θ and the curve y = sin θ meet at the origin; since the slope is greater than 2/π (the chord to (π/2, 1)), the line also meets the curve once on each side, so 3 intersections.

Q6. Example 21(iv): cos[Tan⁻¹(−1)] (with cos the general cosine) equals

• A. 1/2
• B. √2/2(correct answer)
• C. −√2/2
• D. undefined

Answer: B.Tan⁻¹(−1) = −π/4 and cos(−π/4) = √2/2. (Part (ii) of the same example is inconsistent in the book: it calls Cos(−π/4) undefined because the principal Cos is restricted to [0, π], while part (iv) gives √2/2.)

Q7. Review Exercise 12 Q4(iii) (with the last term added, since the printed minus sign gives a false statement): Sin⁻¹(4/5) + Sin⁻¹(5/13) + Sin⁻¹(16/65) equals

• A. π
• B. π/4
• C. π/3
• D. π/2(correct answer)

Answer: D.Sin⁻¹(4/5) + Sin⁻¹(5/13) = Sin⁻¹(63/65) because (4/5)(12/13) + (5/13)(3/5) = 63/65; and Sin⁻¹(63/65) + Sin⁻¹(16/65) = π/2 since 63² + 16² = 65² (numerically 0.9273 + 0.3948 + 0.2487 = 1.5708). With a minus sign the value would be 1.0734, not π/2.

Q8. Example 40: the equation √3 cos θ − sin θ = 0 has the solution set

• A. {2π/3 + mπ}, m ∈ Z
• B. {π/3 + mπ}, m ∈ Z(correct answer)
• C. {−π/3 + 2mπ}, m ∈ Z
• D. {π/6 + mπ}, m ∈ Z

Answer: B.√3 cos θ = sin θ gives tan θ = √3, so θ = π/3 + mπ. (The book's final line writes {2π/3 + mπ}, but its previous line θ = −2π/3 + mπ is the same set as π/3 + mπ; 2π/3 + mπ would be the solution of tan θ = −√3.)

Q9. Review Exercise 12 Q5(ii) as printed, Tan⁻¹A − Tan⁻¹B = Tan⁻¹(A − B), is incorrect. The correct formula is

• A. Tan⁻¹(A + B)
• B. Tan⁻¹((A + B)/(1 + AB))
• C. Tan⁻¹((A − B)/(1 − AB))
• D. Tan⁻¹((A − B)/(1 + AB))(correct answer)

Answer: D.From tan(x − y) = (tan x − tan y)/(1 + tan x tan y). For A = 1, B = 1/2 the printed formula would give Tan⁻¹(1/2) ≈ 0.4636 whereas the true difference is Tan⁻¹1 − Tan⁻¹(1/2) ≈ 0.3218.

Q10. In Exercise 12.5 Q8(vii) the book states Tan⁻¹(x − 1) + Tan⁻¹(x + 1) = Tan⁻¹3x. Using the addition formula the left side is actually

• A. Tan⁻¹(2x)
• B. Tan⁻¹(x² − 1)
• C. Tan⁻¹(2x/(2 − x²)) (for suitable x)(correct answer)
• D. Tan⁻¹(3x)

Answer: C.(A + B)/(1 − AB) with A = x − 1, B = x + 1 gives 2x/(1 − (x² − 1)) = 2x/(2 − x²), not 3x (the printed identity is wrong, e.g. at x = 1: Tan⁻¹0 + Tan⁻¹2 = 1.107 while Tan⁻¹3 = 1.249).