Trigonometric Functions and their Graphs

Maths, HSSC-Punjab First Year. This chapter is split into 28 topics and has 719 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Comparing the properties of the sine, cosine and tangent graphs 23 questions
2. Periods of secant and cosecant functions 28 questions
3. Amplitude, vertical shift, period and phase shift 26 questions
4. Graph of y = tan x - features and table values 22 questions
5. Maximum and minimum values of a + b cos(cx + d) 27 questions
6. Periods of sin kx and cos kx 28 questions
7. Maximum and minimum values of a + b sin(cx + d) 27 questions
8. Periods of tan kx and cot kx 28 questions
9. Applications - angles of elevation and depression 27 questions
10. Graph of y = tan kx - period, asymptotes and intercepts 26 questions
11. Domain and range of tangent and cotangent 21 questions
12. Applications - tides, temperature, swings and populations 28 questions
13. Periodicity - definition and the periods of the six functions 24 questions
14. Solving equations by the graphs of trigonometric functions 22 questions
15. Classifying trigonometric expressions as even, odd or neither 28 questions
16. Applications - Ferris wheel and rider height 27 questions
17. Even and odd functions - definitions and examples 22 questions
18. Graph of y = cos x - features and table values 23 questions
19. Graph of y = sin x - features and table values 24 questions
20. Domain and range of secant and cosecant 22 questions
21. Values of sine, cosine and tangent used to plot graphs 28 questions
22. Domain and range of sine and cosine 20 questions
23. Periods of general forms a + b f(cx + d) 27 questions
24. Counting zeros, maxima and asymptotes on an interval 28 questions
25. Excluded values in the domain of trigonometric expressions 30 questions
26. Range of simple trigonometric expressions 29 questions
27. Maximum and minimum values of reciprocal forms 26 questions
28. Graphs of y = a sin bx and y = a cos bx - amplitude, period and key points 28 questions

Sample questions: Trigonometric Functions and their Graphs

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

Q1. Is f(x) = sin x + cos x even, odd or neither? It is

• A. an even function
• B. an odd function
• C. both even and odd
• D. neither even nor odd(correct answer)

Answer: D.Replacing x by −x (cos, sec even; sin, tan, cot, cosec odd): f(−x) is neither f(x) nor −f(x), so the function is neither even nor odd.

Q2. The function f(x) = tan x + sec x is

• A. neither even nor odd(correct answer)
• B. an odd function
• C. an even function
• D. both even and odd

Answer: A.Replacing x by −x (cos, sec even; sin, tan, cot, cosec odd): f(−x) is neither f(x) nor −f(x), so the function is neither even nor odd.

Q3. The function f(x) = x + cos x is

• A. an even function
• B. both even and odd
• C. an odd function
• D. neither even nor odd(correct answer)

Answer: D.Replacing x by −x (cos, sec even; sin, tan, cot, cosec odd): f(−x) is neither f(x) nor −f(x), so the function is neither even nor odd.

Q4. The period of the function y = 19 sin(πx/20) is

• A. 40(correct answer)
• B. 80
• C. 20
• D. 1/10

Answer: A.The coefficient of x inside the sin is π/20, so the period is 2π/(π/20) = 40; the constants and the phase do not change the period (Exercise 11.1, Q2).

Q5. The period of the function y = cosec(2x/5) is

• A. 10π
• B. 4π/5
• C. 5π(correct answer)
• D. 5π/2

Answer: C.The coefficient of x inside the cosec is 2/5; period = 2π/k = 5π; the constants and the phase do not change the period (Exercise 11.1, Q2).

Q6. The period of the function y = (1/2) sin(3x/2 − π/2) is

• A. 2π/3
• B. 4π/3(correct answer)
• C. 8π/3
• D. 3π

Answer: B.The coefficient of x inside the sin is 3/2; period = 2π/k = 4π/3; the constants and the phase do not change the period (Exercise 11.1, Q2).

Q7. The period of the function y = −5 − 3 sec(7πx + π/4) is

• A. 14
• B. 1/7
• C. 4/7
• D. 2/7(correct answer)

Answer: D.The coefficient of x inside the sec is 7π, so the period is 2π/(7π) = 2/7; the constants and the phase do not change the period (Exercise 11.1, Q2).

Q8. The period of the function y = 12 + 10 tan(πx/30) is

• A. 1/30
• B. 30(correct answer)
• C. 60
• D. 15

Answer: B.The coefficient of x inside the tan is π/30, so the period is 1π/(π/30) = 30; the constants and the phase do not change the period (Exercise 11.1, Q2).

Q9. The period of the function y = 6 − 4 cot(7x/4 + π/4) is

• A. 7π/4
• B. 4π/7(correct answer)
• C. 8π/7
• D. 2π/7

Answer: B.The coefficient of x inside the cot is 7/4; period = 1π/k = 4π/7; the constants and the phase do not change the period (Exercise 11.1, Q2).

Q10. The period of the function y = 9 + 30 sec(x/15 + 2π/15) is

• A. 60π
• B. 30π(correct answer)
• C. 2π/15
• D. 15π

Answer: B.The coefficient of x inside the sec is 1/15; period = 2π/k = 30π; the constants and the phase do not change the period (Exercise 11.1, Q2).