Solution of Trigonometric Equations

Maths, HSSC-Punjab Second Year. This chapter is split into 29 topics and has 735 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Principal solutions of sin x = k 28 questions
2. General solution of tan, cot, sec and csc equations 27 questions
3. Equations quadratic in a trigonometric function 23 questions
4. Equations reduced by sum and double-angle identities 22 questions
5. Equations solved by taking out a common factor 27 questions
6. Real life: range of a projectile 25 questions
7. Real life: work done by a force at an angle 21 questions
8. General solution of cos x = k 26 questions
9. Counting solutions in a given interval 28 questions
10. Reference angles and quadrants 27 questions
11. Principal solutions of cos x = k 28 questions
12. General solution of sin x = k 26 questions
13. Equations reduced by fraction, cosine and tangent identities 24 questions
14. Graphical solutions of mixed trigonometric equations 29 questions
15. Equations with a multiple angle 25 questions
16. Periods and signs of trigonometric functions 28 questions
17. Real life: tower with an antenna 22 questions
18. Trigonometric equations: terms and definitions 21 questions
19. Graphical solution of trigonometric equations (method) 22 questions
20. Equations in squares of a trigonometric function 28 questions
21. Real life: heights and angles of elevation 22 questions
22. Principal solutions of sec, csc and cot equations 26 questions
23. Squaring and extraneous roots 30 questions
24. Principal solutions of tan x = k 26 questions
25. Graphical solutions: counting and estimating roots (cosine, tangent, mixed) 23 questions
26. Graphical solutions: counting and estimating roots (sine and cosine) 28 questions
27. Real life: equal angles for a lantern on a cliff 21 questions
28. Solving equations with sum-to-product formulas 26 questions
29. Equal-angle equations: sin A = sin B, cos A = cos B, tan A = tan B 26 questions

Sample questions: Solution of Trigonometric Equations

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

Q1. The solutions of sin 3x = −1/2 in [0, 2π) are

• A. 7π/18 and 11π/18
• B. 7π/18, 11π/18, 19π/18, 23π/18, 31π/18 and 35π/18(correct answer)
• C. 7π/6 and 11π/6
• D. 3π/2

Answer: B.Write sin u = −1/2 with u = 3x: u = 7π/6 and 11π/6 + multiples of the period; dividing by 3 and keeping x in [0, 2π) gives 7π/18, 11π/18, 19π/18, 23π/18, 31π/18 and 35π/18.

Q2. The solutions of sin 4x = √3/2 in [0, 2π) are

• A. π/12, π/6, 7π/12, 2π/3, 13π/12, 7π/6, 19π/12 and 5π/3(correct answer)
• B. π/3 and 2π/3
• C. 2π/3 and 4π/3
• D. π/12 and π/6

Answer: A.Write sin u = √3/2 with u = 4x: u = π/3 and 2π/3 + multiples of the period; dividing by 4 and keeping x in [0, 2π) gives π/12, π/6, 7π/12, 2π/3, 13π/12, 7π/6, 19π/12 and 5π/3.

Q3. The solutions of sin 3x = 1/√2 in [0, 2π) are

• A. π/4 and 3π/4
• B. π/12, π/4, 3π/4, 11π/12, 17π/12 and 19π/12(correct answer)
• C. π/12, π/4 and 3π/4
• D. π/12 and π/4

Answer: B.Write sin u = 1/√2 with u = 3x: u = π/4 and 3π/4 + multiples of the period; dividing by 3 and keeping x in [0, 2π) gives π/12, π/4, 3π/4, 11π/12, 17π/12 and 19π/12.

Q4. The solutions of sin 4x = −1/2 in [0, 2π) are

• A. 7π/6 and 11π/6
• B. 2π/3 and 4π/3
• C. 7π/24, 11π/24, 19π/24, 23π/24, 31π/24, 35π/24, 43π/24 and 47π/24(correct answer)
• D. 7π/24 and 11π/24

Answer: C.Write sin u = −1/2 with u = 4x: u = 7π/6 and 11π/6 + multiples of the period; dividing by 4 and keeping x in [0, 2π) gives 7π/24, 11π/24, 19π/24, 23π/24, 31π/24, 35π/24, 43π/24 and 47π/24.

Q5. How many solutions does cos 4x = 1/√2 have in [0, 2π)?

• A. 8(correct answer)
• B. 2
• C. 10
• D. 4

Answer: A.u = 4x runs through [0, 8π), which covers 4 periods of the function (or 8 of tan); the solutions of x are π/16, 7π/16, 9π/16, 15π/16, 17π/16, 23π/16, 25π/16 and 31π/16.

Q6. Squaring √3 cos x − sin x = 1 leads to cos x (2 cos x − √3) = 0. The solutions of the original equation in [0, 2π) are

• A. π/6 and 3π/2(correct answer)
• B. π/6, π/2 and 3π/2
• C. π/6, π/2, 3π/2 and 11π/6
• D. π/2 and 11π/6

Answer: A.Squaring gives the candidates π/6, π/2, 3π/2 and 11π/6; substituting each in √3 cos x − sin x = 1 shows that π/2 and 11π/6 are extraneous, so only π/6 and 3π/2 remain.

Q7. Squaring √3 cos x + sin x = 1 leads to cos x (2 cos x − √3) = 0. The solutions of the original equation in [0, 2π) are

• A. π/6, π/2 and 11π/6
• B. π/6, π/2, 3π/2 and 11π/6
• C. π/2 and 11π/6(correct answer)
• D. π/6 and 3π/2

Answer: C.Squaring gives the candidates π/6, π/2, 3π/2 and 11π/6; substituting each in √3 cos x + sin x = 1 shows that π/6 and 3π/2 are extraneous, so only π/2 and 11π/6 remain.

Q8. Squaring √3 cos x + sin x = −1 leads to cos x (2 cos x + √3) = 0. The solutions of the original equation in [0, 2π) are

• A. π/2 and 7π/6
• B. π/2, 5π/6, 7π/6 and 3π/2
• C. 5π/6 and 3π/2(correct answer)
• D. π/2, 5π/6 and 3π/2

Answer: C.Squaring gives the candidates π/2, 5π/6, 7π/6 and 3π/2; substituting each in √3 cos x + sin x = −1 shows that π/2 and 7π/6 are extraneous, so only 5π/6 and 3π/2 remain.

Q9. Squaring √3 cos x − sin x = −1 leads to cos x (2 cos x + √3) = 0. The solutions of the original equation in [0, 2π) are

• A. π/2, 5π/6 and 7π/6
• B. 5π/6 and 3π/2
• C. π/2, 5π/6, 7π/6 and 3π/2
• D. π/2 and 7π/6(correct answer)

Answer: D.Squaring gives the candidates π/2, 5π/6, 7π/6 and 3π/2; substituting each in √3 cos x − sin x = −1 shows that 5π/6 and 3π/2 are extraneous, so only π/2 and 7π/6 remain.

Q10. Squaring √(2 sin²x + 1) = sin x + cos x leads to sin x (sin x − cos x) = 0. The solutions of the original equation in [0, 2π) are

• A. π and 5π/4
• B. 0, π/4, π and 5π/4
• C. 0 and π/4(correct answer)
• D. 0, π/4 and π

Answer: C.Squaring gives the candidates 0, π/4, π and 5π/4; substituting each in √(2 sin²x + 1) = sin x + cos x shows that π and 5π/4 are extraneous, so only 0 and π/4 remain.