HEC USAT-E (Pre-Engineering) Mathematics Trigonometric Equations — Set 2

HEC USAT-E (Pre-Engineering) Mathematics Trigonometric Equations — Set 2

1. Question 1

   Q1. If tan(x) = 1, then x =

   • A) π/4 + nπ
   • B) π/4 + 2nπ
   • C) nπ + π/2
   • D) 2nπ

   Answer: π/4 + nπ

   Explanation: The tangent function equals 1 at π/4 and has a period of π, so the general solution is π/4 + nπ.

2. Question 2

   Q2. The solution of cos(x) = 0 is

   • A) nπ
   • B) (2n + 1)π/2
   • C) 2nπ
   • D) (n + 1/2)π/3

   Answer: (2n + 1)π/2

   Explanation: The cosine function equals zero at odd multiples of π/2, hence the general solution is (2n + 1)π/2.

3. Question 3

   Q3. The general solution of 2sin(x) + √3 = 0 is

   • A) nπ + (-1)^n * π/3
   • B) nπ + (-1)^(n+1) * π/3
   • C) 2nπ ± π/3
   • D) nπ + (-1)^(n+1) * 2π/3

   Answer: nπ + (-1)^(n+1) * π/3

   Explanation: First, isolate sin(x) to get sin(x) = -√3/2, then use the general solution for sin(x) = sin(α).

4. Question 4

   Q4. The general solution of the equation sin(2x) = 1/2 is

   • A) nπ + (-1)^n * π/12
   • B) nπ/2 + (-1)^n * π/12
   • C) nπ + (-1)^(n+1) * π/6
   • D) nπ + π/6

   Answer: nπ/2 + (-1)^n * π/12

   Explanation: Use the identity sin(2x) = sin(α), then 2x = nπ + (-1)^n * α, and solve for x.

5. Question 5

   Q5. The general solution of tan(2x) = √3 is

   • A) nπ/2 + π/6
   • B) nπ + π/3
   • C) nπ/2 + π/3
   • D) nπ + π/6

   Answer: nπ/2 + π/3

   Explanation: First, find the principal solution, then use the periodicity of the tangent function to find the general solution.

6. Question 6

   Q6. The general solution of cos(3x) = 1/2 is

   • A) 2nπ/3 ± 2π/9
   • B) 2nπ ± π/9
   • C) nπ/3 + (-1)^n * π/9
   • D) 2nπ/3 ± π/9

   Answer: 2nπ/3 ± 2π/9

   Explanation: First, isolate cos(3x), then use the general solution for cos(x) = cos(α).

7. Question 7

   Q7. The general solution of 3tan²(x) - 1 = 0 is

   • A) nπ ± π/6
   • B) nπ ± arctan(1/√3)
   • C) nπ ± arctan(√3)
   • D) nπ/2 ± arctan(1/√3)

   Answer: nπ ± arctan(1/√3)

   Explanation: First, solve for tan(x), then use the general solution for tan(x) = tan(α).

8. Question 8

   Q8. The solution of 2sin(x)cos(x) = 1 is

   • A) nπ + (-1)^n * π/4
   • B) nπ/2 + (-1)^n * π/4
   • C) nπ + π/4
   • D) nπ + (-1)^n * π/2

   Answer: nπ/2 + (-1)^n * π/4

   Explanation: Use the identity sin(2x) = 2sin(x)cos(x), then simplify and solve.

9. Question 9

   Q9. The general solution of sin(x) + sin(3x) = 0 is

   • A) nπ/2
   • B) nπ
   • C) nπ/4
   • D) nπ or nπ ± π/4

   Answer: nπ or nπ ± π/4

   Explanation: Use sum-to-product identity, then simplify and solve the resulting equation.

10. Question 10

    Q10. The general solution of cos(x) = cos(2x) is

    • A) 2nπ/3
    • B) 2nπ
    • C) nπ or 2nπ ± 2π/3
    • D) 2nπ/3 or 2nπ ± π/3

    Answer: nπ or 2nπ ± 2π/3

    Explanation: Use the identity cos(2x) = 2cos²(x) - 1, then simplify and solve the resulting quadratic equation.

11. Question 11

    Q11. If sin(3x) = sin(x), then x =

    • A) nπ or nπ ± π/2
    • B) nπ
    • C) nπ/2
    • D) nπ or 2nπ ± π/4

    Answer: nπ

    Explanation: Use the identity sin(3x) - sin(x) = 0, then apply sum-to-product identity and solve.

12. Question 12

    Q12. The general solution of tan(x) = tan(2x) is

    • A) nπ/2
    • B) nπ
    • C) nπ or nπ ± π/4
    • D) nπ or 2nπ ± π/3

    Answer: nπ

    Explanation: First, simplify the equation to tan(2x) - tan(x) = 0, then use the identity for tan(A) - tan(B).

13. Question 13

    Q13. The general solution of 2sin²(x) + sin(x) = 0 is

    • A) nπ or nπ + (-1)^n * π/6
    • B) nπ or nπ + (-1)^(n+1) * π/6
    • C) nπ
    • D) nπ or 2nπ ± 7π/6

    Answer: nπ

    Explanation: Factor out sin(x), then solve for sin(x) = 0 or sin(x) = -1/2.

14. Question 14

    Q14. The solution of cos(2x) + cos(x) = 0 is

    • A) 2nπ ± 2π/3
    • B) 2nπ/3 or 2nπ ± π/3
    • C) nπ or 2nπ ± 2π/3
    • D) 2nπ ± π/3

    Answer: nπ or 2nπ ± 2π/3

    Explanation: Use the identity cos(2x) = 2cos²(x) - 1, then simplify and solve the resulting quadratic equation.

15. Question 15

    Q15. The general solution of sin(2x) = cos(x) is

    • A) nπ/2 + (-1)^n * π/6
    • B) nπ + (-1)^n * π/6
    • C) nπ/2 + π/6 or nπ - π/2
    • D) nπ + π/6

    Answer: nπ/2 + π/6 or nπ - π/2

    Explanation: Use the identity sin(2x) = 2sin(x)cos(x) and cos(x) = cos(x), then simplify and solve.

16. Question 16

    Q16. If tan(x) = √3, then x =

    • A) nπ + π/3
    • B) 2nπ + π/3
    • C) nπ - π/3
    • D) 2nπ - π/3

    Answer: nπ + π/3

    Explanation: The tangent function has a period of π, and tan(π/3) = √3, hence the general solution is nπ + π/3.

17. Question 17

    Q17. The general solution of sin(2x) = 1 is

    • A) nπ + π/4
    • B) nπ/2 + (-1)^n * π/4
    • C) nπ + (-1)^n * π/4
    • D) nπ + π/2

    Answer: nπ/2 + (-1)^n * π/4

    Explanation: For sin(2x) = 1, 2x = nπ + (-1)^n * π/2, hence x = nπ/2 + (-1)^n * π/4.

18. Question 18

    Q18. The solution of 2cos²(x) - 1 = 0 is

    • A) nπ ± π/4
    • B) 2nπ ± π/4
    • C) nπ ± π/3
    • D) 2nπ ± π/3

    Answer: nπ ± π/4

    Explanation: 2cos²(x) - 1 = cos(2x) = 0, hence 2x = (2n+1)π/2, giving x = nπ ± π/4.

19. Question 19

    Q19. The general solution of tan(3x) = 1 is

    • A) nπ/3 + π/12
    • B) nπ + π/12
    • C) nπ/3 - π/12
    • D) nπ - π/12

    Answer: nπ/3 + π/12

    Explanation: For tan(3x) = 1, 3x = nπ + π/4, hence x = nπ/3 + π/12.

20. Question 20

    Q20. If sin(x) + cos(x) = √2, then x =

    • A) 2nπ + π/4
    • B) 2nπ - π/4
    • C) nπ + (-1)^n * π/4
    • D) nπ + π/4

    Answer: 2nπ + π/4

    Explanation: Dividing by √2 gives sin(x+π/4) = 1, hence x+π/4 = 2nπ + π/2, so x = 2nπ + π/4.