FAST-NUCES Entry Test Mathematics Trigonometric Identities — Set 2

Trigonometric Identities MCQs set 2 for FAST-NUCES Entry Test Mathematics — 20 solved questions.

FAST-NUCES Entry Test Mathematics Trigonometric Identities — Set 2

  1. Question 1

    Q1. What is the value of sin(2x) in terms of sin(x) and cos(x)?

    • A) 2 sin(x) cos(x)
    • B) sin(x) + cos(x)
    • C) sin(x) - cos(x)
    • D) sin(x) cos(x)

    Answer: 2 sin(x) cos(x)

    Explanation: Using the double angle formula, sin(2x) = 2 sin(x) cos(x), derived from sum and difference identities.

  2. Question 2

    Q2. If sin(x) + cos(x) = √2, then what is the value of sin(x) cos(x)?

    • A) 1/2
    • B) 1
    • C) 0
    • D) -1/2

    Answer: 1/2

    Explanation: Squaring both sides, we get sin²(x) + 2 sin(x) cos(x) + cos²(x) = 2. Using sin²(x) + cos²(x) = 1, we simplify to 2 sin(x) cos(x) = 1, so sin(x) cos(x) = 1/2.

  3. Question 3

    Q3. What is the value of cos(A + B) + cos(A - B)?

    • A) 2 cos(A) cos(B)
    • B) 2 sin(A) sin(B)
    • C) 2 cos(A) sin(B)
    • D) 2 sin(A) cos(B)

    Answer: 2 cos(A) cos(B)

    Explanation: Using sum and difference formulas, cos(A + B) + cos(A - B) = cos(A)cos(B) - sin(A)sin(B) + cos(A)cos(B) + sin(A)sin(B) = 2 cos(A) cos(B).

  4. Question 4

    Q4. If tan(x) = 1/√3, then what is the value of x in the range 0 to π?

    • A) π/6
    • B) π/3
    • C) π/4
    • D) π/2

    Answer: π/6

    Explanation: tan(π/6) = 1/√3, so x = π/6, as tan(x) is positive in the first quadrant.

  5. Question 5

    Q5. What is the value of sin(π/2 - x)?

    • A) cos(x)
    • B) sin(x)
    • C) -cos(x)
    • D) -sin(x)

    Answer: cos(x)

    Explanation: Using the co-function identity, sin(π/2 - x) = cos(x), as sine and cosine are co-functions.

  6. Question 6

    Q6. If sin(x) = 3/5, then what is the value of cos(2x)?

    • A) 7/25
    • B) -7/25
    • C) 24/25
    • D) -24/25

    Answer: 7/25

    Explanation: cos(2x) = 1 - 2sin²x = 1 - 2(9/25) = 7/25 when sin x = 3/5.

  7. Question 7

    Q7. What is the value of tan(A + B) in terms of tan(A) and tan(B)?

    • A) (tan(A) + tan(B)) / (1 - tan(A) tan(B))
    • B) (tan(A) - tan(B)) / (1 + tan(A) tan(B))
    • C) (tan(A) + tan(B)) / (1 + tan(A) tan(B))
    • D) (tan(A) - tan(B)) / (1 - tan(B) tan(A))

    Answer: (tan(A) + tan(B)) / (1 - tan(A) tan(B))

    Explanation: Using the sum formula for tangent, tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A) tan(B)), derived from sum and difference identities.

  8. Question 8

    Q8. If cos(x) = 1/2, then what is the value of sin(3x)?

    • A) 1
    • B) 0
    • C) -1
    • D) 1/2

    Answer: 0

    Explanation: Using the triple angle formula, sin(3x) = 3 sin(x) - 4 sin³(x). First, find sin(x) = √(1 - cos²(x)) = √(1 - (1/2)²) = √3/2. Then, sin(3x) = 3(√3/2) - 4(√3/2)³ = 0.

  9. Question 9

    Q9. What is the value of sin(A) cos(B) + cos(A) sin(B)?

    • A) sin(A + B)
    • B) cos(A + B)
    • C) sin(A - B)
    • D) cos(A - B)

    Answer: sin(A + B)

    Explanation: Using the sum formula for sine, sin(A) cos(B) + cos(A) sin(B) = sin(A + B), a standard trigonometric identity.

  10. Question 10

    Q10. If sin(x) = 1/2, then what is the value of cos(x/2)?

    • A) √3/2
    • B) 1/2
    • C) √(3/4 + 1/4)/√2
    • D) √(3/4 - 1/4)/√2

    Answer: √(3/4 + 1/4)/√2

    Explanation: Using the half angle formula, cos(x/2) = √((1 + cos(x))/2). First, find cos(x) = √(1 - sin²(x)) = √(1 - (1/2)²) = √3/2. Then, cos(x/2) = √((1 + √3/2)/2).

  11. Question 11

    Q11. What is the value of cos(2x) in terms of sin(x)?

    • A) 1 - 2 sin²(x)
    • B) 2 sin²(x) - 1
    • C) 1 + 2 sin²(x)
    • D) 2 sin(x) cos(x)

    Answer: 1 - 2 sin²(x)

    Explanation: Using the double angle formula, cos(2x) = 1 - 2 sin²(x), derived from the Pythagorean identity.

  12. Question 12

    Q12. If tan(x) = 3/4, then what is the value of sin(2x)?

    • A) 24/25
    • B) 12/25
    • C) 7/25
    • D) 1/25

    Answer: 24/25

    Explanation: Using the double angle formula, sin(2x) = 2 tan(x) / (1 + tan²(x)) = 2(3/4) / (1 + (3/4)²) = (3/2) / (25/16) = 24/25.

  13. Question 13

    Q13. What is the value of sin(x + π/2)?

    • A) cos(x)
    • B) -cos(x)
    • C) sin(x)
    • D) -sin(x)

    Answer: cos(x)

    Explanation: Using the sum formula for sine, sin(x + π/2) = sin(x)cos(π/2) + cos(x)sin(π/2) = cos(x), as cos(π/2) = 0 and sin(π/2) = 1.

  14. Question 14

    Q14. If cos(x) = -1/2, then what is the value of sin(x/2)?

    • A) √3/2
    • B) 1/2
    • C) √2/2
    • D) 1/√2

    Answer: √2/2

    Explanation: Using the half angle formula, sin(x/2) = √((1 - cos(x))/2). Then, sin(x/2) = √((1 - (-1/2))/2) = √(3/4) = √2/2 (approx), but the exact value should be calculated directly.

  15. Question 15

    Q15. If sin(x) + sin(y) = 1 and cos(x) + cos(y) = 1, then what is the value of sin(x + y)?

    • A) 1
    • B) 0
    • C) 1/2
    • D) -1/2

    Answer: 1

    Explanation: For n equal coplanar forces in equilibrium, adjacent forces are separated by 360°/n, which is 2π/n radians.

  16. Question 16

    Q16. What is the value of tan(π/4 + x)?

    • A) (1 + tan(x)) / (1 - tan(x))
    • B) (1 - tan(x)) / (1 + tan(x))
    • C) 1 + tan(x)
    • D) 1 - tan(x)

    Answer: (1 + tan(x)) / (1 - tan(x))

    Explanation: Using the sum formula for tangent, tan(π/4 + x) = (tan(π/4) + tan(x)) / (1 - tan(π/4) tan(x)) = (1 + tan(x)) / (1 - tan(x)), as tan(π/4) = 1.

  17. Question 17

    Q17. If cos(2x) = 1/2, then what is the value of sin(x)?

    • A) 1/2
    • B) 1/√2
    • C) √3/2
    • D) 1/√3

    Answer: 1/√2

    Explanation: Using the double angle formula, cos(2x) = 1 - 2 sin²(x) = 1/2. Then, 2 sin²(x) = 1/2, so sin²(x) = 1/4, giving sin(x) = 1/√2.

  18. Question 18

    Q18. What is the value of sin(3x) in terms of sin(x)?

    • A) 3 sin(x) - 4 sin³(x)
    • B) 3 sin(x) + 4 sin³(x)
    • C) 4 sin³(x) - 3 sin(x)
    • D) 4 sin(x) - 3 sin³(x)

    Answer: 3 sin(x) - 4 sin³(x)

    Explanation: Using the triple angle formula, sin(3x) = 3 sin(x) - 4 sin³(x), derived from sum and difference identities.

  19. Question 19

    Q19. If tan(x) = 1, then what is the value of tan(2x)?

    • A) undefined
    • B) 0
    • C) 1
    • D) -1

    Answer: undefined

    Explanation: Using the double angle formula, tan(2x) = 2 tan(x) / (1 - tan²(x)) = 2(1) / (1 - 1²) = 2 / 0, which is undefined.

  20. Question 20

    Q20. The value of sin(2x) is

    • A) 2sin(x)cos(x)
    • B) sin²(x) - cos²(x)
    • C) cos²(x) - sin²(x)
    • D) 1 - 2sin²(x)

    Answer: 2sin(x)cos(x)

    Explanation: Using the double-angle identity, sin(2x) = 2sin(x)cos(x).

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What is the value of sin(2x) in terms of sin(x) and cos(x)?