FAST-NUCES Entry Test Mathematics Trigonometric Identities — Set 3

Trigonometric Identities MCQs set 3 for FAST-NUCES Entry Test Mathematics — 20 solved questions.

FAST-NUCES Entry Test Mathematics Trigonometric Identities — Set 3

  1. Question 1

    Q1. If tan(x) = 1, what is tan(2x)?

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

    Answer: undefined

    Explanation: Using tan(2x) = 2tan(x) / (1 - tan²(x)), we get tan(2x) = 2 / (1 - 1) which is undefined.

  2. Question 2

    Q2. The expression sin(x) + sin(π - x) simplifies to

    • A) 2sin(x)
    • B) 0
    • C) 2cos(x)
    • D) sin(x)cos(x)

    Answer: 2sin(x)

    Explanation: Using sin(π - x) = sin(x), the expression simplifies to 2sin(x).

  3. Question 3

    Q3. What is the value of cos(2x) if cos(x) = 1 / √2?

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

    Answer: 0

    Explanation: Using cos(2x) = 2cos²(x) - 1, we find cos(2x) = 2 * (1 / 2) - 1 = 0.

  4. Question 4

    Q4. The value of sin(x + π / 2) is

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

    Answer: cos(x)

    Explanation: Using the sum identity, sin(x + π / 2) = sin(x)cos(π / 2) + cos(x)sin(π / 2) = cos(x).

  5. Question 5

    Q5. If sin(x) = -1 / 2, what is cos(2x)?

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

    Answer: 1 / 2

    Explanation: Using cos(2x) = 1 - 2sin²(x), we find cos(2x) = 1 - 2 * (-1 / 2)² = 1 / 2.

  6. Question 6

    Q6. The expression cos(x)cos(y) - sin(x)sin(y) simplifies to

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

    Answer: cos(x + y)

    Explanation: Using the cosine sum identity, the expression simplifies to cos(x + y).

  7. Question 7

    Q7. What is the value of tan(π / 4 + x) if tan(x) = 1?

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

    Answer: undefined

    Explanation: Using tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)), we find tan(π / 4 + x) is undefined.

  8. Question 8

    Q8. If cos(x) = 1 / 2, what is sin(2x)?

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

    Answer: √3 / 2

    Explanation: Using sin(2x) = 2sin(x)cos(x) and sin(x) = √3 / 2, we find sin(2x) = √3 / 2.

  9. Question 9

    Q9. The value of cos(π / 2 + x) is

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

    Answer: -sin(x)

    Explanation: Using the cosine sum identity, cos(π / 2 + x) = -sin(x).

  10. Question 10

    Q10. If tan(x) = √3, what is tan(2x)?

    • A) -√3
    • B) √3
    • C) undefined
    • D) 1 / √3

    Answer: undefined

    Explanation: Using tan(2x) = 2tan(x) / (1 - tan²(x)), we find tan(2x) is undefined.

  11. Question 11

    Q11. The expression sin(x)cos(y) + cos(x)sin(y) simplifies to

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

    Answer: sin(x + y)

    Explanation: Using the sine sum identity, the expression simplifies to sin(x + y).

  12. Question 12

    Q12. If cos(x) = -1 / 2, what is cos(2x)?

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

    Answer: -1 / 2

    Explanation: Using cos(2x) = 2cos²(x) - 1, we find cos(2x) = -1 / 2.

  13. Question 13

    Q13. The value of tan(x + π) is

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

    Answer: tan(x)

    Explanation: Using the tangent sum identity and tan(π) = 0, we find tan(x + π) = tan(x).

  14. Question 14

    Q14. If sin(x) = 1, what is sin(2x)?

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

    Answer: 0

    Explanation: Using sin(2x) = 2sin(x)cos(x) and cos(x) = 0, we find sin(2x) = 0.

  15. Question 15

    Q15. The expression cos²(x) - sin²(x) simplifies to

    • A) cos(2x)
    • B) sin(2x)
    • C) 1
    • D) 0

    Answer: cos(2x)

    Explanation: Using the cosine double-angle identity, the expression simplifies to cos(2x).

  16. Question 16

    Q16. What is the value of sin(x - y) if sin(x) = 1 / 2 and cos(y) = 1 / 2?

    • A) 0
    • B) 1
    • C) 1 / 2
    • D) √3 / 2 - 1 / 2

    Answer: √3 / 2 - 1 / 2

    Explanation: Using sin(x - y) = sin(x)cos(y) - cos(x)sin(y), we find the value.

  17. Question 17

    Q17. If tan(x) = 1 / √3, what is tan(2x)?

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

    Answer: √3

    Explanation: Using tan(2x) = 2tan(x) / (1 - tan²(x)), we find tan(2x) = √3.

  18. Question 18

    Q18. If tan(x) = 1 / 2 and tan(y) = 1 / 3, what is tan(x + y)?

    • A) 1
    • B) 1 / 7
    • C) 7
    • D) 1 / 5

    Answer: 1

    Explanation: Using the tangent addition formula, tan(x + y) = (1/2 + 1/3) / (1 - (1/2)*(1/3)) = 1

  19. Question 19

    Q19. What is the value of cos(2x) if sin(x) = 3/5?

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

    Answer: 7/25

    Explanation: Using the double angle formula, cos(2x) = 1 - 2sin²(x) = 1 - 2(3/5)² = 7/25

  20. Question 20

    Q20. The expression cos(20°)cos(40°)cos(80°) equals

    • A) 1/8
    • B) 1/4
    • C) 1/2
    • D) 1

    Answer: 1/8

    Explanation: Using the formula for cos(A)cos(B)cos(C), we simplify to 1/8

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If tan(x) = 1, what is tan(2x)?