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Trigonometric Identities MCQs — Mathematics

42 multiple-choice questions on Trigonometric Identities from Mathematics, each with the correct answer and, where available, a teacher's explanation. Read through them to revise the chapter, then practise the same questions one by one to test yourself.

1. Without a calculator, sec^2 20^circ - tan^2 20^circ =

  1. 1
  2. 0
  3. 2
  4. sec 40^circ

Answer: A) 1

Explanation: sec^2theta - tan^2theta = 1 for every angle. SLO M-09-B-24 · Application

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2. If sectheta = 54 (theta acute), then tantheta =

  1. 53
  2. 35
  3. 43
  4. 34

Answer: D) 34

Explanation: tan^2theta = sec^2theta - 1 = 2516 - 1 = 916, so tantheta = 34. SLO M-09-B-24 · Application

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3. If sintheta = 2425 and theta is acute, find sectheta.

  1. 725
  2. 2425
  3. 247
  4. 257

Answer: D) 257

Explanation: Draw a right triangle with opposite = 24, adjacent = 7, hypotenuse = 25 (Pythagoras: 24^2 + 7^2 = 25^2). Then sectheta = 257. SLO M-09-B-24 · Application

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4. A surveyor measures costheta = 0.8 for an acute angle. Then sintheta is:

  1. 0.8
  2. 0.2
  3. 0.6
  4. 1.25

Answer: C) 0.6

Explanation: sintheta = 1 - 0.64 = 0.36 = 0.6. SLO M-09-B-24 · Application

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5. A slope has tantheta = 34. A path up the slope is 50 m long. The height gained is:

  1. 37.5 m
  2. 50 m
  3. 30 m
  4. 40 m

Answer: C) 30 m

Explanation: sintheta = 35; height = 50 35 = 30 m. SLO M-09-B-24 · Application

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6. Which expression is identical to sin^4theta - cos^4theta?

  1. sin^2theta + cos^2theta
  2. 1
  3. sin^2theta - cos^2theta
  4. (sintheta - costheta)^2

Answer: C) sin^2theta - cos^2theta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, sin^4theta - cos^4theta = sin^2theta - cos^2theta. SLO M-09-B-23 · Knowledge

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7. Which expression is identical to tantheta + cottheta?

  1. 1
  2. sinthetacostheta
  3. sectheta + csctheta
  4. secthetacsctheta

Answer: D) secthetacsctheta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, tantheta + cottheta = secthetacsctheta. SLO M-09-B-23 · Knowledge

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8. Which expression is identical to csc^2theta - 1?

  1. sec^2theta
  2. tan^2theta
  3. cot^2theta
  4. 1

Answer: C) cot^2theta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, csc^2theta - 1 = cot^2theta. SLO M-09-B-23 · Knowledge

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9. Which expression is identical to 11 + sintheta + 11 - sintheta?

  1. 2
  2. 2sec^2theta
  3. 2csc^2theta
  4. sec^2theta

Answer: B) 2sec^2theta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, 11 + sintheta + 11 - sintheta = 2sec^2theta. SLO M-09-B-23 · Knowledge

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10. If sintheta = 2029 and theta is acute, find tantheta.

  1. 2921
  2. 2129
  3. 2021
  4. 2029

Answer: C) 2021

Explanation: Draw a right triangle with opposite = 20, adjacent = 21, hypotenuse = 29 (Pythagoras: 20^2 + 21^2 = 29^2). Then tantheta = 2021. SLO M-09-B-24 · Application

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11. Without tables, sin^2 37^circ + cos^2 37^circ =

  1. sin 74^circ
  2. 2
  3. 1
  4. 0

Answer: C) 1

Explanation: sin^2theta + cos^2theta = 1 for every angle. SLO M-09-B-24 · Application

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12. If costheta = 1213 and theta is acute, find sintheta.

  1. 1213
  2. 1312
  3. 512
  4. 513

Answer: D) 513

Explanation: Draw a right triangle with opposite = 5, adjacent = 12, hypotenuse = 13 (Pythagoras: 5^2 + 12^2 = 13^2). Then sintheta = 513. SLO M-09-B-24 · Application

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13. Which expression is identical to sectheta - costheta?

  1. costhetacottheta
  2. 0
  3. tantheta
  4. sinthetatantheta

Answer: D) sinthetatantheta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, sectheta - costheta = sinthetatantheta. SLO M-09-B-23 · Knowledge

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14. sectheta is defined as:

  1. costhetasintheta
  2. 1sintheta
  3. 1tantheta
  4. 1costheta

Answer: D) 1costheta

Explanation: secant is the reciprocal of cosine. SLO M-09-B-23 · Knowledge

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15. Which expression is identical to (1 - sintheta)(1 + sintheta)?

  1. cos^2theta
  2. 1
  3. sin^2theta
  4. 1 - sintheta

Answer: A) cos^2theta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, (1 - sintheta)(1 + sintheta) = cos^2theta. SLO M-09-B-23 · Knowledge

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16. Which expression is identical to 1 + cot^2theta?

  1. tan^2theta
  2. csc^2theta
  3. sin^2theta
  4. sec^2theta

Answer: B) csc^2theta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, 1 + cot^2theta = csc^2theta. SLO M-09-B-23 · Knowledge

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17. The identity sin^2theta + cos^2theta = 1 comes from applying Pythagoras' theorem to:

  1. a circle of radius 2
  2. a square
  3. a right triangle with hypotenuse 1 (a point on the unit circle)
  4. an equilateral triangle

Answer: C) a right triangle with hypotenuse 1 (a point on the unit circle)

Explanation: With hypotenuse 1, the legs are costheta and sintheta, so cos^2theta + sin^2theta = 1^2. SLO M-09-B-23 · Knowledge

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18. Which expression is identical to cotthetasintheta?

  1. costheta
  2. 1
  3. tantheta
  4. sintheta

Answer: A) costheta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, cotthetasintheta = costheta. SLO M-09-B-23 · Knowledge

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19. Which expression is identical to 1 + tan^2theta?

  1. cos^2theta
  2. cot^2theta
  3. csc^2theta
  4. sec^2theta

Answer: D) sec^2theta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, 1 + tan^2theta = sec^2theta. SLO M-09-B-23 · Knowledge

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20. Which expression is identical to sinthetasectheta?

  1. sin^2theta
  2. 1
  3. tantheta
  4. cottheta

Answer: C) tantheta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, sinthetasectheta = tantheta. SLO M-09-B-23 · Knowledge

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21. Which expression is identical to 1 - cos^2theta?

  1. sin^2theta
  2. 1
  3. cos^2theta
  4. tan^2theta

Answer: A) sin^2theta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, 1 - cos^2theta = sin^2theta. SLO M-09-B-23 · Knowledge

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22. Which expression is identical to sinthetacostheta?

  1. cottheta
  2. sectheta
  3. 1
  4. tantheta

Answer: D) tantheta

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, sinthetacostheta = tantheta. SLO M-09-B-23 · Knowledge

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23. If tantheta = 724 and theta is acute, find costheta.

  1. 2524
  2. 725
  3. 2425
  4. 724

Answer: C) 2425

Explanation: Draw a right triangle with opposite = 7, adjacent = 24, hypotenuse = 25 (Pythagoras: 7^2 + 24^2 = 25^2). Then costheta = 2425. SLO M-09-B-24 · Application

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24. Which expression is identical to (sectheta + tantheta)(sectheta - tantheta)?

  1. 0
  2. 1
  3. 2tantheta
  4. sec^2theta

Answer: B) 1

Explanation: Using sin^2theta + cos^2theta = 1 and the definitions of the ratios, (sectheta + tantheta)(sectheta - tantheta) = 1. SLO M-09-B-23 · Knowledge

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25. If sintheta = 35 and theta is acute, find tantheta.

  1. 35
  2. 34
  3. 45
  4. 54

Answer: B) 34

Explanation: Draw a right triangle with opposite = 3, adjacent = 4, hypotenuse = 5 (Pythagoras: 3^2 + 4^2 = 5^2). Then tantheta = 34. SLO M-09-B-24 · Application

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