Trigonometry (Class 10 Mathematics)

Trigonometry relates the sides and angles of right triangles. The ratios sine, cosine, and tangent are fundamental to solving problems in geometry, physics

Trigonometry (Class 10 Mathematics)

Trigonometry relates the sides and angles of right triangles. The ratios sine, cosine, and tangent are fundamental to solving problems in geometry, physics

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TL;DR: Trigonometry relates the sides and angles of right triangles. The ratios sine, cosine, and tangent are fundamental to solving problems in geometry, ph…

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Trigonometry relates the sides and angles of right triangles. The ratios sine, cosine, and tangent are fundamental to solving problems in geometry, physics

Trigonometry MCQs with Answers & Explanations

Q1. In a right triangle, sin(θ) is defined as:

  1. Opposite/Adjacent
  2. Hypotenuse/Opposite
  3. Opposite/Hypotenuse ✓ (correct)
  4. Adjacent/Hypotenuse

Explanation: sin(θ) = opposite/hypotenuse. It's abbreviated as the ratio of the side opposite to angle θ and the hypotenuse.

Q2. What is the value of sin(90°)?

  1. 0
  2. √3/2
  3. 0.5
  4. 1 ✓ (correct)

Explanation: sin(90°) = 1 because at 90°, the opposite side equals the hypotenuse.

Q3. tan(θ) equals:

  1. sin(θ)/cos(θ) ✓ (correct)
  2. sin(θ) × cos(θ)
  3. sin(θ) + cos(θ)
  4. cos(θ)/sin(θ)

Explanation: tan(θ) = sin(θ)/cos(θ) = opposite/adjacent.

Q4. The value of cos(0°) is:

  1. √3/2
  2. 1 ✓ (correct)
  3. 0.5
  4. 0

Explanation: cos(0°) = 1 because at 0°, the adjacent side equals the hypotenuse.

Q5. Which of the following is an identity?

  1. sin(θ) × cos(θ) = 1
  2. sin(θ) + cos(θ) = 1
  3. sin²(θ) + cos²(θ) = 1 ✓ (correct)
  4. sin(θ) - cos(θ) = 0

Explanation: sin²(θ) + cos²(θ) = 1 is the fundamental Pythagorean identity in trigonometry.

Q6. In a right triangle with angle 30°, if the hypotenuse is 10, the side opposite to 30° is:

  1. 5√3
  2. 10
  3. 10/√3
  4. 5 ✓ (correct)

Explanation: sin(30°) = 1/2 = opposite/hypotenuse. So opposite = 10 × (1/2) = 5.

Q7. The angle of elevation is measured from:

  1. Above the horizontal ✓ (correct)
  2. Below the horizontal
  3. The vertical downward
  4. Above the vertical

Explanation: The angle of elevation is the angle above the horizontal when looking upward at an object.

Q8. What is tan(45°)?

  1. 1/√2
  2. 1 ✓ (correct)
  3. √3
  4. 0

Explanation: tan(45°) = 1 because in a 45-45-90 triangle, the opposite and adjacent sides are equal.

Q9. If sin(θ) = 3/5, then cos(θ) could be:

  1. 5/3
  2. 2/5
  3. 4/5 ✓ (correct)
  4. 3/5

Explanation: Using sin²(θ) + cos²(θ) = 1: (3/5)² + cos²(θ) = 1, so cos²(θ) = 1 - 9/25 = 16/25, thus cos(θ) = 4/5.

Q10. cot(θ) equals:

  1. tan(θ)
  2. sin(θ)
  3. cos(θ)
  4. 1/tan(θ) ✓ (correct)

Explanation: cot(θ) = 1/tan(θ) = cos(θ)/sin(θ) = adjacent/opposite.

Q11. In a right-angled triangle, sin A =

  1. Opposite / Hypotenuse ✓ (correct)
  2. Opposite / Adjacent
  3. Hypotenuse / Opposite
  4. Adjacent / Hypotenuse

Explanation: Sin A = Opposite / Hypotenuse, which is one of the basic trigonometric ratios.

Q12. If sin A = 3/5 and angle A is acute, then cos A =

  1. 5/4
  2. 4/5 ✓ (correct)
  3. 3/4
  4. 5/3

Explanation: Using sin^2 A + cos^2 A = 1: (3/5)^2 + cos^2 A = 1, so cos^2 A = 16/25, thus cos A = 4/5.

Q13. The value of sin 0 degrees is:

  1. 1/2
  2. 1
  3. 0 ✓ (correct)
  4. -1

Explanation: sin 0 = 0, which is a standard trigonometric value.

Q14. tan A = cot A when A =

  1. 60 degrees
  2. 90 degrees
  3. 30 degrees
  4. 45 degrees ✓ (correct)

Explanation: tan A = cot A when tan A = 1/tan A, which means tan^2 A = 1, so A = 45 degrees.

Q15. If sin A = cos B and A and B are acute angles, then A + B =

  1. 90 degrees ✓ (correct)
  2. 45 degrees
  3. 180 degrees
  4. 60 degrees

Explanation: If sin A = cos B, then A and B are complementary angles, so A + B = 90 degrees.

Q16. The value of sec 60 degrees is:

  1. sqrt(3)/2
  2. 2 ✓ (correct)
  3. sqrt(2)
  4. 2/sqrt(3)

Explanation: sec 60 = 1/cos 60 = 1/(1/2) = 2.

Q17. If tan A = 1/sqrt(3), then A =

  1. 90 degrees
  2. 45 degrees
  3. 30 degrees ✓ (correct)
  4. 60 degrees

Explanation: tan 30 = 1/sqrt(3), so A = 30 degrees.

Q18. sin^2 A + cos^2 A =

  1. 0
  2. -1
  3. 2
  4. 1 ✓ (correct)

Explanation: sin^2 A + cos^2 A = 1 is the fundamental trigonometric identity.

Q19. If a ladder of length 10 m leans against a wall making 60 degrees with the ground, the height at which it touches the wall is:

  1. 5sqrt(3) m ✓ (correct)
  2. 5 m
  3. 10 m
  4. 10sqrt(3) m

Explanation: Height = 10 x sin 60 = 10 x sqrt(3)/2 = 5sqrt(3) m.

Q20. The angle of elevation from a point on the ground to the top of a 20 m building is 30 degrees. The distance from the point to the base of the building is:

  1. 10 m
  2. 20sqrt(3) m ✓ (correct)
  3. 40/sqrt(3) m
  4. 10sqrt(3) m

Explanation: Distance = 20 / tan 30 = 20 / (1/sqrt(3)) = 20sqrt(3) m.

Frequently Asked Questions

What are the trigonometric ratios for 30°, 45°, and 60°?

For 30°: sin = 1/2, cos = √3/2, tan = 1/√3. For 45°: sin = cos = 1/√2, tan = 1. For 60°: sin = √3/2, cos = 1/2, tan = √3. These standard angles are frequently used in calculations.

How are angles of elevation and depression used in real-world problems?

Angles of elevation (when looking up) and depression (when looking down) help solve problems involving heights and distances. For example, finding the height of a building from a known distance using the angle of elevation.

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