Trigonometry - Previous Year Questions Solved Examples (Class 10 Mathematics)
Trigonometry deals with relationships between angles and sides in triangles. These questions test knowledge of trigonometric ratios, angles of elevation an
TL;DR: Trigonometry deals with relationships between angles and sides in triangles. These questions test knowledge of trigonometric ratios, angles of elevati…
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Trigonometry deals with relationships between angles and sides in triangles. These questions test knowledge of trigonometric ratios, angles of elevation an
Trigonometry - Previous Year Questions — Solved Numerical Examples (Step by Step)
Example 1: If sin(A) = 3/5 and A is an acute angle, find cos(A) and tan(A). [3 marks]
Solution: Given sin(A) = 3/5. Using sin²(A) + cos²(A) = 1: (3/5)² + cos²(A) = 1. 9/25 + cos²(A) = 1. cos²(A) = 16/25. cos(A) = 4/5 (positive since A is acute). tan(A) = sin(A)/cos(A) = (3/5)/(4/5) = 3/4.
Example 2: A ladder of length 10 m leans against a wall, making an angle of 60 degrees with the ground. How high does it reach on the wall? [2 marks]
Solution: Let the height be h. The ladder, wall, and ground form a right triangle. sin(60°) = h/10. h = 10 × sin(60°) = 10 × (√3/2) = 5√3 m ≈ 8.66 m.
Example 3: From a point 50 m away from the base of a building, the angle of elevation to the top is 30 degrees. Find the height of the building. [3 marks]
Solution: Let height be h. Distance from base = 50 m. tan(30°) = h/50. h = 50 × tan(30°) = 50 × (1/√3) = 50/√3 = 50√3/3 m ≈ 28.87 m.
Example 4: Prove that tan²(A) + 1 = sec²(A). [2 marks]
Solution: Starting from left side: tan²(A) + 1 = sin²(A)/cos²(A) + 1 = [sin²(A) + cos²(A)] / cos²(A) = 1/cos²(A) = sec²(A).
Example 5: Evaluate sin(45°) × cos(45°) + sin(30°) × cos(60°). [2 marks]
Solution: sin(45°) = cos(45°) = 1/√2. sin(30°) = 1/2. cos(60°) = 1/2. Expression = (1/√2)(1/√2) + (1/2)(1/2) = 1/2 + 1/4 = 3/4.
Example 6: A person standing on ground observes a bird at the top of a tree with angle of elevation 45 degrees. After moving 10 m away, the angle becomes 30 degrees. Find the height of the tree. [3 marks]
Solution: Let height = h. From first position: tan(45°) = h/x, so x = h. From second position: tan(30°) = h/(x + 10). 1/√3 = h/(h + 10). h + 10 = h√3. 10 = h(√3 - 1). h = 10/(√3 - 1) = 10(√3 + 1)/2 = 5(√3 + 1) ≈ 13.66 m.
Example 7: If A and B are complementary angles, prove that sin(A) = cos(B). [2 marks]
Solution: If A and B are complementary, then A + B = 90°, so B = 90° - A. cos(B) = cos(90° - A) = sin(A). Therefore sin(A) = cos(B).
Tips
- Memorize values for standard angles: 0°, 30°, 45°, 60°, 90° for all six trigonometric ratios.
- For angle of elevation/depression problems, draw a clear diagram identifying the angle and sides.
- Use trigonometric identities to simplify expressions: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ.
- In inverse trigonometry, ensure angles are in the correct range: sin⁻¹ gives angles in [-90°, 90°].
Frequently Asked Questions
What is the difference between angle of elevation and angle of depression?
Angle of elevation is measured upward from horizontal when looking at something above. Angle of depression is measured downward when looking at something below. They are equal by alternate angles.
Why must I rationalize the denominator?
Rationalization simplifies expressions and makes them easier to compute. For example, 1/√3 becomes √3/3.
More Mathematics Solved Examples
- Real Numbers (HCF LCM Euclid)
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