Introduction to Trigonometry — Previous Year Questions (Class 10 Mathematics)
Trigonometry relates angles to the sides of triangles. Understanding trigonometric ratios and identities is crucial for solving real-world problems involvi
TL;DR: Trigonometry relates angles to the sides of triangles. Understanding trigonometric ratios and identities is crucial for solving real-world problems in…
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Trigonometry relates angles to the sides of triangles. Understanding trigonometric ratios and identities is crucial for solving real-world problems involvi
Introduction to Trigonometry — Previous Year Questions with Solutions
Q (2023, 2 marks): If sin A = 3/5, find the values of cos A and tan A, where A is an acute angle.
Answer: Given sin A = 3/5
Using sin^2 A + cos^2 A = 1:
(3/5)^2 + cos^2 A = 1
9/25 + cos^2 A = 1
cos^2 A = 16/25
cos A = 4/5 (positive, as A is acute)
tan A = sin A / cos A = (3/5) / (4/5) = 3/4
Q (2022, 2 marks): Prove that tan^2 A + 1 = sec^2 A.
Answer: LHS = tan^2 A + 1
= sin^2 A / cos^2 A + 1
= sin^2 A / cos^2 A + cos^2 A / cos^2 A
= (sin^2 A + cos^2 A) / cos^2 A
= 1 / cos^2 A
= sec^2 A = RHS
Hence proved.
Q (2023, 2 marks): Find the value of sin 60° × cos 30° + sin 30° × cos 60°.
Answer: sin 60° = sqrt(3)/2, cos 30° = sqrt(3)/2
sin 30° = 1/2, cos 60° = 1/2
sin 60° × cos 30° + sin 30° × cos 60°
= (sqrt(3)/2) × (sqrt(3)/2) + (1/2) × (1/2)
= 3/4 + 1/4
= 4/4
= 1
Q (2022, 3 marks): A ladder is leaning against a wall. The ladder is 13 m long and the distance from the wall to the bottom of the ladder is 5 m. Find the angle the ladder makes with the ground.
Answer: Let the angle be θ.
cos θ = adjacent / hypotenuse = 5 / 13
θ = cos^(-1)(5/13)
θ ≈ 67.38° or approximately 67°
Q (2023, 3 marks): Prove that (sin A + cos A)^2 + (sin A - cos A)^2 = 2.
Answer: LHS = (sin A + cos A)^2 + (sin A - cos A)^2
= (sin^2 A + 2 sin A cos A + cos^2 A) + (sin^2 A - 2 sin A cos A + cos^2 A)
= sin^2 A + cos^2 A + sin^2 A + cos^2 A + 2 sin A cos A - 2 sin A cos A
= 1 + 1
= 2 = RHS
Hence proved.
Q (2021, 3 marks): If tan A = 5/12 and tan B = 1/3, find tan(A + B).
Answer: Using tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
tan(A + B) = (5/12 + 1/3) / (1 - (5/12) × (1/3))
= (5/12 + 4/12) / (1 - 5/36)
= (9/12) / (36/36 - 5/36)
= (3/4) / (31/36)
= (3/4) × (36/31)
= 27/31
Frequently Asked Questions
What are the three primary trigonometric ratios?
The three primary trigonometric ratios are: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. These are often remembered as SOH-CAH-TOA.
Why is the value of sin θ always between 0 and 1 for acute angles?
For acute angles, sin θ = opposite/hypotenuse. Since the opposite side is always shorter than the hypotenuse in a right triangle, the ratio is always less than 1. Also, both sides are positive, so the ratio is greater than 0.
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