THE GOAL • CBSE CLASS 10

TRIGONOMETRY

Chapters 8 & 9 • Basic Mathematics • 2025 PYQs

CBSE 2025 Basic Mathematics Previous Year Questions Chapter-wise Collection

Chapters 8 & 9: Trigonometry

Unique Previous Year Questions extracted from the CBSE Class 10 Basic Mathematics 2025 question paper sets. Repetitive questions have been removed while retaining the major question types and variations.

Chapter 8: Introduction to Trigonometry

Trigonometric ratios, identities, standard values and identity-based questions.

I

Multiple Choice Questions

1. Value Comparison
1 Mark
Which of the following statements is false?
(A) \(\tan45^\circ=\cot45^\circ\)
(B) \(\sin90^\circ=\tan45^\circ\)
(C) \(\sin30^\circ=\cos30^\circ\)
(D) \(\sin45^\circ=\cos45^\circ\)
Set 430/1/1
2. Identity Evaluation
1 Mark
The value of \[ \tan^2 A-\frac{1}{\cos^2 A} \] is:
(A) More than 1
(B) 1
(C) 0
(D) \(-1\)
Set 430/1/1
3. Identity Application
1 Mark
The value of \[ \cot^2 A-\frac{1}{\sin^2 A} \] is:
(A) More than 1
(B) 1
(C) 0
(D) \(-1\)
Set 430/1/2
4. Reciprocal Identity
1 Mark
If \[ \sec\theta-\tan\theta=2, \] then \(\sec\theta+\tan\theta\) is equal to:
(A) \(\frac12\)
(B) \(\sqrt2\)
(C) \(\frac1{\sqrt2}\)
(D) 2
Set 430/4/2
5. Ratio Calculation
1 Mark
If \[ \tan A=\frac12, \] then \(\sin A\) is equal to:
(A) \(\frac2{\sqrt5}\)
(B) \(\frac1{\sqrt3}\)
(C) \(\frac1{\sqrt5}\)
(D) 1
Set 430/4/2
6. Complex Evaluation
1 Mark
The value of \[ \frac{\sec^2 30^\circ+\tan^2 30^\circ} {\sin^2 45^\circ+\cos^2 45^\circ} \] is:
(A) 1
(B) \(\frac53\)
(C) \(\frac{13}{3}\)
(D) 7
Set 430/3/1
7. Variable Solving
1 Mark
The value of \(\theta\) for which \[ \sin2\theta=\tan45^\circ \] is:
(A) \(22.5^\circ\)
(B) \(30^\circ\)
(C) \(45^\circ\)
(D) \(90^\circ\)
Set 430/3/3
8. Algebraic Identity
1 Mark
\[ (\sin\theta+\cos\theta)^2+ (\sin\theta-\cos\theta)^2 \] is equal to:
(A) 1
(B) 2
(C) \(2+2\sin\theta\cos\theta\)
(D) \(2+4\sin\theta\cos\theta\)
Set 430/2/2
9. Standard Table Value
1 Mark
The value of \[ \frac{2\tan60^\circ}{1-\tan^260^\circ} \] is:
(A) \(-3\)
(B) \(\sqrt3\)
(C) \(-\frac1{\sqrt3}\)
(D) \(-\sqrt3\)
Set 430/2/1
II

Assertion – Reason Questions

10. Sine and Cosine Relation
1 Mark
Assertion (A): For \(\sin\theta=1\), \(\cos\theta\) must be 0.
Reason (R): \[ \sin^2\theta-\cos^2\theta=1. \]
Set 430/3/1
11. Cotangent and Cosecant
1 Mark
Assertion (A): For an acute angle \(\theta\), \[ \cot\theta=1\implies\csc\theta=\sqrt2. \]
Reason (R): \[ \csc^2\theta-\cot^2\theta=1. \]
Set 430/3/2
III

Very Short Answer Type Questions

12. Finding Unknown Angles
2 Marks
Find the values of \(A\) and \(B\), where \[ 0^\circ\le A<90^\circ,\qquad 0^\circ\le B<90^\circ, \] if \[ \tan(A+B)=1 \] and \[ \tan(A-B)=\frac1{\sqrt3}. \]
Set 430/1/1
13. Geometric Proof
2 Marks
Prove that \[ \tan45^\circ=1 \] geometrically.
Set 430/1/1
14. Expression Evaluation
2 Marks
Find the value of \(k\), if \[ (\sec A+\tan A)(1-\sin A)=k\cos A. \]
Set 430/3/1
15. Standard Values
2 Marks
Evaluate \[ \frac{\cos45^\circ} {\sec30^\circ+\csc30^\circ}. \]
Set 430/2/1
16. Identity Verification
2 Marks
Verify that \[ \cos2A= \frac{1-\tan^2A}{1+\tan^2A} \] for \(A=30^\circ\).
Set 430/5/1
IV

Short Answer Type Questions

17. Proving Identity – Variant 1
3 Marks
Prove the identity: \[ \sqrt{\frac{\csc A-1}{\csc A+1}} = \sec A-\tan A. \]
Set 430/1/2
18. Proving Identity – Variant 2
3 Marks
Prove the identity: \[ (\sin A-\csc A)(\cos A-\sec A) = \frac1{\tan A+\cot A}. \]
Set 430/1/3
19. Proving Identity – Variant 3
3 Marks
Prove the identity: \[ \frac{\cos\theta}{1+\sin\theta} + \frac{1+\sin\theta}{\cos\theta} = 2\sec\theta. \]
Set 430/2/1
20. Proving Identity – Variant 4
3 Marks
Prove the identity: \[ \frac{1+\sec A}{\sec A} = \frac{\sin^2A}{1-\cos A}. \]
Set 430/3/2
21. Proving Identity – Variant 5
3 Marks
Prove the identity: \[ \frac{\tan\theta}{1-\cot\theta} + \frac{\cot\theta}{1-\tan\theta} = \sec\theta\csc\theta+1. \]
Set 430/6/1
22. Proving Identity – Variant 6
3 Marks
Prove the identity: \[ \left(1+\frac1{\tan^2\theta}\right) \left(1+\frac1{\cot^2\theta}\right) = \frac1{\sin^2\theta-\sin^4\theta}. \]
Set 430/6/2

Chapter 9: Some Applications of Trigonometry

Heights and distances, angles of elevation and depression, shadows and real-life applications of trigonometry.

I

Multiple Choice Questions

23. Figure Interpretation – Angle of Depression
1 Mark
In the given figure, which of the following angles represents the angle of depression?
(A) \(x\)
(B) \(y\)
(C) \(z\)
(D) \(a\)
Note: Figure shows an observer at a height looking down at an object, with \(z\) marked between the horizontal line and line of sight.
Set 430/1/1
24. Conceptual Shadow
1 Mark
When the sun's altitude changes from \(30^\circ\) to \(60^\circ\), the length of the shadow of a tower will:
(A) Become shorter
(B) Become longer
(C) Remain same
(D) Be doubled
Set 430/2/1
25. Angle of Elevation – Figure Based
1 Mark
In the given figure, the angle of elevation of point \(A\) from point \(C\) is:
\[ AB=2\text{ cm},\qquad BC=2\sqrt3\text{ cm},\qquad AC=4\text{ cm} \]
(A) \(30^\circ\)
(B) \(45^\circ\)
(C) \(60^\circ\)
(D) Cannot be determined
Set 430/3/1
II

Long Answer Type Questions

26. Double Ships Problem
5 Marks
As observed from the top of a \(70\) m high lighthouse from the sea level, the angles of depression of two ships are \(30^\circ\) and \(45^\circ\).
If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
\[ \text{Use }\sqrt3=1.73 \]
Set 430/4/2
27. Building and Tower
5 Marks
The angle of elevation of the top of a building from the foot of a tower is \(30^\circ\) and the angle of elevation of the top of the tower from the foot of the building is \(60^\circ\).
If the tower is \(30\) m high, find the height of the building.
Set 430/5/1
III

Case Study Based Questions

CASE STUDY • 4 MARKS
28. Kite Festival – Figure/Data Based
4 Marks
Reena and Ravi are flying kites. The height of Reena's kite is \(60\) m above the ground. The string is inclined at \(30^\circ\) to the ground.

Ravi is flying a kite from a \(10\) m high building; his kite is also \(60\) m above the ground. The string used by Ravi is the same length as Reena's.
(i) Find the length of string used by Reena. [1 Mark]
(ii) Find the value of \(\sin\theta\) for Ravi's kite. [1 Mark]
(iii) If \(\theta\) changes to \(60^\circ\) without changing the string length, find the new height of Ravi's kite. OR Find Ravi's kite height if the string had a \(30^\circ\) inclination with the roof. [2 Marks]
Set 430/2/2