Introduction to Trigonometry Test Class 10 Maths — 40 Marks
PERIODIC TEST — Introduction to Trigonometry (2026-27) | CLASS 10 | MATHEMATICS
Max Marks: 40 · Time: 1 hour 30 minutes (Time left: 90:00)
Name: ______________________________ Roll No: ____________ Date: ____________
| Section | Marks | Questions | Time | Strategy |
|---|---|---|---|---|
| A | 12 | 12 MCQ | 15 min | Recall the standard-angle values |
| B | 8 | 4 Short | 20 min | Write the ratio, then substitute |
| C | 12 | 4 Long | 30 min | Take the side that is common and simplify |
| D | 8 | 2 Case | 20 min | Draw the right triangle and label the sides |
| Review | — | — | 5 min | Check every identity step |
General Instructions: All questions are compulsory. Section A has 12 MCQs of 1 mark each. Internal choice is given in Q16 of Section B, Q19 and Q20 of Section C, and Q22 of Section D — attempt any ONE of the two options. Give reasons for each step in proofs. Use the exact values of the trigonometric ratios of 0°, 30°, 45°, 60° and 90°. A rough sketch is enough for the case-study parts.
The other two complementary relations follow the same pattern: cos(90° − θ) = sin θ and tan(90° − θ) = cot θ.
Section A — Multiple Choice (12 × 1 = 12 marks)
Choose the single best option. Every solution names the common trap behind the tempting wrong option.
Total: 1 Mark
The side opposite ∠A is BC, and the hypotenuse is AC.
sin A = BC / AC
Note: BC / AB is tan A, not sin A.
Total: 1 Mark
cos2A = 1 − sin2A
cos2A = 1 − (3/5)2
cos2A = 1 − 9/25
cos2A = 16/25
cos A = 4/5 (A is acute, so cos A > 0)
Total: 1 Mark
cos 60° = 1/2
So sin 30° + cos 60°
= 1/2 + 1/2
= 1
Total: 1 Mark
cot 45° = 1
So tan 45° + cot 45°
= 1 + 1
= 2
Total: 1 Mark
cos 30° = √3/2
So sin 60° × cos 30°
= (√3/2) × (√3/2)
= (3)/(4)
= 3/4
Total: 1 Mark
Given sin A = cos B
So cos(90° − A) = cos B
Since both angles are acute, 90° − A = B
Therefore A + B = 90°
Total: 1 Mark
This is a standard identity, true for all values of θ.
Total: 1 Mark
Rearranging: sec2θ − tan2θ = 1
So the value is 1 for every angle θ.
Total: 1 Mark
Take opposite = 3 and adjacent = 4.
hypotenuse = √(32 + 42) = √(9 + 16) = √25 = 5
sin A = opposite / hypotenuse = 3/5
Total: 1 Mark
sin 30° = 1/2
cosec 30° = 1 ÷ (1/2)
= 2
Total: 1 Mark
cos 0° = 1
So sin 0° + cos 0°
= 0 + 1
= 1
Total: 1 Mark
So sin θ = 1 / cosec θ
sin θ = 1 / 2
Section B — Short Answer (4 × 2 = 8 marks)
Total: 2 Marks
Take opposite = 4k and adjacent = 3k.
= √(16k2 + 9k2)
= √(25k2)
= 5k
cos A = adjacent / hypotenuse = 3k / 5k = 3/5
Total: 2 Marks
cos 30° = √3/2 · sin 60° = √3/2
= 1/4 + 3/4
= 4/4
= 1
Total: 2 Marks
Divide both sides by cos θ:
tan θ = 1
So θ = 45°
sin 45° = 1/√2, so sin245° = 1/2
= 2(1/2) − 1
= 1 − 1
= 0
Total: 2 Marks
= 1 − (5/13)2
= 1 − 25/169
= 144/169
cos A = 12/13
= (5/13) ÷ (12/13)
= 5/12
Total: 2 Marks
cos 45° = 1/√2, so cos245° = 1/2
tan 60° = √3, so tan260° = 3
= 1/2 − 3/2 + 3
= −1 + 3
= 2
[Attempt any ONE of the two options of Q16]
Section C — Long Answer (4 × 3 = 12 marks)
Total: 3 Marks
= [sin2θ + (1 + cos θ)2] / [sin θ (1 + cos θ)]
= sin2θ + 1 + 2cos θ + cos2θ
= (sin2θ + cos2θ) + 1 + 2cos θ
= 1 + 1 + 2cos θ
= 2 + 2cos θ
= 2(1 + cos θ)
= 2 / sin θ
= 2 cosec θ
= RHS
Total: 3 Marks
cot A = 4/3
tan A = 1 / cot A = 3/4
(1 − 9/16) / (1 + 9/16)
= (7/16) / (25/16)
= 7/25
Total: 3 Marks
cosec θ − cot θ = (1 − cos θ)/sin θ
So the expression = (1 − cos θ)2 / [(1 − cos θ)(1 + cos θ)]
= (1 − cos θ)/(1 + cos θ)
Total: 3 Marks
= (sin2θ + cos2θ)2 − 2 sin2θ cos2θ
= (1)2 − 2 sin2θ cos2θ
= RHS
[Attempt any ONE of the two options of Q19]
Total: 3 Marks
Let t = tan θ.
t + 1/t = 2
Multiply by t: t2 + 1 = 2t
(t − 1)2 = 0
t = 1
So tan θ = 1 and θ = 45°
cos 45° = 1/√2
Total: 3 Marks
cosec2θ = 1/sin2θ
= (sin2θ + cos2θ) / (sin2θ cos2θ)
LHS = 1 / (sin2θ cos2θ)
= (1/cos2θ) · (1/sin2θ)
= sec2θ cosec2θ
= RHS
[Attempt any ONE of the two options of Q20]
Section D — Case Study (2 × 4 = 8 marks)
(a) Find the length of the hypotenuse AC.
(b) Write the values of sin A and cos A.
(c) Find the value of sin2A + cos2A and name the identity you used.
Schematic — not to scale
Total: 4 Marks
= 72 + 242
= 49 + 576
= 625
AC = √625 = 25 cm
Side adjacent to ∠A = AB = 7 cm
Hypotenuse = AC = 25 cm
sin A = 24/25
cos A = 7/25
= 576/625 + 49/625
= 625/625
= 1
Identity used: sin2θ + cos2θ = 1
(a) Find QR.
(b) Write sin P, cos P and tan P.
(c) Find the value of 2 sin P cos P.
Schematic — not to scale
(a) Find AC.
(b) Write sin A, cos A and tan A.
(c) Find the value of (sin A)/(cos A) and state what it equals.
Schematic — not to scale
Total: 4 Marks
QR2 = 132 − 52
= 169 − 25
= 144
QR = 12 cm
Adjacent to ∠P = PQ = 5 cm
Hypotenuse = PR = 13 cm
sin P = 12/13
cos P = 5/13
tan P = 12/5
= (2 × 12 × 5) / (13 × 13)
= 120/169
Total: 4 Marks
= 82 + 152
= 64 + 225
= 289
AC = √289 = 17 cm
Adjacent to ∠A = AB = 8 cm
Hypotenuse = AC = 17 cm
sin A = 15/17
cos A = 8/17
tan A = 15/8
= (15/17) × (17/8)
= 15/8
This equals tan A.
[Attempt any ONE of the two options of Q22]
Answer Key & Marks (tap to open)
| Q | Section | Marks | Answer |
|---|---|---|---|
| 1 | A | 1 | (b) BC / AC |
| 2 | A | 1 | (c) 4/5 |
| 3 | A | 1 | (b) 1 |
| 4 | A | 1 | (d) 2 |
| 5 | A | 1 | (a) 3/4 |
| 6 | A | 1 | (c) 90° |
| 7 | A | 1 | (d) 1 |
| 8 | A | 1 | (c) 1 |
| 9 | A | 1 | (a) 3/5 |
| 10 | A | 1 | (d) 2 |
| 11 | A | 1 | (b) 1 |
| 12 | A | 1 | (a) 1/2 |
| 13–16 | B | 8 | Written — see the marking scheme under each question (Q16 has a choice) |
| 17–20 | C | 12 | Written — see the marking scheme under each question (Q19, Q20 have a choice) |
| 21–22 | D | 8 | Written — see the marking scheme under each question (Q22 has a choice) |
| Total | 40 | Section A 12 + Sections B–D 28 | |
Self-assessment for Sections B, C and D (28 marks)
The MCQ section is scored for you. For the written questions, tick each one you got right out of the marks shown, and add them up honestly.
Written total: 0 / 28
Paper total: score Section A above, then add your 0 written marks for the total out of 40.
Questions students ask about this paper
- Is this the actual CBSE board question paper?
- No. It is original board-pattern practice built to the CBSE Class 10 Maths syllabus 2026-27 (NCERT Chapter 8, Introduction to Trigonometry). It is not a prediction of any examination paper.
- How is the paper scored?
- Section A (12 MCQs) is scored automatically when you press Submit & Score. Sections B, C and D are written, so you check them against the marking scheme under each question and tick the self-assessment box; the box then shows your total out of 40.
- What does the test cover?
- Trigonometric ratios, the values of the ratios at 0°, 30°, 45°, 60° and 90°, ratios of complementary angles, and the three basic identities — exactly the scope of NCERT Chapter 8.
- Do my answers stay if I reload the page?
- Yes. Your selected options and your mode are kept for this browser tab, so an interrupted session resumes where you left off.
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Written and checked by Pandey Sir · Pandey Sir’s BEST Classes, Dammaiguda, Hyderabad. Every answer on this paper is worked out and checked before publishing.
This is board-pattern practice for Introduction to Trigonometry, built to the current CBSE Class 10 Maths syllabus 2026-27. It is not a prediction of any examination paper.
Curriculum scope checked against cbseacademic.nic.in. Spotted a mistake? Tell us here.
