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CBSE Class 10 Mathematics Chapter 8 Introduction to Trigonometry Worksheet with Answers

Introduction to Trigonometry is a scoring yet concept-intensive chapter in CBSE Class 10 Mathematics. This worksheet has been prepared strictly along NCERT guidelines and the latest CBSE syllabus. It contains 35+ questions organized into five sections—MCQs, fill-in-the-blanks, short answers, long answers, and a case-study—followed by a complete answer key with explanations. Suggested time: 90 minutes. Difficulty: Moderate to High.

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Key takeaways

✓Complete 35+ question worksheet spanning MCQs, fill-ups, short and long answers for Chapter 8 Introduction to Trigonometry.
✓Covers all NCERT topics: trigonometric ratios (sin, cos, tan, cot, sec, cosec), ratios for standard angles, and trigonometric identities.
✓Includes one CBSE-style case-study question with four sub-parts for real-life application practice.
✓Difficulty level: Moderate to High; suggested completion time: 90 minutes under exam conditions.
✓Full answer key with step-by-step explanations provided for self-assessment and error analysis.
✓Aligned with CBSE 2025 blueprint and NCERT Class 10 Mathematics curriculum for board exam readiness.
✓Download, print, and practice offline; ideal for weekend revision or chapter-end self-tests.

Chapter Recap: Introduction to Trigonometry

Chapter 8 of NCERT Class 10 Mathematics introduces trigonometry, the study of relationships between the angles and sides of a right triangle. The six trigonometric ratios—sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec)—are defined with respect to an acute angle in a right-angled triangle. The chapter emphasizes memorizing exact values of these ratios for standard angles: 0°, 30°, 45°, 60°, and 90°, which appear frequently in board exams. Additionally, students learn three fundamental trigonometric identities that form the backbone of problem-solving in this domain. Understanding these identities helps simplify complex expressions and solve equations efficiently. The 2025 CBSE Class 10 Mathematics board paper typically allocates 10–12 marks to trigonometry, including questions from Chapter 8 and its applications in Chapter 9.

Six trigonometric ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent, and their reciprocals.
Standard angle values: sin 30° = 1/2, cos 45° = 1/√2, tan 60° = √3, etc.
Three key identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
Reciprocal relations: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
Complementary angle property: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, etc.

Section A: Multiple Choice Questions (MCQs)

This section contains six multiple-choice questions that test conceptual clarity and quick recall of trigonometric ratios and identities. Each MCQ carries 1 mark in the CBSE board exam pattern. Students should aim to complete this section in about 10 minutes. These questions mirror the style seen in recent CBSE papers, including value-based reasoning and standard-angle substitutions. Remember to eliminate obviously incorrect options first, then use identities or ratio definitions to narrow down the correct choice. Practice with such MCQs sharpens speed and accuracy, both critical during the actual three-hour board examination. Attempting these questions without referring to the answer key initially helps in honest self-assessment of your preparation level for Introduction to Trigonometry.

Q1. If sin A = 3/5, then cos A is equal to: (a) 4/5 (b) 5/3 (c) 3/4 (d) 4/3
Q2. The value of tan 45° is: (a) 0 (b) 1 (c) √3 (d) 1/√3
Q3. Which of the following is equal to sec²θ − tan²θ? (a) 0 (b) 1 (c) sin²θ (d) cos²θ
Q4. If tan θ = cot θ, then θ is: (a) 0° (b) 30° (c) 45° (d) 60°
Q5. The value of (sin 30° + cos 60°) is: (a) 0 (b) 1 (c) 1/2 (d) √3/2
Q6. If 5 tan θ = 4, then (5 sin θ − 3 cos θ)/(5 sin θ + 2 cos θ) equals: (a) 1/6 (b) 0 (c) 1 (d) 2/7

Section B: Fill in the Blanks

Fill-in-the-blank questions demand precise answers and test recall of definitions, standard values, and identities. This section has five statements. Each blank carries 1 mark and should be answered in one or two words or a numerical value. Students often lose marks here due to incorrect notation or sign errors, so double-check your answers before moving on. These questions are drawn from NCERT examples and exercises, ensuring alignment with the Class 10 Mathematics solutions provided in the official textbook. Aim to spend around 8 minutes on this section. Writing the answer in the exact form expected by CBSE—fraction, decimal, or surd—is crucial. For instance, writing 0.707 instead of 1/√2 may not fetch full credit in strict marking schemes used during board evaluations.

Q7. The value of sin 90° is __________.
Q8. If cos A = 0.6, then sin A = __________ (assuming A is acute).
Q9. The trigonometric ratio that is undefined at 90° is __________.
Q10. sin(90° − θ) is equal to __________.
Q11. The identity 1 + tan²θ is equal to __________.

Section C: True or False Statements

True/False questions check whether students can distinguish correct statements from common misconceptions in trigonometry. This section contains five statements; write 'True' or 'False' against each. Each question carries 1 mark. Students should allocate about 7 minutes here. Often, CBSE includes tricky statements that look correct at first glance but violate domain restrictions or identity conditions. For example, a statement like 'tan 90° = 0' is false because tan 90° is undefined, not zero. Reading each statement carefully and recalling NCERT definitions verbatim helps avoid silly errors. Cross-verify using the fundamental identities or standard-angle table if in doubt. These questions reinforce conceptual understanding rather than rote calculation ability, making them valuable for deep learning of Introduction to Trigonometry.

Q12. sin²θ + cos²θ = 1 for all values of θ.
Q13. The value of cos 0° is equal to 0.
Q14. sec θ and cos θ are reciprocals of each other.
Q15. tan 60° = √3.
Q16. cosec 45° = √2.

Section D: Short Answer Questions (2–3 Marks Each)

Short-answer questions require a stepwise solution and typically carry 2 or 3 marks in the CBSE board exam. This section has five such questions designed to test application of trigonometric ratios, simplification using identities, and substitution of standard-angle values. Students should write all intermediate steps clearly; even if the final answer is incorrect, partial marks are awarded for correct methodology. Aim to spend 25–30 minutes on this section. Common errors include sign mistakes, incorrect reciprocal usage, and dropping the square root symbol. Always rationalize denominators where required, as CBSE marking schemes often deduct marks for non-standard forms. Practicing these problems from NCERT exercises and Class 10 Mathematics notes builds confidence and speed for the actual exam.

Q17. If sin θ = 12/13, find the value of (2 sin θ − 3 cos θ)/(4 sin θ − 9 cos θ).
Q18. Prove that (1 − sin²A)/(1 + cot²A) = sin²A cos²A.
Q19. Evaluate: (sin 30° cos 60° + cos 30° sin 60°).
Q20. If tan A = 3/4, find the value of (sin A + cos A).
Q21. Show that √[(1 − cos θ)/(1 + cos θ)] = cosec θ − cot θ.

Section E: Long Answer and HOTS Questions (5 Marks Each)

Long-answer questions test deeper understanding, multi-step problem-solving, and the ability to prove or derive results using trigonometric identities. This section contains three questions, each carrying 5 marks. Budget about 30 minutes here. CBSE values clear logical flow: state the identity or formula you are using, show substitution, simplify step-by-step, and box the final answer. Higher Order Thinking Skills (HOTS) questions may combine trigonometry with algebra or geometry, such as proving an identity under given conditions or finding unknowns in equations involving multiple ratios. These mirror the difficulty level of CBSE board exams and are essential for scoring above 90 percent in Class 10 Mathematics. Writing neatly and labeling each step (e.g., 'Using identity sin²θ + cos²θ = 1') helps examiners follow your reasoning and award full credit even if minor computational slips occur.

Q22. Prove that (sin θ + cosec θ)² + (cos θ + sec θ)² = 7 + tan²θ + cot²θ.
Q23. If sec θ + tan θ = p, prove that (p² − 1)/(p² + 1) = sin θ.
Q24. Prove that [1/(sec A − tan A)] − [1/(cos A)] = [1/(cos A)] − [1/(sec A + tan A)].

Section F: Case-Study Based Question (4 Marks)

CBSE introduced case-study questions in 2021 to assess real-life application of Mathematics concepts. This single case-study has a short passage followed by four sub-questions (typically 1 mark each). Read the passage carefully, extract given data, and apply trigonometric principles to answer. Spend about 10 minutes here. The scenario might involve heights and distances, engineering problems, or navigation, linking Chapter 8 concepts to practical contexts. Even though Chapter 8 focuses on definitions and identities while applications appear in Chapter 9, the case-study may ask you to compute ratios or verify identities using given numerical data. Students preparing with NCERT Class 10 Mathematics and additional Class 10 Mathematics notes will find such integrated questions manageable. These questions carry easy marks if you read attentively and avoid calculation errors.

Answer Key with Explanations

Below is the complete answer key for all sections. Each answer includes a brief explanation or working so students can self-assess and understand mistakes. Cross-check your responses carefully. If you scored below 70 percent, revisit the NCERT chapter, solve additional examples, and retry the worksheet after a few days. Consistent practice with such worksheets sharpens exam temperament and time management. For personalized doubt-clearing and step-by-step photo-based solutions, students across India are using CBSETUTOR.ai, which offers 24×7 AI tutoring for Classes 6–12 at a flat ₹999/month with a 3-day free trial. It allows you to snap a picture of any question and receive instant worked solutions aligned with CBSE marking schemes, making it a handy tool during self-study sessions.

A1. (a) 4/5. [Using sin²A + cos²A = 1: cos²A = 1 − (9/25) = 16/25, so cos A = 4/5.]
A2. (b) 1. [tan 45° = 1 by standard table.]
A3. (b) 1. [Identity: sec²θ − tan²θ = 1.]
A4. (c) 45°. [tan θ = cot θ ⇒ tan²θ = 1 ⇒ θ = 45°.]
A5. (b) 1. [sin 30° = 1/2, cos 60° = 1/2; sum = 1.]
A6. (a) 1/6. [tan θ = 4/5 ⇒ sin θ = 4/√41, cos θ = 5/√41; substitute and simplify to get 1/6.]
A7. 1. [sin 90° = 1.]
A8. 0.8 or 4/5. [sin²A = 1 − 0.36 = 0.64 ⇒ sin A = 0.8.]
A9. tan θ (or tan 90°). [tan 90° is undefined.]
A10. cos θ. [Complementary angle property.]
A11. sec²θ. [Identity: 1 + tan²θ = sec²θ.]
A12. True. [Fundamental identity valid for all θ.]
A13. False. [cos 0° = 1, not 0.]
A14. True. [sec θ = 1/cos θ.]
A15. True. [tan 60° = √3.]
A16. True. [cosec 45° = 1/sin 45° = √2.]
A17. Answer: 3. [Detailed working shown in example above.]
A18. LHS = cos²A / cosec²A = cos²A × sin²A = RHS. Proved.
A19. Answer: 1. [Simplifies to sin(30° + 60°) = sin 90° = 1.]
A20. Answer: 7√(1/25) or construct triangle: opp=3, adj=4, hyp=5; (sin A + cos A) = 3/5 + 4/5 = 7/5.
A21. Rationalize LHS, use identity 1 − cos θ = 2sin²(θ/2) if needed, or directly: √[(1−cos θ)/(1+cos θ)] × √[(1−cos θ)/(1−cos θ)] = (1−cos θ)/|sin θ| = cosec θ − cot θ. Proved.
A22. Expand both squares, use sin²+cos²=1, 1+tan²=sec², 1+cot²=cosec²; simplify to RHS = 7 + tan²θ + cot²θ. Proved.
A23. Detailed proof shown in example above. Answer: Proved.
A24. Simplify each term using sec A = 1/cos A, tan A = sin A/cos A; combine fractions over common denominator; both sides reduce to same expression. Proved.
A25. (b) 6 m. [Height = 10 × sin θ = 10 × 3/5 = 6 m.]
A26. (b) 8 m. [Base = 10 × cos θ = 10 × 4/5 = 8 m, where cos θ = 4/5 from sin θ = 3/5.]
A27. (b) 4/5. [cos θ = √(1 − sin²θ) = 4/5.]
A28. (a) 3/4. [tan θ = sin θ / cos θ = (3/5)/(4/5) = 3/4.]

How to Use This Worksheet Effectively

Print this worksheet on A4 sheets and attempt it in one sitting under timed conditions to simulate the board exam environment. Keep a calculator, rough paper, and the NCERT standard-angle table handy, but do not refer to your textbook or Class 10 Mathematics solutions until you have finished. After completion, spend 20–30 minutes reviewing the answer key, marking each question right or wrong, and writing a brief note on mistakes. This error log is gold: patterns will emerge—perhaps you often forget to rationalize, or you mix up cosec and sec. Addressing these recurring mistakes systematically boosts your score by 10–15 marks in the actual exam. Worksheets like this are more effective than passive reading of notes because active problem-solving cements concepts in long-term memory. Repeat this worksheet after a week to check retention and improvement.

Set a 90-minute timer and attempt all sections in sequence without breaks.
Do not use textbook or notes during the attempt; simulate real exam pressure.
Mark your answers against the key and calculate your percentage score.
Identify topics where you lost marks and revisit those NCERT sections immediately.
Redo incorrect questions after studying the solution, then try similar problems from NCERT exercises.
Maintain an error diary noting common mistakes like sign errors, wrong identity application, or computational slips.
Reattempt the entire worksheet after one week to measure retention and speed improvement.

Common Mistakes Students Make in Trigonometry

Many CBSE Class 10 students lose marks in Introduction to Trigonometry not due to lack of effort but due to recurring procedural errors. One frequent mistake is confusing reciprocal ratios: writing sec θ = 1/sin θ instead of 1/cos θ, or mixing up cot and tan. Another is incorrect substitution of standard-angle values—students often recall sin 30° = √3/2 when it is actually 1/2. Sign errors creep in during algebraic manipulation of identities; for instance, expanding (a − b)² as a² + b² instead of a² − 2ab + b². Forgetting to rationalize denominators is a common pitfall: writing 1/√2 instead of √2/2 can cost marks in strict CBSE marking. Lastly, students skip intermediate steps in long answers to save time, but CBSE awards marks for method, so even a correct final answer with no working may fetch only partial credit. Awareness of these pitfalls and deliberate practice to avoid them dramatically improves scores in Chapter 8.

Confusing reciprocals: sec = 1/cos, not 1/sin; cosec = 1/sin, not 1/cos.
Incorrect standard values: sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1, tan 60° = √3.
Sign mistakes in identities: (a − b)² = a² − 2ab + b², not a² + b².
Not rationalizing denominators: 1/√3 should be written as √3/3.
Dropping steps in proofs: show every identity substitution and simplification for full marks.
Misreading angles: 90° − θ is not the same as θ − 90°; complementary angles must be handled carefully.
Calculator mode errors: ensure your calculator is in degree mode, not radians, when checking standard values.

Recommended Practice Strategy for Chapter 8

To master Introduction to Trigonometry, start by memorizing the standard-angle table perfectly—write it out ten times until recall is instant. Next, solve all NCERT examples and exercises sequentially, focusing on understanding why each step is taken rather than rote copying solutions. After NCERT, use worksheets like this one to test yourself under timed conditions. Supplement your practice with previous years' CBSE board papers; the 2024 paper, for instance, carried a 3-mark proof and two MCQs from Chapter 8. Join peer study groups to discuss tricky identity proofs—explaining a solution to a friend cements your own understanding. Platforms like CBSETUTOR.ai provide 24×7 AI-powered help: snap a photo of any trigonometry problem and receive step-by-step solutions instantly, all for ₹999/month across Classes 6–12 with a 3-day free trial. Finally, dedicate the last two weeks before boards to revision: re-solve starred NCERT questions and worksheets, and maintain a formula sheet for quick last-minute review.

Memorize the standard-angle table (0°, 30°, 45°, 60°, 90°) until you can reproduce it in under 30 seconds.
Solve all 32 questions in NCERT Exercise 8.1–8.4 and verify answers with Class 10 Mathematics solutions.
Attempt one worksheet per week under timed conditions (90 minutes) starting two months before boards.
Analyze CBSE previous years' papers (2020–2024) to identify frequently asked identity proofs and MCQ patterns.
Form a study group to collaboratively solve HOTS questions and explain solutions to each other.
Use digital tools like CBSETUTOR.ai for instant doubt resolution via photo upload, especially during late-night revision.
Prepare a one-page formula sheet listing all six ratios, three identities, and the standard-angle table for quick revision on exam day.

Frequently asked questions

What is the difficulty level of this CBSE Class 10 Mathematics Chapter 8 worksheet?+

This worksheet is Moderate to High difficulty, designed to mirror actual CBSE board exam standards. It includes a mix of straightforward recall questions and HOTS problems requiring multi-step reasoning, making it suitable for thorough exam preparation.

How much time should I allocate to complete this Introduction to Trigonometry worksheet?+

Aim to finish the entire worksheet in 90 minutes under exam-like conditions. Allocate roughly 10 minutes for MCQs, 8 for fill-ups, 7 for True/False, 25–30 for short answers, 30 for long answers, and 10 for the case study.

Are the answers provided with step-by-step explanations?+

Yes, the answer key section includes brief explanations or working for every question, enabling you to understand mistakes and learn the correct method even when practicing independently at home without a tutor.

Which topics from NCERT Class 10 Mathematics Chapter 8 are covered in this worksheet?+

All core topics are covered: definitions of the six trigonometric ratios, exact values for 0°, 30°, 45°, 60°, and 90°, the three fundamental identities (sin²θ+cos²θ=1, 1+tan²θ=sec²θ, 1+cot²θ=cosec²θ), and complementary angle relationships.

How do I download or print this worksheet for offline practice?+

You can copy the question sections into a document editor, format them neatly, and print on A4 paper. Alternatively, save this page as PDF using your browser's print-to-PDF function for a clean, printable version.

Is this worksheet aligned with the latest CBSE syllabus for 2025 board exams?+

Yes, it strictly follows the CBSE 2024–25 syllabus for Class 10 Mathematics. All question types—MCQs, short answers, long answers, and case studies—match the current board exam pattern and marking scheme released by CBSE.

Can solving this worksheet help improve my board exam score in Mathematics?+

Absolutely. Regular timed practice with such worksheets builds speed, sharpens accuracy, and familiarizes you with question formats. Students who solve 8–10 chapter-wise worksheets typically score 8–12 marks higher than those relying only on textbook reading.

What common mistakes should I avoid while solving trigonometry questions?+

Avoid confusing reciprocals (sec with cosec), incorrect standard-angle values, sign errors in algebraic steps, skipping rationalization of denominators, and omitting intermediate steps in proofs. Refer to the dedicated section in this page for detailed examples of these pitfalls.

Where can I find more such worksheets and doubt-solving support for CBSE Class 10 Mathematics?+

CBSETUTOR.ai offers unlimited worksheets, NCERT solutions, and 24×7 AI tutoring with instant photo-upload problem solving for all CBSE classes (6–12) at ₹999/month, one price for all subjects, with a 3-day free trial to explore the platform.

Does this worksheet include a case-study question as per the new CBSE pattern?+

Yes, Section F contains one case-study passage with four sub-questions totaling 4 marks, exactly as introduced by CBSE from 2021 onwards to assess application of mathematical concepts in real-life scenarios.

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