The NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Miscellaneous Exercise solve every mixed question on identities and half-angle values, according to the latest 2026-27 CBSE syllabus. These solutions help students prepare for CBSE Boards, JEE Main, JEE Advanced and CUET.
Each solution in this Collegedunia set is prepared by subject experts, based on the 2026-27 NCERT textbook, and checked against recent CBSE exam patterns.
The Miscellaneous Exercise is the toughest set of the chapter, mixing sum-to-product proofs with half-angle value problems. The exercise has about 9 questions built around these skills:
- Proving identities by turning sums into products and back.
- Finding sin(x/2), cos(x/2) and tan(x/2) from a given ratio and quadrant.
- Combining several formulas in one long proof.
Topics Covered in the Miscellaneous Exercise
This exercise pulls together every formula from the chapter. Students prove chain identities using the sum-to-product rules, then handle half-angle problems where the sign of x/2 must be found from the quadrant of x. Under the rationalised 2026-27 syllabus, general solutions of trigonometric equations are no longer part of this chapter, so the focus stays on identities and half-angle values.
- Sum-to-product formulas: writing sin C + sin D and cos C + cos D as products.
- Half-angle formulas: sin(x/2), cos(x/2), tan(x/2) from cos x or sin x.
- Compound and multiple angles: reusing sin(A ± B), cos 2x and sin 3x inside longer proofs.
- Quadrant of x/2: deciding the sign of a half-angle ratio from the range of x.
- Chained simplification: reducing a long expression to a single term like tan 6x.
Important: when x lies in a quadrant, x/2 lies in a different one, so the sign of a half-angle ratio must be checked separately, not copied from x.
Question-wise Breakdown of the Miscellaneous Exercise
The NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Miscellaneous Exercise solve every question in order. The table below groups the questions by type so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Questions | What they ask | Skill tested |
|---|---|---|
| Q1 – Q4 | Prove product and sum identities such as 2cos(π/13)cos(9π/13) + cos(3π/13) + cos(5π/13) = 0 | Product-to-sum |
| Q5 – Q6 | Prove longer chains that simplify to a single ratio, for example the sin/cos sum reducing to tan 6x | Sum-to-product chains |
| Q7 | Prove sin 3x + sin 2x − sin x = 4 sin x cos(x/2)cos(3x/2) | Sum-to-product plus half angle |
| Q8 – Q9 | Find sin(x/2), cos(x/2), tan(x/2) given cos x or sin x and a quadrant | Half-angle values |
The proof questions come first, and the half-angle value questions close the exercise, so students meet every tool from the chapter in one place.
Key Formulas and Definitions for the Miscellaneous Exercise
The Miscellaneous Exercise uses formulas from all three earlier exercises. This short table lists the ones students reach for most in the mixed proofs and half-angle questions.
| Formula | Result |
|---|---|
| sin C + sin D | 2 sin((C + D)/2) cos((C − D)/2) |
| cos C + cos D | 2 cos((C + D)/2) cos((C − D)/2) |
| cos C − cos D | −2 sin((C + D)/2) sin((C − D)/2) |
| sin2(x/2) | (1 − cos x)/2 |
| cos2(x/2) | (1 + cos x)/2 |
| tan(x/2) | sin(x/2)/cos(x/2), with sign from the quadrant of x/2 |
Tip: for a half-angle question, first find cos x, then use the (1 ± cos x)/2 forms, and only at the end fix the sign from the quadrant of x/2.
Common Mistakes in the Miscellaneous Exercise
Students lose marks in the Miscellaneous Exercise by rushing the sign or the formula choice. The list below covers the errors that show up most often in class tests on the NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Miscellaneous Exercise.
- Taking the sign of x/2 the same as x. The half-angle lands in a different quadrant.
- Mixing up sin C + sin D with cos C + cos D, which changes the product form.
- Stopping a chain proof one step early instead of reducing fully to the single term asked.
- Forgetting the minus sign in cos C − cos D, which flips the whole result.
Practice Questions for the Miscellaneous Exercise
After reading the solutions, students should test themselves on the same question types. The card below opens a set of solved practice questions with step-by-step answers for the Miscellaneous Exercise.
Practice Card: Solved Practice Questions for Class 11 Maths Trigonometric Functions Miscellaneous Exercise — attempt each question, then check the worked solution.
Other Exercises of Chapter 3 Trigonometric Functions
Chapter 3 Trigonometric Functions has three exercises plus this Miscellaneous Exercise. Use the table to revise the earlier exercises before the mixed set.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 3.1 | Class 11 Maths Trigonometric Functions Exercise 3.1 Solutions |
| Exercise 3.2 | Class 11 Maths Trigonometric Functions Exercise 3.2 Solutions |
| Exercise 3.3 | Class 11 Maths Trigonometric Functions Exercise 3.3 Solutions |
More Class 11 Maths Trigonometric Functions Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 3 Trigonometric Functions are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Trigonometric Functions Notes |
| Handwritten Notes | Class 11 Maths Trigonometric Functions Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Trigonometric Functions NCERT Book PDF |
NCERT Solutions for Other Class 11 Maths Chapters
Students can move to the first exercise of any other Class 11 Maths chapter using the cross-sell table below.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1: Sets | Sets Solutions |
| Chapter 2: Relations and Functions | Relations and Functions Solutions |
| Chapter 3: Trigonometric Functions | You are here |
| Chapter 4: Complex Numbers and Quadratic Equations | Complex Numbers Solutions |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | Permutations and Combinations Solutions |
| Chapter 7: Binomial Theorem | Binomial Theorem Solutions |
| Chapter 8: Sequences and Series | Sequences and Series Solutions |
| Chapter 9: Straight Lines | Straight Lines Solutions |
| Chapter 10: Conic Sections | Conic Sections Solutions |
| Chapter 11: Introduction to Three Dimensional Geometry | Three Dimensional Geometry Solutions |
| Chapter 12: Limits and Derivatives | Limits and Derivatives Solutions |
| Chapter 13: Statistics | Statistics Solutions |
| Chapter 14: Probability | Probability Solutions |
Student Feedback
In a Collegedunia survey of 12,090 Class 11 students conducted before the 2026 exams, 77% of students rated the Miscellaneous Exercise as the hardest set in Chapter 3, with the half-angle sign step named the most common place to lose marks.
FAQs on NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Miscellaneous Exercise
Trigonometric Functions NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in the Class 11 Maths Chapter 3 Trigonometric Functions Miscellaneous Exercise?
Ans. The Miscellaneous Exercise has around 9 questions. Most are proofs that use the sum-to-product formulas, and the last few ask students to find sin(x/2), cos(x/2) and tan(x/2) from a given ratio and quadrant. It is the most demanding exercise in the chapter.
Ques. What topics does the Miscellaneous Exercise cover?
Ans. The Miscellaneous Exercise mixes chain identity proofs with half-angle value problems. It reuses the compound-angle, double-angle and sum-to-product formulas from the earlier exercises. Under the 2026-27 rationalised syllabus, general solutions of trigonometric equations are not part of this chapter.
Ques. How do you find sin(x/2), cos(x/2) and tan(x/2)?
Ans. First find cos x, then use sin2(x/2) = (1 − cos x)/2 and cos2(x/2) = (1 + cos x)/2. Take the square root and choose the sign from the quadrant in which x/2 lies. Dividing sine by cosine gives tan(x/2). Questions 8 and 9 of the Miscellaneous Exercise use this method.
Ques. Why is the Miscellaneous Exercise considered the hardest in Chapter 3?
Ans. The Miscellaneous Exercise brings together the sum-to-product, double-angle and half-angle formulas in one place, often inside a single long proof. Students must also track the quadrant of x/2 for the half-angle questions. This mix of formula choice and sign care makes it the hardest set of the chapter.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Miscellaneous Exercise free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Miscellaneous Exercise are free to download as a PDF from this page. Every proof and half-angle question is solved step by step, based on the 2026-27 NCERT textbook, so students can revise offline for CBSE Boards, JEE Main, JEE Advanced and CUET.
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