Exercise 10.4 Solutions - Class 11 Mathematics Unit 10 Trigonometric Identities

Step-by-step solved problems and identity proofs for Exercise 10.4, Unit 10: Trigonometric Identities.Comprehensive solutions covering Product-to-Sum and Sum-to-Product conversions of sines and cosines for Punjab Board Class 11 (PCTB).

This page provides complete, step-by-step solved solutions for Exercise 10.4 of Unit 10: Trigonometric Identities for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All problems converting products of sines and cosines into sums or differences, factorization of trigonometric sums into products, and constant numerical proofs (such as evaluating sin 20° sin 40° sin 60° sin 80° = 3/16) are thoroughly solved for HSSC Part 1 board exam preparation.
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Exercise Overview

Exercise 10.4 concludes Unit 10: Trigonometric Identities in Class 11 Mathematics, equipping students with essential bidirectional transformation formulas connecting trigonometric products with sums and differences.

The exercise systematically explores two symmetrical sets of formulas: the Product-to-Sum identities (2 sin α cos β, 2 cos α sin β, 2 cos α cos β, and -2 sin α sin β), which linearize trigonometric products; and the Sum-to-Product identities (sin P ± sin Q and cos P ± cos Q), which factorize sums into compact multiplicative terms. Students apply these identities to evaluate celebrated product constants (such as sin 20° sin 40° sin 60° sin 80° = 3/16 and cos 20° cos 40° cos 60° cos 80° = 1/16) and prove elegant algebraic identities involving multiple angles.

In Punjab Board HSSC Part 1 examinations, Exercise 10.4 is a hallmark topic for high-scoring long questions in Section C and conceptual short questions in Section B across Lahore, Rawalpindi, Faisalabad, Gujranwala, Multan, and other BISE boards. Mastering these transformation techniques provides essential mathematical dexterity for calculus, Fourier series, and physics wave mechanics.

FAQs

Are these Exercise 10.4 solutions according to the Punjab Board syllabus?
Yes, these solutions strictly follow the official Punjab Curriculum and Textbook Board (PCTB / PECTAA) Class 11 Mathematics syllabus for Unit 10 (Trigonometric Identities) for the 2026-2027 academic session.
What are the key formulas covered in Exercise 10.4 Class 11 Mathematics?
Exercise 10.4 covers Product-to-Sum identities (converting products of sines and cosines into sums/differences) and Sum-to-Product factorization identities.
What are the Product-to-Sum identities in trigonometry?
Key identities are 2 sin α cos β = sin(α + β) + sin(α - β), 2 cos α sin β = sin(α + β) - sin(α - β), 2 cos α cos β = cos(α + β) + cos(α - β), and -2 sin α sin β = cos(α + β) - cos(α - β).
What are the Sum-to-Product (factorization) identities for sine and cosine?
They are sin P + sin Q = 2 sin((P+Q)/2) cos((P-Q)/2), sin P - sin Q = 2 cos((P+Q)/2) sin((P-Q)/2), cos P + cos Q = 2 cos((P+Q)/2) cos((P-Q)/2), and cos P - cos Q = -2 sin((P+Q)/2) sin((P-Q)/2).
How do you prove constant numerical products like sin 20° sin 40° sin 60° sin 80° = 3/16?
Substitute known standard values like sin 60° = √3/2, pair remaining terms using 2 sin α sin β = cos(α - β) - cos(α + β), distribute, and simplify to obtain the exact constant.
Can students read these Exercise 10.4 solutions online?
Yes, students can read all step-by-step proofs and solved questions for Exercise 10.4 directly online on MaryamNotes.pk on any web browser without needing to download files.
Are these product-to-sum transformations important for Punjab Board exams?
Yes, questions involving numerical constant proofs (such as 20°-40°-60°-80° products) and quotient simplifications are standard Section B and Section C board exam problems.
Are these Class 11 Exercise 10.4 notes completely free?
Yes, all solved notes, identity derivations, and practice solutions on MaryamNotes.pk are 100% free and open to all students and educators.