Exercise 10.1 Solutions - Class 11 Mathematics Unit 10 Trigonometric Identities

Solved problems and step-by-step derivations for Exercise 10.1, Unit 10: Trigonometric Identities.Complete solutions covering the Distance Formula, Fundamental Law of Trigonometry, Deductions, and Allied Angles for Punjab Board Class 11 (PCTB).

This page provides complete, step-by-step solved solutions for Exercise 10.1 of Unit 10: Trigonometric Identities for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All proofs involving the distance formula, the Fundamental Law of Trigonometry cos(α - β) = cos α cos β + sin α sin β, deductions for sum/difference identities, and evaluations of trigonometric functions of allied angles without using tables or calculators are thoroughly worked out for HSSC Part 1 exam success.
Back
Exercise 9.2
Next
Exercise 10.2
Advertisement

Exercise Overview

Exercise 10.1 opens Unit 10: Trigonometric Identities in Class 11 Mathematics, establishing the foundational analytical framework for angle addition, subtraction, and periodic transformations in circular trigonometry.

The exercise begins with the rigorous geometric proof of the Fundamental Law of Trigonometry, cos(α - β) = cos α cos β + sin α sin β, constructed on a unit circle using the Cartesian distance formula between terminal points. From this fundamental relation, deductive proofs unfold for cos(α + β), sin(α ± β), and tan(α ± β). Students also master the algebra of Allied Angles (angles of the form n(π/2) ± θ), learning to quickly determine function changes (co-function identities for odd multiples of π/2) and sign allocation based on standard quadrant analysis.

In Punjab Board HSSC Part 1 examinations, Exercise 10.1 is among the most heavily tested sections. The geometric proof of the Fundamental Law and allied angle simplifications without calculators appear regularly in Section B short conceptual questions and Section C theorem-based long questions across Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, and other BISE boards.

FAQs

Are these Exercise 10.1 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 is the Fundamental Law of Trigonometry?
The Fundamental Law of Trigonometry states that cos(α - β) = cos α cos β + sin α sin β for any two angles α and β, proven geometrically using unit circles and the distance formula.
How is the distance formula used to prove the Fundamental Law of Trigonometry?
By placing angles on a standard unit circle, coordinates of terminal points are defined as (cos θ, sin θ). Equating chord lengths between rotated arcs using the Euclidean distance formula proves the identity.
What are Allied Angles in trigonometry?
Allied Angles are angles whose sum or difference with angle θ is an integer multiple of 90° (π/2 radians), such as (π/2 ± θ), (π ± θ), (3π/2 ± θ), and (2π ± θ).
What deductions are derived from the Fundamental Law cos(α - β)?
Key deductions include cos(α + β) = cos α cos β - sin α sin β, sin(α ± β) = sin α cos β ± cos α sin β, and tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β).
Can students read these Exercise 10.1 notes online?
Yes, students can read all Exercise 10.1 step-by-step solved questions and identity proofs online on MaryamNotes.pk across all digital devices without downloading.
Are these solutions helpful for Punjab Board Class 11 examinations?
Yes, proving the Fundamental Law and evaluating trigonometric ratios of allied angles without calculators are essential Section B and Section C questions in Punjab Board exams.
Are these Class 11 Exercise 10.1 notes completely free to access?
Yes, all solved exercises, textbook proofs, and revision guides on MaryamNotes.pk are 100% free of charge for students and teachers across Punjab.