Exercise 10.2 Class 12 Math Solutions - Unit 10 Solution of Trigonometric Equations, Punjab Board
Step-by-step solutions to Exercise 10.2, Unit 10: Solution of Trigonometric Equations, for Class 12 Punjab Board students. Covers graphical solutions of transcendental and trigonometric equations by plotting simultaneous curves to determine intersection coordinates, along with applied real-life modeling problems involving simple harmonic motion, oscillating springs, alternating current (AC) waveforms, and tidal water height variations, worked out question by question.
Exercise Overview
Exercise 10.2 concludes Unit 10: Solution of Trigonometric Equations in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum by connecting theoretical trigonometric equations with graphical methods and real-world periodic modeling. When a trigonometric equation involves mixed transcendental terms (such as (\sin x = x), (\cos x = 2x), or (\tan x = 1 - x)), exact algebraic solutions cannot be obtained using elementary algebraic manipulation. In such cases, the graphical method provides an intuitive, highly accurate geometric solution.
To solve an equation (f(x) = g(x)) graphically, students construct a table of values for two separate functions: (y_1 = f(x)) and (y_2 = g(x)) across a specified domain (such as ([-\pi, \pi]) or ([0, 2\pi])). Both curves are plotted on the same coordinate axes using equal scaling. The points where the two curves intersect satisfy (y_1 = y_2), and the corresponding (x)-coordinates of these intersection points represent the real roots of the given equation.
Furthermore, Exercise 10.2 showcases practical applications of trigonometric equations to real-life physical phenomena. Trigonometric and sinusoidal functions naturally model cyclical behaviors such as Simple Harmonic Motion (SHM), mechanical vibrations of spring-mass systems, oscillation periods of clock pendulums, instantaneous voltage and current in AC electrical circuits ((V(t) = V_0 \sin(\omega t + \phi))), acoustic sound waves, and seasonal tidal water depths. Students learn how to set up trigonometric models from physical boundary conditions and solve for unknown time intervals or critical positions.
For Punjab Board HSSC Part 2 examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 10.2 provides high-yield conceptual questions, graphical interpretation MCQs, and Section C real-life word problems. These step-by-step solutions present clear table-of-values breakdowns, intersection explanations, and complete mathematical modeling formulations for every textbook exercise question.