Exercise 9.1 Class 12 Math Solutions - Unit 9 Inverse Trigonometric Functions, Punjab Board
Step-by-step solutions to Exercise 9.1, Unit 9: Inverse Trigonometric Functions and Their Graphs, for Class 12 Punjab Board students. Covers restricted domains and principal value branches for inverse sine, cosine, tangent, cosecant, secant, and cotangent functions, evaluating exact values, and plotting their graphs, worked out question by question.
Exercise Overview
Exercise 9.1 opens Unit 9: Inverse Trigonometric Functions and Their Graphs in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum. Because all six standard trigonometric functions are periodic, they are many-to-one relations over their unrestricted domains and thus lack inverse functions in the strict mathematical sense. To establish unique, single-valued inverse functions, the domain of each trigonometric function is suitably restricted to a principal interval where the function is strictly monotonic (and hence bijective: one-to-one and onto).
The exercise systematically defines the principal value branches, domains, and ranges for all six inverse trigonometric functions: 1. Inverse Sine ((y = \sin^{-1} x)) with Domain ([-1, 1]) and Range ([-\frac{\pi}{2}, \frac{\pi}{2}]); 2. Inverse Cosine ((y = \cos^{-1} x)) with Domain ([-1, 1]) and Range ([0, \pi]); 3. Inverse Tangent ((y = \tan^{-1} x)) with Domain ((-\infty, \infty)) and Range ((-\frac{\pi}{2}, \frac{\pi}{2})); 4. Inverse Cosecant ((y = \csc^{-1} x)) with Domain ((-\infty, -1] \cup [1, \infty)) and Range ([-\frac{\pi}{2}, \frac{\pi}{2}] \setminus \{0\}); 5. Inverse Secant ((y = \sec^{-1} x)) with Domain ((-\infty, -1] \cup [1, \infty)) and Range ([0, \pi] \setminus \{\frac{\pi}{2}\}); 6. Inverse Cotangent ((y = \cot^{-1} x)) with Domain ((-\infty, \infty)) and Range ((0, \pi)). The graph of each inverse function is the reflection of the restricted function's curve across the line (y = x).
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 9.1 provides high-frequency Section B Short Questions (evaluating exact principal values such as (\sin^{-1}(-\frac{1}{2})), (\cos^{-1}(-\frac{\sqrt{3}}{2})), (\tan^{-1}(-1)); stating domain and range boundaries) and multiple choice questions testing graph symmetries. These step-by-step solutions provide rigorous radian evaluations, quadrant analysis, and clear graphical insights for every textbook problem.