Mathematics · Chapter 2
Study notes aligned to the official NEB syllabus.
Grade 12 trigonometry has two closely linked themes: the inverse circular functions, which turn a ratio back into an angle, and the general solution of trigonometric equations, which finds every angle satisfying a given trigonometric condition. Because sine, cosine and tangent are periodic, an equation such as $\sin x = \tfrac12$ has infinitely many solutions, and the aim is to describe all of them by a single formula. Mastery of both themes rests on a careful understanding of the domains and ranges on which each trigonometric function can be inverted.
A function has an inverse only where it is one-to-one. Sine, cosine and tangent are periodic and so are not one-to-one over the whole real line; we therefore restrict each to a principal branch on which it is one-to-one, and the inverse defined there is the principal value.
| Function | Domain | Principal range |
|---|---|---|
| $\sin^{-1}x$ | $[-1,1]$ | $\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$ |
| $\cos^{-1}x$ | $[-1,1]$ | $[0,\pi]$ |
| $\tan^{-1}x$ | $(-\infty,\infty)$ | $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$ |
| $\cot^{-1}x$ | $(-\infty,\infty)$ | $(0,\pi)$ |
| $\sec^{-1}x$ | $ | x |
| $\csc^{-1}x$ | $ | x |
Thus $\sin^{-1}x$ answers the question "which angle in $\left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]$ has sine $x$". For example $\sin^{-1}\tfrac12 = \tfrac{\pi}{6}$ and $\cos^{-1}\left(-\tfrac12\right) = \tfrac{2\pi}{3}$.
Grade 12 trigonometry has two closely linked themes: the inverse circular functions, which turn a ratio back into an angle, and the general solution of trigonometric equations, which finds every angle satisfying a given trigonometric condition. Because sine, cosine and tangent are periodic, an equation such as has infinitely many solutions, and the aim is to describe all of them by a single formula. Mastery of both themes rests on a careful understanding of the domains and ranges on which each trigonometric function can be inverted.
A function has an inverse only where it is one-to-one. Sine, cosine and tangent are periodic and so are not one-to-one over the whole real line; we therefore restrict each to a principal branch on which it is one-to-one, and the inverse defined there is the principal value.
| Function | Domain | Principal range |
|---|---|---|
| $ | x | |
| $ | x |
Thus answers the question "which angle in has sine ". For example and .