Inverse Trigonometric Functions (Class 12): 25 flashcards
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1. What does $\sin^{-1}x$ actually mean?
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It is the **inverse sine** (arcsine) — the angle whose sine is $x$. It does **not** mean $\frac{1}{\sin x}$.
2. What is the reciprocal of $\sin x$, and how is it written?
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$\csc x$, also written as $(\sin x)^{-1}$. It is completely different from $\sin^{-1}x$.
3. What does an inverse trigonometric function take as input and what does it return?
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It takes a **ratio** (a number) and returns an **angle**.
4. Why must we restrict the domain of $\sin x$ before defining $\sin^{-1}x$?
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Because $\sin x$ is periodic and not one-to-one over $\mathbb{R}$. Restricting to $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ makes it one-to-one so the inverse is a function.
5. Principal value range of $\sin^{-1}x$?
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$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
6. Principal value range of $\cos^{-1}x$?
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$[0, \pi]$
7. Principal value range of $\tan^{-1}x$?
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$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ — open interval, endpoints excluded.
8. Domain of $\sin^{-1}x$ and $\cos^{-1}x$?
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$[-1, 1]$
9. Domain of $\tan^{-1}x$ and $\cot^{-1}x$?
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$\mathbb{R}$ (all real numbers)
10. Domain of $\sec^{-1}x$ and $\csc^{-1}x$?
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$(-\infty,-1]\cup[1,\infty)$
11. State the identity $\sin^{-1}x + \cos^{-1}x = ?$
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$\frac{\pi}{2}$ for all $x \in [-1,1]$.
12. State the identity $\tan^{-1}x + \cot^{-1}x = ?$
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$\frac{\pi}{2}$ for all $x \in \mathbb{R}$.
13. When is $\sin(\sin^{-1}x) = x$ valid?
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For all $x \in [-1,1]$ — always true inside the domain.
14. When is $\sin^{-1}(\sin x) = x$ valid?
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Only when $x$ lies in the principal branch $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Otherwise the answer is the equivalent angle inside that range.
15. Formula for $\tan^{-1}x + \tan^{-1}y$ when $xy < 1$?
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$\tan^{-1}\frac{x+y}{1-xy}$
16. What happens to $\tan^{-1}x + \tan^{-1}y$ when $xy > 1$?
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The formula needs a $\pm\pi$ correction depending on the signs of $x$ and $y$.
17. Formula for $2\tan^{-1}x$ when $|x| < 1$?
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$\tan^{-1}\frac{2x}{1-x^2}$
18. How is the graph of $y = \sin^{-1}x$ related to $y = \sin x$?
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It is the reflection of $y = \sin x$ (restricted to $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$) across the line $y = x$.
19. What are the horizontal asymptotes of $y = \tan^{-1}x$?
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$y = \frac{\pi}{2}$ and $y = -\frac{\pi}{2}$
20. Evaluate $\tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3}$.
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$\frac{\pi}{4}$. Since $xy = \frac{1}{6} < 1$, apply the sum formula: $\tan^{-1}\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{6}} = \tan^{-1}1 = \frac{\pi}{4}$.
21. Why is the range of an inverse trig function called a 'cage'?
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Because every computed answer must lie inside that range. If your answer falls outside, it is wrong — the range catches the error.
22. Evaluate $\sin^{-1}\left(\sin\frac{5\pi}{6}\right)$.
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$\frac{\pi}{6}$. Since $\frac{5\pi}{6}$ is outside $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, first compute $\sin\frac{5\pi}{6} = \frac{1}{2}$, then $\sin^{-1}\frac{1}{2} = \frac{\pi}{6}$.
23. Where do inverse trig functions appear in real applications?
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Projectile motion (finding launch angles), AC circuit phase angles ($\tan^{-1}\frac{X_L - X_C}{R}$), computer graphics (`atan2` for direction angles), and statistics (Cauchy distribution CDF).
24. What is `atan2` and why is it useful?
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It is a two-argument version of $\tan^{-1}$ used in programming to recover direction angles from a vector, handling all four quadrants correctly.
25. State the identity $\sec^{-1}x + \csc^{-1}x = ?$
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$\frac{\pi}{2}$ for $|x| \geq 1$.
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