Inverse Trig: y = a · arcsin(b x)
Inverse Trigonometric Functions · Class XII
Domain of arcsin is [-1, 1]. Scale x with b and watch where the curve is defined.
- Domain1
- Range1.5708
- y at x = 1/(2b)0.5236
- y = a · arcsin(b x)
Formulas in this lab
- Domain
- Range
- y at x = 1/(2b)
Frequently asked questions
▶What are the principal-value domains of $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$?
$\sin^{-1}: [-1, 1] \to [-\pi/2, \pi/2]$. $\cos^{-1}: [-1, 1] \to [0, \pi]$. $\tan^{-1}: \mathbb{R} \to (-\pi/2, \pi/2)$. These restricted ranges make the inverses single-valued functions, not relations.
▶How do I use the inverse trig domains lab?
Slide an input $x$ and the lab returns $\sin^{-1}(x)$, $\cos^{-1}(x)$ and $\tan^{-1}(x)$ in radians and degrees, plus the identity $\sin^{-1}(x) + \cos^{-1}(x) = \pi/2$. Try $x = 0.5$ to see $\sin^{-1}(0.5) = \pi/6$ and $\cos^{-1}(0.5) = \pi/3$.
▶Why does $\sin^{-1}(\sin(2\pi)) = 0$ and not $2\pi$?
Because $\sin^{-1}$ only outputs values in $[-\pi/2, \pi/2]$, and $\sin(2\pi) = 0$, so the inverse returns $0$. Students often write $2\pi$ and lose marks. The composition rule is $\sin^{-1}(\sin\theta) = \theta$ only when $\theta\in[-\pi/2, \pi/2]$.
▶Where does $\tan^{-1}$ appear in real applications?
Computing the angle of elevation: if a kite is at height $h$ and horizontal distance $d$, angle is $\theta = \tan^{-1}(h/d)$. Smartphone level apps, drone tilt sensors, and the autopilot in commercial aircraft all use the inverse tangent to recover angles from ratios.