Q: 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.

Question 1

What are the principal-value domains of $\sin^{-1}$, $\cos^{-1}$ and $	an^{-1}$?

Accepted Answer

$\sin^{-1}: [-1, 1] 	o [-\pi/2, \pi/2]$. $\cos^{-1}: [-1, 1] 	o [0, \pi]$. $	an^{-1}: \mathbb{R} 	o (-\pi/2, \pi/2)$. These restricted ranges make the inverses single-valued functions, not relations.

Question 2

How do I use the inverse trig domains lab?

Accepted Answer

Slide an input $x$ and the lab returns $\sin^{-1}(x)$, $\cos^{-1}(x)$ and $	an^{-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 $\cos^{-1}(0.5) = \pi/3$.

Question 3

Why does $\sin^{-1}(\sin(2\pi)) = 0$ and not $2\pi$?

Accepted Answer

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 $\sin^{-1}(\sin	heta) = 	heta$ only when $	heta\in[-\pi/2, \pi/2]$.

Question 4

Where does $	an^{-1}$ appear in real applications?

Accepted Answer

Computing the angle of elevation: if a kite is at height $h$ and horizontal distance $d$, angle is $	heta = 	an^{-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.

Inverse Trig: y = a · arcsin(b x)

Formulas in this lab

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