Inverse Trig FunctionsJEE Main

Concept

Trig functions repeat, so to invert them we restrict each to one stretch where it is one-to-one. The output of an inverse trig function is its principal value — the single angle that lands in that agreed range.

Key formula

Derivation

$\sin$ is increasing on $[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$, so its inverse arcsin uses exactly that range. $\cos$ is decreasing on $[0,\pi]$, giving arccos its range.

$\tan$ runs over all reals on $(-\tfrac{\pi}{2},\tfrac{\pi}{2})$, so arctan accepts any input but its output stays strictly inside that open interval — the dashed asymptotes it never reaches.

Scenarios to explore

• Inverse Trig Functions — Principal values, domains and ranges of arcsin/arccos/arctan.

Real-world applications

• Recovering angles from ratios in geometry and physics.
• Phase angles in AC circuits and oscillations.
• Computer graphics (atan2 for direction).

JEE exam tips

• $\arcsin x + \arccos x = \tfrac{\pi}{2}$ for all $x\in[-1,1]$.
• $\arctan x + \operatorname{arccot} x = \tfrac{\pi}{2}$.
• Use the principal range to resolve sign ambiguities in identities.

Common mistakes

• Giving a value outside the principal range (e.g. arcsin returning $\tfrac{5\pi}{6}$).
• Assuming $\arcsin(\sin\theta)=\theta$ for every $\theta$ — only on the principal range.
• Forgetting arcsin/arccos are undefined outside $[-1,1]$.

Exam traps to avoid

• $\arccos$ is decreasing — larger input, smaller angle.
• $\arctan$ approaches $\pm\tfrac{\pi}{2}$ but never equals it.