Step by Step
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arcsin — restricted range
arcsin has domain [−1,1] and range [−90°,90°], covering Quadrants IV and I. This restriction means arcsin only ever returns one specific angle, even though multiple angles could have the same sine value.
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arccos — restricted range
arccos has domain [−1,1] and range [0°,180°], covering Quadrants I and II.
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arctan — restricted range
arctan has domain of all real numbers and range (−90°,90°), covering Quadrants IV and I, but — notably — never actually reaching the endpoints ±90° themselves.
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Composition rules
sin(arcsin(x)) always equals x, for any valid x in the domain. But arcsin(sin(x)) only equals x if x itself is already within arcsin's restricted range [−90°,90°] — otherwise, arcsin(sin(x)) gives you the equivalent angle within that restricted range instead of x itself.
Applied Walkthrough
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Evaluate arcsin(1/2). Even though both 30° and 150° have a sine of 1/2, arcsin can only return one value — the one within its restricted range of [−90°,90°].
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Since 30° falls within [−90°,90°] but 150° does not, arcsin(1/2) = 30° only.
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Now consider arcsin(sin(150°)). Since 150° is outside arcsin's restricted range, this does NOT simply return 150° back.
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Instead, since sin(150°) = 1/2, and arcsin(1/2) = 30° (the value within the restricted range), arcsin(sin(150°)) = 30° — not 150°, illustrating exactly why the restricted range matters for composition problems.
Exam Application
Exams test whether you know the specific restricted range for each inverse trig function, and whether you understand that arcsin(sin(x)) only returns x directly when x is already within arcsin's own restricted range — otherwise it returns a different, equivalent angle.
⚠ Common Trap
The most common trap is assuming arcsin(sin(x)) always simply returns x. This is only true when x is already within arcsin's restricted range [−90°,90°] — for any angle outside that range, the composition returns a different, equivalent value instead.
✓ Quick Self-Check
1. What is the range of arcsin?
[−90°, 90°].
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2. What is the range of arccos?
[0°, 180°].
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3. What is the range of arctan?
(−90°, 90°), open (never reaching the endpoints).
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4. What is arcsin(1/2)?
30° (not 150°, since only 30° falls within arcsin's restricted range).
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5. Does arcsin(sin(x)) always equal x?
No — only when x is already within arcsin's restricted range of [−90°,90°]; otherwise it returns a different, equivalent angle.
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