Interactive Simulators
Interactive Principal Values Zone
Use this tool to visualize the valid ranges (principal values) for each inverse function on the unit circle.
Principal Values
Inverse Sine (arcsin x)
Output Range[ -π/2, π/2 ]
Input Domain[-1, 1]
Valid QuadrantsI and IV
The inverse function returns the unique angle in the highlighted range that corresponds to the given ratio.
Interactive Inverse Composition Explorer
Interact with this tool to see the effect of composing trig and inverse trig functions, especially outside the principal value range.
Function Composition: $\arcsin(\sin(x))$
Trace how an angle passes through a function and its inverse. Notice what happens when the angle is outside the principal domain $[-\pi/2, \pi/2]$.
Input $x$
150°
Outside Principal Domain
Inner: $\sin(x)$
0.5000
Ratio (y-value)
Outer: $\arcsin(\textratio)$
30°
Mapped to Reference Angle
-180°-90° (Domain Limit)0°90° (Domain Limit)180°360°
Note: Because 150° is outside the principal domain of $[-\pi/2, \pi/2]$ (or $[-90^\circ, 90^\circ]$), the inverse sine function cannot return the original angle. It returns the equivalent angle in Quadrant I or IV that has the same sine value.
Solved Problems
Example
Problem 1: Find the exact value of .
Step-by-Step Solution
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Example
Problem 2: Find the exact value of .
Step-by-Step Solution
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Example
Problem 3: Evaluate .
Step-by-Step Solution
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Example
Problem 4: Evaluate .
Step-by-Step Solution
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Example
Problem 5: Find the exact value of .
Step-by-Step Solution
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