Inverse Trigonometric Functions

The inverse trigonometric functions arcsin, arccos and arctan "undo" the standard trig functions. Since sin, cos and tan are not one-to-one, we must restrict their domains to define proper inverses. This topic is tested at A-Level (9MA0).

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Why We Need Restricted Domains

sin(x) = 0.5 has infinitely many solutions: x = pi/6, 5pi/6, pi/6 + 2pi, ...

To define a unique inverse, we restrict each trig function to a domain where it is one-to-one (passes the horizontal line test).

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The Three Inverse Functions

arcsin (or sin^(-1))

Property	Value
Domain	-1 <= x <= 1
Range (principal values)	-pi/2 <= y <= pi/2
arcsin(0)	0
arcsin(1/2)	pi/6
arcsin(1)	pi/2
arcsin(-1)	-pi/2

arccos (or cos^(-1))

Property	Value
Domain	-1 <= x <= 1
Range (principal values)	0 <= y <= pi
arccos(1)	0
arccos(1/2)	pi/3
arccos(0)	pi/2
arccos(-1)	pi

arctan (or tan^(-1))