1. Inverse Trig Functions
I. Properties of Inverse Functions.
A) Inverse function: swap the x & y values.
1) (x , y)  (y , x) ) y = mx + b  x = my + b.
B) They reflect across the line y = x (45° angle line)
C) Symbol for inverse is f –1(x).
D) To be an inverse function, both the original [ f(x) ] and its inverse [ f –1(x) ] must be functions.
1) They must both pass the vertical line test.
2) Shortcut: f(x) must pass vertical & horizontal line test.
3) If they are both functions, they are called one-to-one.

2. Inverse Trig Functions
II. Inverse Sine Function.
A) The sine graph does not pass the one-to-one test
1) It fails the Horizontal line test.
B) It will pass if we restrict the domain.
y = sin x Domain = Range = [–1 , 1]
1) Now the sine function is one-to-one. y x
y = sin x

3. Inverse Trig Functions
II. Inverse Sine Function.
C) Inverse sin function: y = sin–1 x or y = arcsin x.
D) y = arcsin x comes from the formula (s = θ r).
where s = arc length, θ = angle (in radians), r = circle radius
1) y = arcsin x: means y is the angle whose arc length is x
E) Since y = sin x Domain = Range = [–1 , 1]
then the inverse y = sin–1 x or y = arcsin x has
1) Domain = [–1 , 1], Range =

4. Inverse Trig Functions
III. Inverse Cosine and Tangent Functions.
A) We must restrict these trig functions domains too.
y = cos x
1) Restrict the domain to
D = [0 , π] R = [–1 ,1]
2) y = arccos x , y = cos –1 x
D = [–1 ,1] R = [0 , π]
y = tan x
D = R = (-∞, +∞)
2) y = arctan x , y = tan–1 x
D = (-∞, +∞) R =

5. Inverse Trig Functions
IV. Evaluating Trig Inverse Functions.
A) For 30°, 45°, 60°, 90° angles, use the Unit Circle’s (x , y)
values to find what angle [ θ ] gives you that arctrig value.
1) sin = y values, cos = x values, tan = y/x values.
B) Remember to stay within the arctrig’s domain.
1) You might have to go in the -θ° direction.
a) =
2) You can subtract 2π from the angle to get a -θ°. a)
3) Some arctrig values are undefined (have no θ answer).