Math 103 – Rimmer
Inverse Trig. Functions
f ( x ) = arcsin x
or f ( x ) = sin −1 x
sin −1 x ≠
1
sin x
arcsin ( sin x ) = x
sin ( arcsin x ) = x
Math 103 – Rimmer
Inverse Trig. Functions
 −π π 
,  for which sin θ = w
Definition : θ = arcsin w is the number in 
 2 2
 3 π
Find arcsin 
 =
2

 3
Find the angle ( b / w
−π
2
and
π
2
) on the unit circle that goes with a point with a y − value of
3
.
2
 −1  −π
Find arcsin   =
6
 2 
Find the angle ( b / w
−π
2
and
π
2
) on the unit circle that goes with a point with a y − value of
Find arcsin ( −1) =
Find the angle ( b / w
−π
2
and
π
2
−1
.
2
−π
2
) on the unit circle that goes with a point with a y − value of
− 1.
Math 103 – Rimmer
Inverse Trig. Functions
f ( x ) = arccos x or f ( x ) = cos −1 x
cos −1 x ≠
1
cos x
arccos ( cos x ) = x
cos ( arccos x ) = x
Math 103 – Rimmer
Inverse Trig. Functions
Definition : θ = arccos w is the number in [ 0, π ] for which cos θ = w
 −1  2π
Find arccos   =
3
 2 
Find the angle ( b / w 0 and π ) on the unit circle that goes with a point with a x − value of
−1
.
2
 3 π
Find arccos 
 =
6
 2 
Find the angle ( b / w 0 and π ) on the unit circle that goes with a point with a x − value of
3
.
2
Find arccos ( −1) = π
Find the angle ( b / w 0 and π ) on the unit circle that goes with a point with a x − value of − 1.
Math 103 – Rimmer
Inverse Trig. Functions
f ( x ) = arctan x or f ( x ) = tan −1 x
lim arctan x =
x →− ∞
1
tan x
−π
2
lim arctan x =
x →∞
tan −1 x ≠
π
2
arctan ( tan x ) = x
tan ( arctan x ) = x
arctan x ≠
arcsin x
arccos x
Math 103 – Rimmer
Inverse Trig. Functions
 −π π 
Definition : θ = arctan w is the number in 
,  for which tan θ = w
 2 2
Add tangent information to the
−π
Find arctan ( −1) =
4
Find the angle ( b / w
−π
2
and
π
2
) such that the tangent of that angle is
unit circle by dividing y by x
tan ( π2 ) = undef
− 1.
tan ( π3 ) = 3
tan ( π4 ) = 1
Find arctan
Find the angle ( b / w
( 3) =
−π
2
and
π
2
π
3
) such that the tangent of that angle is
3.
tan ( 0 ) = 0
 −1 
−π
Find arctan 
 =
6
 3
Find the angle ( b / w
−π
2
and
π
2
) such that the tangent of that angle is
1
3
tan ( π6 ) =
tan ( −6π ) =
−1
.
3
−1
3
tan ( −4π ) = −1
tan ( −3π ) = − 3
tan ( −2π ) = undef
Why is the word arc used?
arclentgh s = rθ
On the unit circle r = 1,so s = θ
Math 103 – Rimmer
Inverse Trig. Functions