7.4
Inverse Trigonometric Functions
From looking at the graphs of the trig functions, we see that they fail the horizontal line
test spectacularly. However, if you restrict their domain, you can find an inverse for these
functions on this domain only.
Inverse Sine
The function sin is one-to-one when restricted to sin : − π2 , π2 → [−1, 1]. Thus there exists
a function sin−1 or arcsin
h π πi
sin−1 : [−1, 1] → − ,
2 2
arcsin x(= sin−1 x)
sin x
The arcsin function satisfies
arcsin(x) = y ⇔ sin(y) = x
π
π
≤x≤
2
2
sin(arcsin x) = x if − 1 ≤ x ≤ 1
arcsin(sin x) = x if −
Examples: Find (a)sin−1 ( 12 ), (b)arcsin(− 21 ), (c)sin−1 ( 32 ) and (d)cos(sin−1 ( 35 )).
solution:
(a) We know that sin( π6 ) = 12 , so sin−1 ( 12 ) = π6 .
(b) Likewise, sin(− π6 ) = − 21 , so arcsin(− 12 ) = − π6 .
(c) We know sin x is never 32 (it is never greater than 1), so sin−1 ( 32 ) is undefined.
p
(d) For u ∈ − π2 , π2 , we have cos(u) = 1 − sin2 u, so
s
3
3
−1
2
−1
cos(sin
)
=
1 − sin sin
5
5
s
2
3
=
1−
5
r
9
1−
=
25
r
16
=
25
4
=
5
Inverse Cosine
The function cos is one-to-one when restricted to cos : [0, π] → [−1, 1]. Thus there exists a
function cos−1 or arccos
cos−1 : [−1, 1] → [0, π]
cos x
arccos x(= cos−1 x)
The arccos function satisfies
arccos(x) = y ⇔ cos(y) = x
arccos(cos x) = x if 0 ≤ x ≤ π
cos(arccos x) = x if − 1 ≤ x ≤ 1
√ 3
−1
and (b) cos−1 (0).
Examples: Find (a) cos
2
solution:
(a) cos( π6 ) =
√
3
,
2
thus cos−1
√ 3
2
(b) cos( π2 ) = 0, thus cos−1 (0) =
= π6 .
π
2
Examples:
√
1. Show that sin(cos−1 (x)) = 1 − x2 .
p
solution: sin u = 1 − cos2 (u) for u ∈ [0, π]. Thus
p
√
sin(cos−1 (x)) = 1 − cos2 (cos−1 (x)) = 1 − x2
2. Show that tan(cos−1 (x)) =
solution: tan(u) =
sin u
,
cos u
√
1−x2
.
x
so
sin(cos−1 (x))
tan(cos (x)) =
=
cos(cos−1 (x))
−1
√
1 − x2
x
√
3. Show that sin(2 cos−1 (x)) = 2x 1 − x2
solution: sin 2u = 2 sin u cos u, so
√
√
sin(2 cos−1 (x) = 2 sin(cos−1 (x)) cos(cos−1 (x)) = 2 1 − x2 x = 2x 1 − x2
Inverse Tangent
The function tan is one-to-one when restricted to tan : − π2 , π2 → R. Thus there exists a
function tan−1 or arctan
h π πi
−1
tan : R → − ,
2 2
arctan x(= tan−1 x)
tan x
The arctan function satisfies
arctan(x) = y ⇔ tan(y) = x
π
π
≤x≤
2
2
tan(arctan x) = x if x ∈ R
arctan(tan x) = x if −
Examples: Find (a) tan−1 (1), (b) tan−1
√
3 and (c)arctan(−20)
solution:
(a) tan( π4 ) = 1, thus tan−1 (1) = π4 .
√
√
(b) tan( π3 ) = 3, thus tan−1 ( 3) =
π
3
(c) Using a calculator, we find arctan(−20) ≈ 1.52084
Other Inverse Trig Functions
The other trig functions cot, csc and sec also have inverses when restricted to suitable
domains, namely cot−1 , csc−1 and sec−1 . You don’t need to worry about graphing these.
Just keep in mind:
1
cot−1 6=
tan−1
1
csc−1 6=
sin−1
1
sec−1 6=
cos−1