INVERSE TRIG REVIEW SHEET
INVERSE SINE FUNCTION:
y = sin −1 x
⇔
sin y = x
Domain:
−1 ≤ x ≤ 1
Range:
−π
π
≤y≤
2
2
Properties:
sin −1 (sin x ) = x
(
)
sin sin −1 x = x
Derivative:
d
1
sin −1 x ) =
, −1 ≤ x ≤ 1
(
dx
1 − x2
INVERSE TANGENT FUNCTION:
y = tan −1 x
1
∫
Integral:
⇔
1 − x2
−∞ ≤ x ≤ ∞
−π
π
<y<
2
2
Range:
Properties:
tan −1 (tan x ) = x
(
)
tan tan −1 x = x
Limits:
lim tan −1 x =
π
x →∞
d
1
tan −1 x ) =
, -∞ ≤ x ≤ ∞
(
dx
1 + x2
Integral:
dx = arcsin x + C
tany = x
Domain:
Derivative:
−π
π
≤x≤
2
2
for -1 ≤ x ≤ 1
for
2
1
∫ 1+ x
2
,
−π
π
<x<
2
2
for all x
for
lim tan −1 x = −
x →−∞
dx = arctan x + C
π
2
INVERSE SECANT FUNCTION: y = sec −1 x
⇔
sec y = x
Domain:
−∞ < x ≤ −1 U 1 ≤ x < ∞
Range:
0≤ y<
Properties:
sec−1 ( sec x ) = x
for
sec ( sec−1 x ) = x
for
Derivative:
d
1
sec−1 x ) =
, x ≥1
(
dx
x x2 −1
Integral:
∫x
1
x2 −1
⎛ 1⎞ π
⎛ 1⎞
sec −1 x = cos −1 ⎜ ⎟ = − sin −1 ⎜ ⎟
⎝ x⎠ 2
⎝ x⎠
π
2
− tan −1 x
⎛ 1⎞
csc −1 x = sin −1 ⎜ ⎟
⎝ x⎠
3π
2
U π ≤ y<
dx = arcsec( x) + C for 0 ≤ arcsec x <
OTHER USEFUL IDENTITIES:
cot −1 x =
2
U π ≤ y<
3π
2
2
− ∞ < x ≤ −1 U 1 ≤ x < ∞
0≤ y<
π
π
π
2
U π ≤ arcsec x <
3π
2
Distinguishing between numbers and angles:
• For y = sin x : Given an angle x, what is sin x? (a number) • For y = arcsin x : Given a number, what is the angle y such that y = arcsin x ? TIP: When in doubt, draw a right triangle. What are you given? What do you want to find?
x
adj
y = arccos x ⇔ cos y =
1
hyp
sin y = 1 − x 2
→ cos y = x
1 − x2
tan y =
x
sec y =
cot y =
1
1
csc y =
1 − x2
1
x
x
1 − x2
y
x