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November 13, 2014
Section 4.7 Inverse Trigonometric Functions
Objective: Evaluate the inverse trig. functions and the composition of trig.
functions.
Does sine or cosine have an inverse function? In other
words, do they pass the Horizontal Line Test?
Definitions:
Inverse Sine Fct.
y = arcsin x iff sin y = x
Domain
-1 ≤ x ≤ 1
Range
-π/2 ≤ y ≤ π/2
Inverse Cosine Fct.
y = arccos x iff cos y = x
-1 ≤ x ≤ 1
0≤y≤π
-∞ < x < ∞
-π/2 < y < π/2
Inverse Tangent Fct.
y = arctan x iff tan y = x
*Note: You can also use sin -1 x, cos -1 x, and tan -1 x for inverse fct.
y
x=siny
-π/2
-1
-π/4 (-√2)/2
-π/6 -1/2
0
is
tor
a
l
u
alc
en
re c
u
s
e wh
d
e
o
B
*
m
dian verses.
a
r
in
in
ing
h
p
a
gr
0
π/6
1/2
π/4
(√2)/2
π/2
1
November 13, 2014
y
x=cosy
0
1
π/6
(√3)/2
π/3
1/2
π/2
0
2π/3 -1/2
5π/6 (-√3)/2
π
-1
y
x=tany
-π/2
undef.
-π/4
-1
-π/6 (-√3)/3
0
0
π/6 (√3)/3
π/4
1
π/2
undef.
November 13, 2014
Remember this...
"The inverse fct. of x is the angle (or number) whose sin, cos, or tan is x"
Ex1: Find the exact value.
a) arcsin (-1)
b) arcsin (1/2)
c) arcsin (2)
Ex2: Find the exact value.
a) arccos (1/2)
b) cos -1 (√3 /2)
c) tan-1 (√3)
Find the angle whose sin is -1.
a) π/3
b) π/6
c) π/3
Ex3: Use a calculator to approximate the value (if possible).
Round to four decimal places.
a) cos-1 0.85
a) 0.5548
-1
b) sin 3.1415
b) Not Possible
*Note (for some calculators): To evaluate sec-1, csc-1 , cot-1,
use cos-1, sin-1, and tan-1.
Ex: sec-1 3.4 = cos-1 (1/3.4) = 1.272264...
From Section 1.6,
f(f-1(x)) = x and f -1(f(x)) = x for x in the domain.
This means for compositions of Trig. functions we have...
Inverse Properties
If -1 ≤ x ≤ 1 and -π/2 ≤ y ≤ π/2, then
sin(sin -1 x) = x and sin -1 (sin y) = y.
If -1 ≤ x ≤ 1 and 0 ≤ y ≤ π, then
cos(cos -1 x) = x and cos -1 (cos y) = y.
If x is a real number and - π/2 < y < π/2, then
tan(tan -1 x) = x and tan -1 (tan y) = y.
a) -π/2
b) π/6
c) undef. D:[-1, 1
November 13, 2014
Ex4: If possible find the exact value...
= sin -1 (sin(π/4)) = π/4
a) sin -1 (sin(3π/4))
b) cos (arccos 0)
=0
c) tan-1 (tan (5π/6))
= tan-1 (tan (-π/6)) = -π/6
Ex5: Write as an algebraic fct. in terms of x.
a) sin (arccos 3x), 0 ≤ x ≤ 1/3
b) cot (arccos 3x), 0 ≤ x ≤ 1/3
Let u = arccos 3x
think "cos ? = 3x"
Draw triangle.
sin u = √ 1-9x2
cot u =
c) sin (tan -1 (-1/2))
sin u = -√5 /5