Math 150 – Fall 2015
Section 8F
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Section 8F – Inverse Trig Functions
Inverse Cosine
y = cos x
To get an inverse of cosine, we will restrict the domain of y = cos x to [0, π] so that the
function is one-to-one on this restricted domain.
Definition. We define arccos as the function with domain [−1, 1] and range [0, π] such
that
arccos x = cos−1 x = y
if and only if
cos y = x
where y ∈ [0, π].
y = cos x, x ∈ [0, π]
y = arccos x
Example 1. Fill in the following table:
y = cos x, x ∈ [0, π]
Domain:
Range:
y = cos−1 x
Math 150 – Fall 2015
Section 8F
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Inverse Tangent
y = tan x
To get an inverse of tangent, we will restrict the domain of y = tan x to − π2 , π2
that the function is one-to-one on this restricted domain.
so
Definition. We define arctan as the function with domain (−∞, ∞) and range − π2 , π2
such that
arctan x = tan−1 x = y
where y ∈ − π2 , π2 .
y = tan x, x ∈ − π2 , π2
Example 2. Fill in the following table:
y = tan x, x ∈ − π2 , π2
Domain:
Range:
Example 3. Evaluate the following:
(a) cos−1
(b) tan−1
√
2
2
√
=
3=
if and only if
y = arctan x
y = tan−1 x
tan y = x
Math 150 – Fall 2015
Section 8F
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(c) cos cos−1 1 =
(d) cos−1 (cos 2π) =
(e) sec tan−1
−4
7
=
(f) cot cos−1
−5
7
=
Example 4. Exactly evaluate sin cos−1 (3x) if 0 ≤ x ≤ 31 .
Example 5. Fully simplify sin cos−1
−3
4
+ tan−1
−1
6
.