Chapter 4
4.7 Inverse Trigonometric Functions
Objectives
O Evaluate and graph inverse trigonometric
functions
O Evaluate compositions of trigonometric
functions
Inverse Functions
O You learned in Section 1.6 that each function has an inverse
O
O
O
O
O
relation, and that this inverse relation is a function only if the
original function is one-to-one.
The six basic trigonometric functions, being periodic, fail the
horizontal line test for one-to-oneness rather spectacularly.
However, you also learned in Section 1.4 that some functions
are
important enough that we want to study their inverse behavior
despite the fact that
they are not one-to-one. We do this by restricting the domain of
the original function
to an interval on which it is one-to-one, then finding the inverse
of the restricted function.
(We did this when defining the square root function, which is
the inverse of the quadratic function)
Properties
O However , when you restrict the domain to
𝜋
−
2
the interval
is increasing
O On the interval
≤𝑥≤
𝜋
2
the function y=sin x
𝜋
−
2
≤𝑥≤
on its full range of values
O On the interval −
to-one
𝜋
2
≤𝑥≤
𝜋
2
y= sinx takes
𝜋
2
y= sinx is one-
Inverse sine
O If you restrict the domain of y=sinx to the
𝜋
2
interval − ≤ 𝑥 ≤
𝜋
2
O , the restricted function is one-to-one. The
inverse sine function is denoted by y=arcsin x or
sin−1 𝑥
O THE NOTATION sin−1 𝑥 IS CONSISTENT WITH THE
INVERSE FUNCTION NOTATION
O The arcsin notation comes from the association
of a central angle with tis intercepted arc length
on a unit circle.
Evaluating inverse sine
function(without calculator)
O A. arcsin(-1/2)
O B.sin−1
3/2
O C. arcsin(0)
Evaluating with calculator
O Use a calculator in radian mode to evaluate
these inverse sine values:
O sin−1 −.81
Graph of inverse sine function
O Sketch a graph of y=arcsin x
y
-𝝅/𝟐
X=sin -1
y
-𝝅/𝟒
-𝝅/𝟔
0
2
-1/2
0
-2
𝝅/𝟔
1/2
𝝅/𝟒
2
2
𝝅/𝟐
1
Inverse cosine and inverse
tangent
O If you restrict the domain of y=cosx to the
interval (0,𝜋), the restricted function is oneto-one. The inverse cosine function is the
inverse of this restricted portion of the
cosine function y=arccos x or 𝑦 = cos −1 𝑥
Inverse tangent function
O you restrict the domain of y= tan x to the
interval(−𝜋/2, 𝜋/2)
O the restricted function is one-to-one. The
inverse tangent function
O is the inverse of this restricted portion of the
tangent function
Evaluating inverse cosine and
inverse tangent
O Find the exact value
O A. arccos 2/2
O B.cos −1 −1
O C. arctan0
O D. tan−1 −1
Composition of function
O We have already seen the need for caution
when applying the Inverse Composition
O Rule to the trigonometric functions and their
inverses (Examples 1e and 3c). The
following
O equations are always true whenever they are
defined:
O sin 1sin-11x22 = x cos (cos-11x22 = x tan
1tan-11x22 = x
Evaluating composition of
functions
Videos
Student guided practice
O Do problems 5-8 ,14 and 23 in your book
page 322
Homework
O Do problems 15, 19-23 in your book page
323