chapter 4 trigonometric functions 1. 4.7 inverse trigonometric functions objectives: evaluate...
TRANSCRIPT
2
4.7 Inverse Trigonometric Functions
Objectives: Evaluate inverse sine functions. Evaluate other inverse trigonometric functions.
Evaluate compositions of trigonometric functions.
3
Inverse Functions Recall that a function and its
inverse reflect over the line y = x. What must be true for a function
to have an inverse? It must be one-to-one, that is, it
must pass the horizontal line test.
4
More Inverse Functions Are sine, cosine, and tangent one-
to-one? If not, what must we do so that
these functions will have inverse functions?
Hint: Consider y = x2. We must restrict the domain of the original function.
5
Sine and Its Inverse f(x) = sin x does not pass the
Horizontal Line Test It must be restricted to find its
inverse. y
2
1
1
x
y = sin x
Sin x has an inverse function on this interval. 22
x
6
Inverse Sine Function The inverse sine function is defined by
y = arcsin x if and only if sin y = x.
The domain of y = arcsin x is [–1, 1]. The range of y = arcsin x is
_____________. Why are the domain and range
defined this way?
Angle whose sine is x
7
What Does “arcsin” Mean? In an inverse function, the x-values
and the y-values are switched. So, arcsin x means the angle (or arc)
whose sin is x. Notation for inverse sine
arcsin x sin -1 x
9
Graphing Arcsine Create a table for sin y = x for –π/2 ≤ y ≤
π/2.
Graph x on horizontal axis and y on vertical axis.
y –π/2 –π/4 –π/6 0 π/6 π/4 π/2x
11
Inverse Cosine Function f(x) = cos x must be restricted to
find its inverse.
Cos x has an inverse function on this interval.
y
2
1
1
x
y = cos x
x0
12
Inverse Cosine Function The inverse cosine function is
defined byy = arccos x if and only if cos y
= x.
The domain of y = arccos x is [–1, 1]. The range of y = arccos x is [0, π]. Notation for inverse cosine:
arccos x or cos -1 x
Angle whose cosine is x
14
Graphing Arccos Create a table for cos y = x for 0 ≤ y ≤ π.
Graph x on horizontal axis and y on vertical axis.
y 0 π/6 π/3 π/2 2π/3 5π/6 πx
16
Inverse Tangent Function f(x) = tan x must be restricted to find
its inverse.
Tan x has an inverse function on this interval.
y
x
2
3
2
32
2
y = tan x
22
x
17
Inverse Tangent Function The inverse tangent function is
defined byy = arctan x if and only if tan y
= x.
The domain of y = arctan x is (–∞, ∞). The range of y = arctan x is (–π/2, π/2). Notation for inverse tangent:
arctan x or tan -1 x
Angle whose tangent is x
20
Examples Evaluate using your calculator.
(What mode should the calculator be in?)
5.2arcsin.4
32.1arctan.3
19.0arcsin.2
75.0cos.1 1
22
Composition of Functions Given the restrictions specified in the
previous slide, we have the following properties of inverse trig functions.
sin(arcsin ) and arcsin(sin )
cos(arccos ) and arccos(cos )
tan(arctan ) and arctan(tan )
x x y y
x x y y
x x y y
24
Example 2Find the exact value of tan arccos .
3
x
y
3
2
adj2 2Let = arccos , then cos .3 hyp 3
u u
2 23 2 5
opp 52tan arccos tan3 adj 2
u
u