4.7 Inverse Trig Functions. Objective Evaluate and graph the inverse sine, cosine and tangent...
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Transcript of 4.7 Inverse Trig Functions. Objective Evaluate and graph the inverse sine, cosine and tangent...
4.7 Inverse Trig Functions
Objective
• Evaluate and graph the inverse sine, cosine and tangent function.
• Evaluate the compositions of trig functions
Inverse Sine Function
y
2
1
1
x
y = sin x
Sin x has an inverse function on this interval.
Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test.
f(x) = sin x does not pass the Horizontal Line Test
and must be restricted to find its inverse.
Take a look at the piece of the graph in the red frame.
x
yIf we restrict the domain to the interval [-π/2, π/2]. The following properties hold:
1. On the interval [-π/2, π/2], the function y = sin x is increasing.
2. On the interval [-π/2, π/2],y = sin x takes on its full range of values, [-1, 1].
Lets zoom in and look at some key points in this section.
x
yy = sin(x)
3. On the interval [-π/2, π/2],y = sin x is one-to-one.
–1.5 1.5
–
The inverse sine function is defined byy = arcsin x if and only if sin y = x.
Angle whose sine is x
The domain of y = arcsin x is [–1, 1].
Example:
1a. arcsin2 6
1 is the angle whose sine is .6 2
1 3b. sin2 3
3sin3 2
This is another way to write arcsin x.
The range of y = arcsin x is [–/2 , /2].
Inverse Cosine Function
Cos x has an inverse function on this interval.
f(x) = cos x must be restricted to find its inverse.
The cosine function is decreasing and one-to-one on the interval [0, π]
y
2
1
1
x
y = cos x
x
yy = arccos(x)
The inverse cosine function is defined byy = arccos x if and only if cos y = x.
Angle whose cosine is x
The domain of y = arccos x is [–1, 1].
Example: 1a.) arccos2 3
1 is the angle whose cosine is .3 2
1 3 5b.) cos2 6
35cos6 2
This is another way to write arccos x.
The range of y = arccos x is [0 , ].
Inverse Tangent Functionf(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
x
y
The inverse tangent function is defined byy = arctan x if and only if tan y = x.
Angle whose tangent is x
Example: 3a.) arctan
3 6 3 is the angle whose tangent is .
6 3
1b.) tan 33 tan 3
3
This is another way to write arctan x.
The domain of y = arctan x is .( , ) The range of y = arctan x is (–/2 , /2).
Composition of Functions:f(f –1(x)) = x and (f –1(f(x)) = x.
If –1 x 1 and – /2 y /2, thensin(arcsin x) = x and arcsin(sin y) = y.
If –1 x 1 and 0 y , thencos(arccos x) = x and arccos(cos y) = y.
If x is a real number and –/2 < y < /2, thentan(arctan x) = x and arctan(tan y) = y.
Example: tan(arctan 4) = 4
Inverse Properties:
Example:
a. sin–1(sin (–/2)) = –/2
1 5b. sin sin3
53 does not lie in the range of the arcsine function, –/2 y /2.
y
x
53
3
5 23 3 However, it is coterminal with
which does lie in the range of the
arcsine function.
1 15sin sin sin sin3 3 3
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