inverse trigonometric functions digital lesson. copyright © by houghton mifflin company, inc. all...
TRANSCRIPT
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
Homework
• 1-9 odd, 16-20 even, 29-45 odd, 48-54 even, 81, 83, 85
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3
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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
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].
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
Inverse Cosine Function
Cos x has an inverse function on this interval.
f(x) = cos x must be restricted to find its inverse.
y
2
1
1
x
y = cos x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
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 , ].
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
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].
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
Graphing Utility: Graph the following inverse functions.
a. y = arcsin x
b. y = arccos x
c. y = arctan x
–1.5 1.5
–
–1.5 1.5
2
–
–3 3
–
Set calculator to radian mode.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
Graphing Utility: Approximate the value of each expression.
a. cos–1 0.75 b. arcsin 0.19
c. arctan 1.32 d. arcsin 2.5
Set calculator to radian mode.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
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:
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12
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