4-6 Inverse Trigonometric Functions
Sketch the graph of each function.
17. y = arcsin x
SOLUTION: First, rewrite y = arcsin x in the form sin y = x.
to make a table of values.
Next, assign values to y on the interval
–
y
x = sin y
–
0
–1
0
1
Plot the points and connect them with a smooth curve.
18. y = sin – 1 2x
SOLUTION: First, rewrite y = sin
–1
2x in the form sin y = x.
to make a table of values.
Next, assign values to y on the interval
–
y
x=
sin y
–
0
0
Plot the points and connect them with a smooth curve.
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4-6 Inverse Trigonometric Functions
18. y = sin – 1 2x
SOLUTION: First, rewrite y = sin
–1
2x in the form sin y = x.
to make a table of values.
Next, assign values to y on the interval
–
y
x=
–
sin y
0
0
Plot the points and connect them with a smooth curve.
19. y = sin – 1 (x + 3)
SOLUTION: First, rewrite y = sin
–1
(x + 3) in the form sin y = x.
to make a table of values.
Next, assign values to y on the interval
y
0
Plot the points and connect them with a smooth curve.
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4-6 Inverse Trigonometric Functions
19. y = sin – 1 (x + 3)
SOLUTION: First, rewrite y = sin
–1
(x + 3) in the form sin y = x.
to make a table of values.
Next, assign values to y on the interval
y
0
Plot the points and connect them with a smooth curve.
20. y = arcsin x – 3
SOLUTION: First, rewrite y = arcsin x – 3 in the form sin y = x.
Next, assign values to y on the interval
y
x = sin(y + 3)
to make a table of values.
0
0.99
0.80
0.14
−0.60
−0.99
Plot the points and connect them with a smooth curve.
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4-6 Inverse Trigonometric Functions
20. y = arcsin x – 3
SOLUTION: First, rewrite y = arcsin x – 3 in the form sin y = x.
to make a table of values.
Next, assign values to y on the interval
y
0
x = sin(y + 3)
0.99
0.80
0.14
−0.60
−0.99
Plot the points and connect them with a smooth curve.
21. y = arccos x
SOLUTION: First, rewrite y = arccos x in the form cos y = x.
Next, assign values to y on the interval
y
0
x = cos y
1
to make a table of values.
0
Plot the points and connect them with a smooth curve.
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eSolutions
22. y = Manual
cos -3x
SOLUTION: Page 4
4-6 Inverse Trigonometric Functions
22. y = cos– 1 3x
SOLUTION: –1
First, rewrite y = cos
3x in the form cos y = x.
Next, assign values to y on the interval
y
to make a table of values.
0
x = cos y
0
Plot the points and connect them with a smooth curve.
23. y = arctan x
SOLUTION: First, rewrite y = arctan x in the form tan y = x.
to make a table of values.
Next, assign values to y on the interval
y
0
x = tan y
0
1
3.08
Plot the points and connect them with a smooth curve.
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4-6 Inverse Trigonometric Functions
23. y = arctan x
SOLUTION: First, rewrite y = arctan x in the form tan y = x.
to make a table of values.
Next, assign values to y on the interval
y
0
x = tan y
0
1
3.08
Plot the points and connect them with a smooth curve.
24. y = tan– 1 3x
SOLUTION: –1
First, rewrite y = tan
3x in the form tan y = x.
to make a table of values. Note that x =
Next, assign values to y on the interval
for y-values of and −
.
y
x=
tan y has no x-values
0
tan y
0
1.03
Plot the points and connect them with a smooth curve.
Further investigation reveals that as x approaches negative infinity, y approaches −
infinity, y approaches
, and as x approaches positive
.
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4-6 Inverse Trigonometric Functions
24. y = tan– 1 3x
SOLUTION: –1
First, rewrite y = tan
3x in the form tan y = x.
to make a table of values. Note that x =
Next, assign values to y on the interval
for y-values of and −
.
y
x=
tan y has no x-values
0
tan y
0
1.03
Plot the points and connect them with a smooth curve.
Further investigation reveals that as x approaches negative infinity, y approaches −
infinity, y approaches
, and as x approaches positive
.
25. y = tan– 1 (x + 1)
SOLUTION: –1
First, rewrite y = tan
(x + 1) in the form tan y = x.
to make a table of values. Note that x = tan y − 1 has no x-values
Next, assign values to y on the interval
for y-values of
and −
.
y
x =Manual
tan y- Powered
–1
eSolutions
by Cognero
0
0
Plot the points and connect them with a smooth curve.
2.08
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4-6 Inverse Trigonometric Functions
25. y = tan– 1 (x + 1)
SOLUTION: –1
First, rewrite y = tan
(x + 1) in the form tan y = x.
to make a table of values. Note that x = tan y − 1 has no x-values
Next, assign values to y on the interval
for y-values of
and −
.
y
0
x = tan y – 1
0
2.08
Plot the points and connect them with a smooth curve.
Further investigation reveals that as x approaches negative infinity, y approaches
infinity, y approaches , and as x approaches positive
.
26. y = arctan x – 1
SOLUTION: First, rewrite y = arctan x – 1 in the form tan y = x.
The range of inverse tangent of x is
. Since we are subtracting 1, the new range should be
.
Next, assign values to y within this range to make a table of values.
y
−2.5
x =Manual
tan (y- Powered
+ 1) by Cognero
−14.1
eSolutions
−
0
−0.64
0.22
1.56
0.5
−4.59
14.1
Page 8
Plot the points and connect them with a smooth curve. Further investigation reveals that as x approaches negative
4-6 Inverse Trigonometric Functions
26. y = arctan x – 1
SOLUTION: First, rewrite y = arctan x – 1 in the form tan y = x.
The range of inverse tangent of x is
. Since we are subtracting 1, the new range should be
.
Next, assign values to y within this range to make a table of values.
y
−2.5
x = tan (y + 1)
−14.1
−
0
−0.64
0.22
1.56
0.5
−4.59
14.1
Plot the points and connect them with a smooth curve. Further investigation reveals that as x approaches negative
infinity, y approaches
− 1 or −2.57, and as x approaches positive infinity, y approaches − 1 or 0.57.
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