Download 4-6 Inverse Trigonometric Functions page 288 1

4-6 Inverse Trigonometric Functions
Find the exact value of each expression, if it exists.
1. sin – 1 0
SOLUTION: with a y-coordinate of 0.
Find a point on the unit circle on the interval
When t = 0, sin t = 0. Therefore, sin
–1
0 = 0.
3. arcsin
SOLUTION: with a y-coordinate of
Find a point on the unit circle on the interval
When t =
, sin t =
. Therefore, arcsin = .
.
5. SOLUTION: with a y-coordinate of
Find a point on the unit circle on the interval
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When t =
, sin t =
. Therefore, sin
–1
.
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=
.
4-6 When
Inverse
Functions
t = Trigonometric
, sin t =
. Therefore, arcsin = .
5. SOLUTION: with a y-coordinate of
Find a point on the unit circle on the interval
When t =
, sin t =
. Therefore, sin
–1
=
.
.
7. cos– 1
SOLUTION: Find a point on the unit circle on the interval
When t =
, cos t =
–1
. Therefore, cos
with an x-coordinate of
= .
.
9. SOLUTION: Find a point on the unit circle on the interval
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with an x-coordinate of
.
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4-6 When
Inverse
Functions
t = Trigonometric
, cos t =
. Therefore,
cos
= .
9. SOLUTION: Find a point on the unit circle on the interval
When t =
, cos t =
. Therefore,
with an x-coordinate of
=
.
.
11. arctan 1
SOLUTION: Find a point on the unit circle on the interval
When t =
, tan t =
. Therefore, arctan 1=
such that =1.
.
13. SOLUTION: Find a point on the unit circle on the interval
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such that =
.
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When t =
, tan t =
. Therefore, arctan 1=
.
4-6 Inverse Trigonometric Functions
13. SOLUTION: such that Find a point on the unit circle on the interval
When t =
, tan t =
. Therefore, tan
–1
= =
.
.
15. ARCHITECTURE The support for a roof is shaped like two right triangles, as shown below. Find θ.
SOLUTION: Use inverse trigonometric functions and the unit circle to solve.
with a y-coordinate of
Find a point on the unit circle on the interval
When t =
, sin t =
. Therefore, sin
–1
= .
.
Sketch the graph of each function.
17. y = arcsin x
SOLUTION: eSolutions Manual - Powered by Cognero
First, rewrite y = arcsin x in the form sin y = x.
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4-6 When
Inverse
Functions
t = Trigonometric
, sin t = . Therefore,
sin
= .
Sketch the graph of each function.
17. y = arcsin x
SOLUTION: First, rewrite y = arcsin x in the form sin y = x.
Next, assign values to y on the interval
to make a table of values.
y
x = sin y
–1
0
0
1
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.
Next, assign values to y on the interval
to make a table of values.
y
x = sin y
–3
–3.89
0
–3.85
–3
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–2.15
–2.11
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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
x = sin y
–3
–3.89
0
–3.85
–3
–2.15
–2.11
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
x = cos y
y
0
to make a table of values.
1
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0
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4-6 Inverse Trigonometric Functions
21. y = arccos x
SOLUTION: First, rewrite y = arccos x in the form cos y = x.
Next, assign values to y on the interval
x = cos y
y
0
to make a table of values.
1
0
-1
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.
Next, assign values to y on the interval
to make a table of values.
y
x = tan y
–3.08
0
–1
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.
Next, assign values to y on the interval
to make a table of values.
y
x = tan y
–3.08
–1
0
0
1
3.08
Plot the points and connect them with a smooth curve.
25. y = tan– 1 (x + 1)
SOLUTION: –1
First, rewrite y = tan
(x + 1) in the form tan y = x.
Next, assign values to y on the interval
for y-values of
to make a table of values. Note that x = tan y − 1 has no x-values
and − .
y
x = tan y –
1
0
–2
–1
0
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Plot the points and connect them with a smooth curve.
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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.
Next, assign values to y on the interval
for y-values of
to make a table of values. Note that x = tan y − 1 has no x-values
and − .
y
x = tan y –
1
0
–2
–1
0
Plot the points and connect them with a smooth curve.
Further investigation reveals that as x approaches negative infinity, y approaches
, and as x approaches positive
infinity, y approaches .
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