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Page 1
Section 7.1 Example 1
Technology Tip
To calculate tan u =
cot u =
1 - 452
1352
1 532
1 - 452
and
, enter
Section 7.2 Example 2
Section 7.2 Example 6
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Technology Tip
Graphs of
1
1
y1 =
+
and y2 = 1.
csc2 x sec2 x
Graphs of y1 =
sin(-x)
cos(-x) tan(-x)
and y2 = 1.
Section 7.2 Example 1
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Graphs of y1 = sin(-x) and
y2 = -sin x.
Section 7.2 Example 7
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Section 7.2 Example 4
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Graphs =
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tan x - cot x
tan x + cot x
and y2 = sin2 x - cos2 x.
Graphs of y1 =
cot2 x
sin x + 1
,y =
,
sin x
1 - csc x
and y = 1 + csc x in the same
[-2p, 2p] by [-5, 5] viewing
rectangle.
Text and Online
The graphs help verify that the
equation is an identity.
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Page 2
Section 7.3 Example 1
Section 7.3 Example 2
(cont.)
Section 7.3 Example 5
b.
Technology Tip
Technology Tip
a. Use a calculator to check the
values of cos15° and
12 + 16
. Be sure the calculator
4
is set in degree mode.
If sin a = - 13 and is in QIII, then
a = p + sin-1 A 13 B (more explanation
can be found in Section 7.7). If
cos b = - 14 and is in QII, then
b = p - cos-1 A 14 B. Now use the
graphing calculator to find tan( ⫹ )
by entering
Section 7.3 Example 3
Technology Tip
b. Use a calculator to
check the values for
7p
12 - 16
cosa b and
.
4
12
Section 7.3 Example 2
Technology Tip
a.
Use a calculator to check the
5p
12 + 16
.
values of sina b and
12
4
Be sure the calculator is in radian
In Step 4, use the graphing calculator
to evaluate both expressions
12
- 115
4
and
12
1 - a
b 1 - 115 2
4
12 - 4115
by entering
4 + 130
mode.
Section 7.3 Example 4
Section 7.4 Example 1
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Technology Tip
Graphs of y1 =
3 sin x cos(3x) + 3 cos x sin(3x)
and y2 = 3 sin(4x).
If cos x =
2
3
and sin x 6 0, then x
is in QIV and one value for x is
x = 2p - cos-1 A 23 B (more
explanation can be found in
Section 7.7). Now use the graphing
calculator to find sin(2x), entering
sin12 C 2p - cos-1 A 23 B D B , and compare
that value to -
4 15
.
9
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Section 7.4 Example 2
Section 7.3 Example 4
(cont.)
Technology Tip
Section 7.5 Example 1
Technology Tip
- 45
Use a TI calculator to check the values
If sin x =
and cos x 6 0, then x
is in QIII and a value for x is
x = p + sin-1 A 45 B (more explanation
can be found in Section 7.7). Now
use the graphing calculator to find
sin(2x), cos(2x), and tan(2x).
2 + 13
.
B
4
Be sure the calculator is in degree
mode.
of cos 15° and
Section 7.5 Example 2
Technology Tip
The graphs help verify that the
equation is an identity.
Use a TI calculator to check
11p
the values of tana
b and
12
13 - 2. Be sure the calculator is in
radian mode.
Section 7.4 Example 4
Section 7.4 Example 3
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Technology Tip
Graphs of y = (sin x - cos x)2 and
y = 1 - sin(2x) in the same
[-p, p] by [-1, 2] viewing
rectangle follow.
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cot x - tan x
Graphs of y =
and
cot x + tan x
y = cos(2x).
Section 7.5 Example 3
Technology Tip
3
3p
and
6 x 6 2p,
5
2
x
then x is in QIV, is in QII, and
2
x = 2p - cos-1 A 35 B (more explanation
can be found in Section 7.7). Now
use the graphing calculator to find
x
x
x
sin , cos , and tan .
2
2
2
If cos x =
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Page 4
Section 7.5 Example 3
(cont.)
Section 7.5 Example 4
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x
Graphs of y1 = cos2 a b and
2
tan x + sin x
y2 =
.
2 tan x
Section 7.6 Example 2
Section 7.7 Example 1
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Technology Tip
Graphs of y1 = cos(4x) cos(3x) and
y2 = 12 3cos(7x) + cos(x)4.
Set the TI/scientific calculator to
Section 7.6 Example 6
Technology Tip
Graphs of
y1 = -9 [sin(2x) - sin(10x)] and
y2 = 18 sin(4x) cos(6x).
degree mode by typing MODE .
a. Using the TI calculator to find
sin-1 a
13
2 b,
type 2nd sin for sin⫺1
and 2nd x2 for 2 .
1
b. arcsin a- b
2
Section 7.7 Sine-Inverse
Sine Identities
Section 7.5 Example 6
Section 7.7 Inverse Sine
Function
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Technology Tip
sin(2px)
1 + cos(2px)
and y2 = tan(px).
To graph y = sin-1 x, use
3-1, 14 as the domain and
p p
c- , d as the range.
2 2
Graphs of y1 =
Technology Tip
Use a TI calculator to find
sin-1 3 and sin-1 0.3. Be sure to set
the calculator in radian mode.
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Page 5
Section 7.7 Example 2
Section 7.7 Example 3
Section 7.7 Example 5
Technology Tip
Technology Tip
Technology Tip
a. Check the answer of
12
sin csin-1a
b d with a
2
calculator.
a. Check the answer of
12
cos-1 ab with a calculator.
2
a. Use a calculator to check the
answer for tan-1 A 13 B.
b. Check the answer of arccos 0.
Radian mode:
b. Check the answer of
3p
sin-1asin b with a calculator in
4
radian mode.
Section 7.7 Example 4
b. Use a calculator to check the
answer for arctan0.
Technology Tip
a. Check the answer of
cos C cos-1 A- 12 B D with a calculator.
Section 7.7 Example 6
Section 7.7 Inverse
Cosine Function
b. Check the answer of
7p
cos-1acos b. Be sure to set
4
the calculator to radian mode.
Technology Tip
Technology Tip
a. Use a calculator to check the
answer for tan(tan-117).
To graph y = cos-1 x, use 3-1, 14 as
the domain and [0, ] as the range.
Section 7.7 Inverse
Tangent Function
Technology Tip
To graph y = tan-1x, use
(- ⬁, ⬁) as the domain and
p p
a - , b as the range.
2 2
b. Use a calculator to check the
2p
answer for tan-1 c tana
b d.
3
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Page 6
Section 7.7 Example 7
Section 7.8 Example 1
Online Only
Technology Tip
Technology Tip
1
cot-1 A13B = tan-1 a
b,
13
1
csc-1 A12B = sin-1a
b , and
12
1
sec-1 A - 12 B = cos-1 a b.
12
a. Use the fact that a solution to the
1
equation sin x = is the same as
2
a point of intersection of y = sin x
1
and y = over one period, [0, 2p).
2
If radian mode is used, then set the
window Xmin = 0, Xmax = 2p,
Xscl = p/6, Ymin = -1,
Ymax = 1, and Yscl = 1.
Section 7.8 Example 1 (cont.)
For the second answer, type
Section 7.7 Example 8
Technology Tip
Use the inverse trigonometry
function identities to find
a. sec-1 2 = cos-1 A 12 B
b. cot 7 = tan A 7 B
-1
-1 1
To find the point of intersection,
use 2nd TRACE for CALC ,
down arrow to
5: Intersect , type
ENTER for the first curve,
ENTER for the second curve, 0
for guess, and ENTER .
p
,
2 for guess, and ENTER .
b. Use the fact that a solution to the
1
equation cos 2x = is the same as
2
the point of intersection of y = cos 2x
1
and y = over one period, [0, 2p).
2
If radian mode is used, then set the
window Xmin = 0, Xmax = p,
Xscl = p/6, Ymin = -1, and
Ymax = 1, and Yscl = 1.
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Page 7
Section 7.8 Example 1 (cont.)
To find the point of intersection,
use 2nd TRACE for CALC ,
down arrow to
5: Intersect , type
ENTER for the first curve,
ENTER for the second curve, 0
for guess, and ENTER .
Section 7.8 Example 2
Online Only
There are multiple ways to use a
calculator to solve the equation
12
over one period,
sin x =
2
[0, 2p) or [0, 360°).
p for guess, and ENTER .
Technology Tip
Method I: Use the inverse
trigonometric function. Be sure to
put the calculator in degree mode.
Technology Tip
For the second answer, type
Section 7.8 Example 3
Use the fact that a solution to the
12
equation sin x =
is the same as
2
a point of intersection of y = sin x
12
and y =
over one period,
2
[0, 2p) or [0, 360°).
To find the point of intersection, use
2nd TRACE for CALC , move
the down arrow to 5: Intersect , type
ENTER for the first curve, ENTER
for the second curve, 0.8 for guess,
and ENTER .
For the second answer, you need a
number close to the answer for the
guess. Type 2.5 for guess, and
ENTER .
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Page 8
Section 7.8 Example 5
Technology Tip
Section 7.8 Example 6
Section 7.8 Example 8
Technology Tip
Technology Tip
Find the points of intersection of
y1 = sin x + cos x and y2 = 1.
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Section 7.8 Example 11
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Section 7.8 Example 7
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