Download sin =15 17 , tan = 8 17 , csc =17 8 cos =15 8 , cot = 8 15 , sec

Name: __________________
1
Class:
Use the definitions to evaluate the six trigonometric functions of denominator.
. In cases in which a radical occurs in a denominator, rationalize the
15 , tan = 8 , csc = 17
17
17
8
15
8
17
, cot =
, sec =
cos =
8
15
15
d. sin 15 , tan = 8 , csc = 1
17
17
17
cos =
, cot = 17 , sec = 1
15
8
e. sin a. sin b. sin =
=
15 , tan = 15 , csc = 17
17
8
15
8
8
cos =
, cot =
, sec = 17
17
15
8
c. sin PAGE 1
Date: _____________
=
1 , tan = 15 , csc = 8
17
17
15
1
17
, cot =
, sec = 15
cos =
15
8
8
=
15 , tan = 8 , csc = 2
17
17
15
cos =
, cot = 17 , sec = 1
8
15
2
=
Name: __________________
2
Suppose that Class:
Date: _____________
ABC is a right triangle with C = 90 . If A C = 3 and BC = 5 , find the following quantities.
cos A , sin A , tan A
3
a. cos A =
34
, sin A =
34
34
, tan A =
34
b. cos A =
34
, sin A =
34
34
, tan A = 3
34
c. cos A =
3 34
, sin A =
34
5 34
, tan A =
34
d. cos A =
34
, sin A =
34
e. cos A =
5 34
, sin A =
34
1
a. yes
5
3
3 34
, tan A =
34
5
3
cos 30
cos 60
b. no
2
Determine whether the equation is correct by evaluating each side. Do not use a calculator. Note: Notation such as sin ( sin )
2
.
2
1
2
tan 30 = sec 30
a. no
5
34
, tan A =
34
5
3
Determine whether the equation is correct by evaluating each side. Do not use a calculator.
cot 30 =
4
3
5
b. yes
Carry out the indicated operation.
sin cos a. PAGE 2
cos sin sin b. cos c. 1
d. tan e. 1
stands for
Name: __________________
6
Class:
Date: _____________
Carry out the indicated operation.
3 2 tan 2 tan 3
3
tan a.
7
2
tan c. 3 2 tan d. 1
e. 1
Factor the expression.
2
tan + 5 tan a. (tan b. (tan 8
b. 3 tan 2
6
1)(tan + 5)
c. (tan 3)(tan + 4)
d. (tan 5)(tan + 3)
4)(tan + 3)
e. (tan 1)(tan + 6)
Use the given information to determine the value of remaining five trigonometric functions. (Assume that the angle is acute.) If a radical
appears in the denominator, rationalize the denominator.
sin =
2
3
a. cos =
cot =
b. cos =
cot =
c. cos =
cot =
PAGE 3
2 5
2 5
, tan =
,
3
5
5
3 5
, sec =
, csc 2
2
5
2 5
, tan =
,
3
5
5
3 5
, sec =
, csc 2
5
5
2 5
, tan =
,
3
5
5
3 5
, sec =
, csc 2
5
=
3
2
=
3
2
=
5
2
d. cos =
cot =
e. cos =
cot =
5
2 5
, tan =
,
3
3
3 5
, sec = 3 , csc 2
5
5
2 5
, tan =
,
3
5
5
3 5
, sec =
, csc 2
5
3
2
=
=
3
5
Name: __________________
9
Class:
Date: _____________
Use the given information to determine the value of remaining five trigonometric functions. (Assume that the angle is acute.) If a radical
appears in the denominator, rationalize the denominator.
cot =
5
3
a. sin =
3 14
, cos 14
=
70
, tan 14
=
3 , sec 5
b. sin =
3 14
, cos 14
=
70
, tan 14
=
3 5
, sec 5
c. sin =
3 , cos 14
d. sin =
3 14
, cos 14
=
70
, tan 14
e. sin =
3 14
, cos 14
=
5
, tan 14
70
, tan 14
=
=
3 5
, sec 5
=
=
70
, csc 5
=
70
, csc 5
=
3 5
, sec 5
3 5
, sec 5
70
, csc 5
=
=
=
14
3
14
3
14
, csc 3
=
70
5
14 5
, csc 5
=
14
3
=
=
14
3
=
10 Rewrite in terms of sine and cosine, and simplify the expression:
cos sec tan a. cos b. sec c. csc d. tan e. sin 11 Simplify the expression and enter the answer in terms of sine and cosine.
(csc A + cot A )(csc A a. 0
b. 1 cot A )
2
c. 1
sin A
d. 2 sin A cos A
2
2
e. csc A + cot A
12 Rewrite in terms of sine and cosine, and simplify the expression:
cos + 1 + 1
cos cos 1 1
cos a. sin cos PAGE 4
b. 1
c. 1
d. sec + csc e. 2 cos 1
Name: __________________
13 Refer to the figure. If Class:
Date: _____________
A = 30 and AB = 60 cm , find A C .
a. AC = 60 2 cm
c. AC = 30 3 cm
b. AC = 60 cm
d. AC = 30 7 cm
e. AC =
14 A ladder 20 ft long leans against a building. The ladder forms and angle of 45
the ladder reach?
a. 10 2 ft
b.
2 ft
c. 20 ft
d.
2 cm
with the ground. How high up the side of the building does
10 ft
e. 20 2
ft
15 From a point level with and 1500 ft away from the base of a monument, the angle of elevation to the top of the monument is 29.45 .
Determine the height of the monument to the nearest foot.
a. 737 ft
b. 1306 ft
c. 847 ft
d. 992 ft
e. 964 ft
16 From a point on ground level, you measure the angle of elevation to the top of a mountain to be 33 . Then you walk 200 m farther away
from the mountain and find that the angle of elevation is now 20 . Find the height of the mountain.
a. 73 m
PAGE 5
b. 130 m
c. 100 m
d. 101.5 m
e. 166 m
Name: __________________
Class:
Date: _____________
17 The radius of the circle in the figure is 10 units. Express the length DC in terms of a. 10 sec b. 10 sin c. 10 tan 18 In the figure, AB = 6 in . Express x
a. 6 cos b. 6 sin PAGE 6
as a function of 3 cot c. 3 tan 3 tan d. 3 sin .
d. 10 cot e. 10 cos .
6 cos 6 cot e. 3 cos 6 tan Name: __________________
Class:
Date: _____________
19 Use the definitions (not a calculator) to evaluate the six trigonometric functions of the angle.
540
a. sin(
540 ) = 0
540 ) = 1
tan(
cot(
540 ) = 0
540 ) is undefined
csc(
sec(
540 ) = 1
540 ) is undefined
b. sin(
540 ) = 1
540 ) = 0
tan(
cot(
540 ) is undefined
540 ) = 0
csc(
sec(
540 ) = 1
540 ) is undefined
c. sin(
540 ) = 0
540 ) = 1
tan(
cot(
540 ) = 0
540 ) is undefined
csc(
sec(
540 ) is undefined
540 ) = 1
d. sin(
540 ) = 0
540 ) = 1
tan(
cot(
540 ) is undefined
540 ) = 0
csc(
sec(
540 ) is undefined
540 ) = 1
e. sin(
540 ) = 1
540 ) = 0
tan(
cot(
540 ) = 0
540 ) is undefined
csc(
sec(
540 ) is undefined
540 ) = 1
cos(
cos(
cos(
cos(
cos(
20 Use the definitions (not a calculator) to evaluate the six trigonometric functions of the angle.
810
a. sin(
810 ) = 1
810 ) = 0
tan(
cot(
810 ) = 0
810 ) is undefined
csc(
sec(
810 ) = 1
810 ) is undefined
b. sin(
810 ) = 0
810 ) = 1
tan(
cot(
810 ) = 0
810 ) is undefined
csc(
sec(
810 ) is undefined
810 ) = 1
c. sin(
810 ) = 0
810 ) = 1
tan(
cot(
810 ) is undefined
810 ) = 0
csc(
sec(
810 ) = 1
810 ) is undefined
d. sin(
810 ) = 1
810 ) = 0
tan(
cot(
810 ) is undefined
810 ) = 0
csc(
sec(
810 ) = 1
810 ) is undefined
e. sin(
810 ) = 1
810 ) = 0
tan(
cot(
810 ) is undefined
810 ) = 0
csc(
sec(
810 ) is undefined
810 ) = 1
cos(
cos(
cos(
cos(
cos(
21 Evaluate the expression using the concept of a reference angle.
cos 225
a.
PAGE 7
2
2
b. 3
2
c. 2
2
d.
1
3
e. 2
3
Name: __________________
Class:
Date: _____________
22 Evaluate the expression using the concept of a reference angle.
cos(
420 )
6
6
a. b.
1
2
1
2
c. 6
2
d.
e.
1
6
23 Evaluate the expression using the concept of a reference angle.
sin 300
1
6
a. b.
3
2
c.
6
6
3
2
d. 6
2
e. 24 Evaluate the expression using the concept of a reference angle.
sin(
a.
330 )
1
3
b.
1
2
c. 1
2
d. 3
2
e. 3
3
25 Use the following information to determine the remaining five trigonometric values. Rationalize any denominators that contain radicals.
7 , 90 < < 180
8
cos = a. sin =
15
, tan = 8
b. sin =
15
, tan =
8
c. sin =
15
, tan = 8
15
, csc =
7
8 15
, cot = 15
7 15
, sec =
15
d. sin =
15
, tan = 8
15
, csc =
7
8 15
, cot = 15
7 15
, sec = 15
e. sin =
15
, tan =
8
PAGE 8
15
, csc = 7
7 15
, csc =
15
7 15
, csc = 15
7 15
, sec =
15
8 , cot = 7
8 15
, cot =
15
8 , cot =
7
15
, sec = 7
15
, sec =
7
8 15
15
8
7
8 15
15
8
7
8
7
Name: __________________
Class:
Date: _____________
26 Use the following information to determine the remaining five trigonometric values. Rationalize any denominators that contain radicals.
csc A = 7 , 270 < A < 360
a. sin A =
4
, cos A = 7
b. sin A =
1 , cos A =
7
c. sin A =
1 , cos A = 7
d. sin A = e. sin A = PAGE 9
4
, tan A =
7
4
, cos A =
7
1 , cos A =
7
1 , tan A = 7
4 , cot A = 4
, cot A =
4
4
, tan A =
7
1 , tan A =
7
4 3
, tan A = 7
4
, cot A =
4
4 , cot A =
4
, sec A =
4
4 , sec A =
7 4
4
7 4
4
4 , sec A = 7 4
4
4
, sec A = 4
7 4
4
3
, cot A = 12
4 3 , sec A =
7 3
12
Name: __________________
Class:
Date: _____________
27 Use the following information to express the remaining five trigonometric values as functions of p . Assume that p
Rationalize any denominators that contain radicals.
cos =
p , 270 < < 360
3
9 p
tan = 9 a.
9 cot = tan = 2
9 p
cot = p
9 tan =
9 p
9 p
csc = sec = 2
p
9 p
3 9 sec = ,
9 p
9 tan =
9 p
p
p
PAGE 10
,
2
9 p
p
2
2
,
3
2
,
9 p
,
3
.
9 p
2
,
3
2
.
3 9 p
p
2
2
,
9 sin = sin =
2
.
p
3
csc = 3 .
p
csc = 3 9 p
2
9 p
2
2
sin = sec = 3 ,
p
,
p
9 2
2
e.
cot = p
p
9 d.
cot = p
2
p
3 ,
p
2
p
9 sin = sin = csc = 3 9 p
2
9 p
,
2
9 2
p
9 ,
c.
tan = ,
2
p
cot = p
2
2
p
,
sec = 3 ,
p
,
b.
2
p
csc = 3 .
p
,
p
3 9 sec = 9 2
p
p
,
2
p
p
9 2
p
9 p
3
2
,
2
,
is positive.
Name: __________________
Class:
Date: _____________
28 Use the following information to express the remaining five trigonometric values as functions of u . Assume that u is positive. Rationalize
any denominators that contain radicals.
cos =
u , 0 < < 90
10
1
tan = u
1
a.
u
1
cot =
2
u
1
cot =
2
2
u
u ,,
10 u
1
u
2
csc =
2
sec =
,
u
1
cot = u
u
1
u
2
u
u
1
1
PAGE 11
u
u
2
u
,
2
,
10u
2
,
10
10 2
1
2
u
2
,
10
2
100 sin =
10u
2
10u
.
100 u
10 2
10
,
u
10u ,
10
,
2
.
, sin =
10 10u
10
2
,
10
.
u
2
u
e.
cot = 1
2
10u
csc =
,
2
1
tan = 10u
10 d.
10 sin =
10 sec =
,
, sin =
10
,
u
2
2
10 tan =
csc =
,
c.
u
2
10
.
u
2
10 u
u
cot = u
1
2
sec =
,
u
tan =
10u
csc =
,
u
1
10 2
u
1
sec =
,
u
tan = u
b.
2
u
10
,
u
sec =
csc = 10 10u
1
2
u
sin = 2
.
10 10u
10
2
,
Name: __________________
Class:
Date: _____________
29 Determine the answer that establishes an identity.
2
tan A + 1 = ?
2
2
a. cot A
2
b. cos A
2
c. sin A
2
d. csc A
e. sec A
30 Determine the answer that establishes an identity.
(1 cos )(csc + cot ) = ?
a. sec b. cos c. 1
d. csc e. sin 31 Determine the answer that establishes an identity.
sin B
+
1 + cos B
1 + cos B = ?
sin B
a. 2 cos B
32 Use the definition a. = 0.56
radians
PAGE 12
b. 2 sec B
=
s
r
c. 2 sin B
d. 2 csc B
e. 1
to determine the radian measure of the angle in the figure below.
b. = 0.83
radians
c. = 0.2
radians
d. = 0.6
radians
e. = 0.3
radians
Name: __________________
Class:
Date: _____________
33 Convert to radian measure. Express your answer in terms of .
90
a.
b.
6
c.
2
7
6
d.
34 Convert to radian measure. Express your answer in terms of 5
3
e.
5
6
.
126
a.
3
10
b.
3
5
c.
7
10
d.
e.
2
2
5
35 Convert the radian measure to degrees.
3
a. 530
b. 490
c. 540
d. 660
e. 620
36 Convert the radian measure to degrees.
6
a. 30
b. 45
c. 35
d. 40
e. 50
37 Convert the radian measure to degrees. Round the answer to two decimal places.
5 radians
a. 286.31
b. 287.64
c. 284.86
d. 286.48
e. 286.91
38 Evaluate the expression.
cos a. PAGE 13
6
3
2
b. 0
c. 2
2
d.
3
2
e.
2
2
Name: __________________
Class:
Date: _____________
39 Evaluate the expression.
sin
11
3
3
2
a. 1
2
b. 1
2
c.
d.
3
2
2
2
e.
40 Evaluate the expression.
5
6
sin a.
3
2
b. 2
2
3
2
c. d. 1
2
e.
1
2
41 Evaluate the expression.
sin
a.
9
4
3
2
b. 1
2
c.
2
2
d. 2
2
e.
1
2
42 Evaluate the expression.
sec
a.
11
3
2 3
3
2 3
3
b. c. d. 2
2
43 Evaluate the expression.
csc a. PAGE 14
15
4
2
b.
2
c. 1
d. 2
e. 2
e. 2
Name: __________________
Class:
Date: _____________
44 Find the arc length s .
a. 5
b.
ft
7
5
ft
c. 35
ft
5
7
d.
ft
45 Find the area of the sector determined by the given radius r and central angle decimal approximation rounded to two decimal places.
r = 14 cm , =
ft
. Express the answer both in terms of 4
7
2
2
2
2
a. 58 cm , 182.21cm
b. 55 cm , 172.79cm
PAGE 15
e. 7
2
2
2
2
c. 56 cm , 175.93cm
d. 59 cm , 185.35cm
2
e. 60 cm , 188.50cm
2
and as a
Name: __________________
Class:
Date: _____________
46 Find the perimeter of the sector and then find the area of the sector.
a. Perimeter
Area
PAGE 16
12.7 in
10.05 in
b. Perimeter
Area
2
.
10.05 in
12.7 in
2
.
c. Perimeter
.
Area
.
11.38 in
11.3 in
d. Perimeter
Area
2
2
e. Perimeter
Area
.
11.38 in
11.34 in
.
.
.
11.34 in
11.3 in
2
.
.
Name: __________________
Class:
Date: _____________
47 You are given the rate of rotation of a wheel as well as its radius. Determine the angular speed. Then determine the linear speed of a point on
the circumference of the wheel and of a point halfway between the center of the wheel and the circumference.
7 revolutions/sec
r = 17 cm
a. Angular speed
14 radians/sec
Linear speed on the circumference
14
cm/sec
Linear speed halfway between the center and edge
b. Angular speed
14 238
cm/sec
Linear speed halfway between the center and edge
238
119
119
Linear speed on the circumference
119
14 cm/sec
238
cm/sec
radians/sec
238
cm/sec
Linear speed halfway between the center and edge
e. Angular speed
cm/sec
cm/sec
Linear speed halfway between the center and edge
119
radians/sec
Linear speed on the circumference
d. Angular speed
cm/sec
radians/sec
Linear speed on the circumference
c. Angular speed
119
radians/sec
Linear speed on the circumference
14 cm/sec
Linear speed halfway between the center and edge
238
cm/sec
48 Suppose that a belt drives two wheels of radii r and R, as indicated in the figure. If r = 6cm , R = 11cm , and the angular speed of
the larger wheel is 160 rpm, determine the angular speed of the smaller wheel in radians per minute. (Hint: Because of the belt, the linear
speed of a point on the circumference of the larger wheel is equal to the linear speed of a point on the circumference of the smaller wheel.)
a.
PAGE 17
1760 3
b.
1760
3
c.
880 3
d. 3520
e.
880
3
Name: __________________
Class:
Date: _____________
49 Compute cos t and tan t .
5 and 0 < t < 13
2
sin t =
13 , tan t =
12
a. cos t =
12
5
13 , tan t =
12
b. cos t = 5
12
26
, tan t =
13
c. cos t =
d. cos t =
12 , tan t =
13
5
12
e. cos t =
5 , tan t =
12
12
13
5
26
50 Compute tan t .
7
3
sin t =
a. tan t =
b. tan t =
51 If sin t =
a.
a. PAGE 18
3
2
< t < c. tan t =
14
2
d. tan t =
6 , find sin ( 7
b. s) =
7
6
c.
b. 1
e. tan t =
3 2
7
18
2
6
7
d. 1
6
e. 6
7
) .
b. 0.01
3
5
7
2
t) .
= 0.99 , find cos ( 0.01
53 If sin ( a.
2
3 2
2
7
6
52 If cos and
d. c. 0.99
< s < 3
2
c. 0
, compute tan s +
d. 5
4
0.99
tan ( e.
e. 0.1
s) .
2
3
Name: __________________
Class:
Date: _____________
54 Use one of the identities below to evaluate the expression.
cos ( t + 2 k ) = cos t or sin ( t + 2 k ) = sin t
cos
2
6
1
2
a. b. 0
d.
c. 1
1
2
e. 3
2
55 Use the Pythagorean identities to simplify the expression.
2
2
cos t + sin t
2
cot t + 1
2
a. sec PAGE 19
b. 1
2
c. csc 2
d. cos 2
e. sin t