Download Trigonometry Section 1.5

Trigonometry Section 1.1
Graph the points on a coordinate system and identify the quadrant or axis for each point.
1.
A  3, 2 
2.
B 8, 3
3.
C  7, 1
4.
D  5,1
5.
E  0, 4 
6.
F  6,0
Use the distance formula to find the distance between the following pairs of points.
7.
W  2,1 and Z  3, 4

8.
X  5, 2  and Y  3, 7 

A triple of positive integers a, b, c is called a Pythagorean Triple if it satisfies the equation of
the Pythagorean Theorem. Determine whether each of the following is a Pythagorean Triple.
9.
9,12,15
10.
6,8,10
5,10,15
11.
Using both the distance formula and the Pythagorean Theorem, determine if the triangle made up
of the following points is a right triangle or not.
12.
J  2,5 , K 1,5 and L 1,9 
Find all values of
13.
x such that the distance between the given points is as indicated.
x,7 and 2,3 is 5
14.
5, x  and 8,1 is 5
Use the midpoint formula to find the midpoint of the line segment joining the two points.
15.
1,3 and 7,5
16.
17.
A line segment has an endpoint at
endpoint.
18.
A line segment has an endpoint at
endpoint.
4,3 and  1,2
3,2 and midpoint 5,3 .
Find the other missing
4,1 and midpoint 5,6 .
Find the other missing
Change the following set notations to interval notations.
19.
Let
22.
x x  6
20.
y y  9
21.
x 3  x  6
f x   2 x 2  4 x  6 . Find each of the following.
f  2
23.
f 3
24.
f a 
25.
f 2  p 
Find the domain and range of the following equations or graphs. Identify any which are functions.
26.
2 x  5 y  10
27.
y  2x2  5
28.
y  4 x
29.
x  y2
30.
31.
Find the domain only of the following equations.
32.
y
1
x
33.
y
34.
y
2
x 1
35.
y
x
4  x2
5x  13x  8
Trigonometry Section 1.2
Find the measure of each angle.
1.
3.
 2y  
 4y  
2.
 5k  5 
 3k  5 
Supplementary angles with measures
6x  4 and 8x  12 degrees.
Perform each calculation.
4.
6218  2141
7.
180  12451
5.
7515  8332
6.
90  5128
Convert each angle measure to decimal degrees. (Round to the nearest thousandth of a degree)
8.
2054
9.
913554
10.
2741859
13.
178.5994
Convert each angle measure to degrees, minutes, and seconds.
11.
59.0854
12.
102.3771
Find the angles of smallest positive measure coterminal with the following angles.
14.
 40
15.
125
16.
539
17.
850
Sketch the angle in standard position. Draw an arrow representing the correct amount of rotation.
Find the measure of two other angles, one positive and one negative, that are coterminal with the
given angle. Give the quadrant of each angle.
18.
89
19.
174
20.
250
22.
 52
23.
159
24.
438
21.
512
Locate the following points in a coordinate system. Draw a ray from the origin through the given
point. Indicate with an arrow the angle in standard position having the smallest positive measure.
Then find the distance r from the origin to the point, using the distance formula.
25.
 3,3
26.
 5,2
27.
 3,1
28.
4
3 ,4

Solve each problem.
29.
A windmill makes 90 revolutions per minute. How many revolutions does it make per
second?
30.
A tire is rotating 600 times per minute. Through how many degrees does a point on the
edge of the tire move in
1
second?
2
Trigonometry Section 1.3
Use the properties of angle measures given in this section to find the measure of each marked
angle.
1.
5x 129 
 2 x  21 
2.
a
 2 x  5 
a b
 x  22 
b
m
3.
4.
 x  1 
n
a
8
6
120
9
7
10
a b
 4 x  56 
4
b
m n
1
5.
55
5
2
3
 2x 120 
6.
 x  20 
x
 210  3x  
7.
 x  30 
Find the measure of the third angle if the other two angles measure:
a)
13650 and 413815
b)
29.6 and 49.7
c)
59.8 and 100.3
d)
4413 and 584
1

 x  15  
2

Classify each triangle as acute, right, equiangular or obtuse and also equilateral, isosceles or
scalene.
8.
130
9.
8
10.
8
r
r
r
11.
12.
10
96
65
65
14
Find all unknown angle measures or sides in each pair of similar triangles.
42
13. A
Q 14.
N
M
B
106
K
C
B
R
P
A
C
30
C
12
85
15.
16.
24
I
P
45
J
10
A 8
F
7
G
14
N
K
M
21
B
10.5
H
8
17.
N
18.
6
Y
B
Q
5
25
A
9
O
M
35
4
Z
P
C
10
R
X
Find the value of the variables.
19.
20.
Solve each problem.
21.
A forest fire lookout tower casts a shadow 180 feet long at the same time that the shadow
of a 9 foot truck is 15 feet long. Find the height of the tower.
22.
On a photograph of a triangular piece of land, the lengths of the three sides are 4cm, 5
cm and 7 cm, respectively. The shortest side of the actual piece of land is 450 m long.
Find the lengths of the other two sides.
23.
Sam built a ramp to a loading dock. The ramp has a vertical support of 2 meters from the
base of the loading dock and 3 meters form the base of the ramp. If the vertical support
is 1.2 meters in height, what is the height of the loading dock?
24.
A tourist that is 192 cm tall wants to estimate the height of an office tower. He places a
mirror on the ground 87.6 meters from the base of the tower and moves back 0.4 meters
to sight the top of the tower in the mirror. How tall is the tower?
25.
Lewis, an adventurous explorer, is trying to determine how wide a river is that he needs
to cross. To do this he first calls the point where he is standing point A. Next, he locates
a tree on the opposite side of the river that is directly across from where he is standing,
which he calls point B. He then walks 32 paces along the river and marks this next spot
as point C. He then walks 10 more paces along the river and marks this next spot as
point D, where he turns and walks directly away from the river until point C lines up with
point B. This takes 8 paces and he marks this final spot as point E. What is the width of
the river in paces?
Trigonometry Section 1.4
Find the values of the six trigonometric functions for the angles in standard position having the
following points on their terminal sides. Rationalize the denominators when applicable.
1.
 3,4
2.
 12,5
3.
6,8
4.
24,7
5.
0,2
6.
 4,0
7.
1, 3 
8.
 2
9.
2
10.

2 , 2 2

5 ,2
3 ,2


An equation with a restriction on x is given. This is an equation of the terminal side of an angle
in standard position. Sketch the smallest positive such angle  , and find the values of the six
trigonometric functions of  .
11.
 4 x  7 y  0, x  0
12.

6 x  5 y  0, x  0
Use the trigonometric functions values of quadrantal angles given in this section to evaluate each
of the following. An expression such as cot 90 means cot 90 which is equal to 0  0 .
2
2
2
13.
tan 0  6 sin 90
14.
4 csc 270  3 cos180
15.
2 sec 0  4 cot 2 90  cos 360
16.
sin 2 360  cos 2 360
17.
sec 2 180  3 sin 2 360  2 cos180
18.
3 csc 2 270  2 sin 2 270  3 sin 270
Trigonometry Section 1.5
Use the appropriate reciprocal identity to find each function value. Rationalize the denominator
when applicable.
1.
sin  if csc  3
3.
cot  if tan  
5.
sec  if cos  
1
5
1
2.
cos if sec  2.5
4.
csc if sin  
2
4
6.
tan  if cot  
 5
3
7
Identify the quadrant or quadrants for the angle satisfying the given conditions.
7.
sin   0, cos   0
8.
sec   0, csc   0
9.
cos   0, sin   0
10.
tan   0, cot   0
11.
cos   0
Give the signs of the six trigonometric functions for each angle.
12.
129
13.
406
14.
 82
121
15.
16.
662
Decide whether each statement is possible or impossible.
17.
sin   2
20.
csc  1  0.2
18.
tan   0.92
21.
cos  
19.
sec  1  1.3
3
4
and sec  
4
3
Find all the other trigonometric functions for each of the following angles.
22.
tan  
 15
, with  in Quad II
8
23.
cos  
3
, with  in Quad III
5
24.
sin  
7
, with  in Quad II
25
25.
csc  2 , with  in Quad II
26.
cot   2 , with  in Quad IV
27.
sec   2 , with cot   0
TRIGONOMETRY
PRACTICE TEST:
Chapter 1
NAME: _______________
Trigonometric Functions
Find the distance between each of the following pairs of points.
1.
3.
A  4, 2 and B 1, 6
2.
C  6,3 and D  2, 5
Using both the distance formula and the Pythagorean Theorem, determine if the triangle
made up of the following points is a right triangle or not.
E  2, 2 , F 8, 4 and G  2,14
Find the domain and range. Write each set using interval notation.
4.
y  9x  2
5.
y x
6.
y
x 1
7.
Find the angle of smallest possible positive measure coterminal with the following angles.
8.
 51
11.
A pulley is rotating 320 times per minute. Through how many degrees does a point on
the pulley move in
9.
174
10.
792
2
seconds?
3
Convert decimal degrees to degrees, minutes, seconds, and convert degrees, minutes, seconds
to decimal degrees. Round to the nearest second or the nearest thousandth of a degree, as
appropriate.
12.
472511
13.
 61.5034
Find the measure of each marked angle.
B
14.
15.
8x  
9 x  4 
12x 14 
 6x  
 4x  
A
C
Find all unknown parts (angles and side lengths) in the pair of similar triangles.
16.
Z
32
T
20
14
X
41
V
11 U
Y
22
Find the values of all the trigonometric functions for an angle in standard position having the
following point on its terminal side. Rationalize the denominator when applicable.
17.
 3,3
18.
1, 3 
19.
 6,0
20.
 8,15
21.
0,1
22.
2
23.
2 ,2 2

Find the values of all the trigonometric functions for an angle in standard position having
its terminal side defined by the equation 5 x  3 y  0, x  0 .
Evaluate each expression.
24.
4 sec 180  2 sin 2 270
25.
 cot 2 90  4 sin 270  3 tan 180
Decide whether each statement is possible or impossible.
3
4
and csc  
4
3
26.
sin  
28.
tan   1.4
2
3
27.
sec  
29.
cos   .25 and sec   4
Find all the other trigonometric function values for each angle. Rationalize the denominators
when applicable.
30.
sin  
3
and cos  0
5
32.
sec  
5
with  in Quad IV
4
31.
cos  
5
with  in Quad III
8
Solutions
EF  2 34, EG  4 17, FG  2 34
1.
AB  5
2.
CD  4 5
3.
272  272
rt 
4.
7.
11.
Domain :  ,  
Range :  ,  
Domain :  5,5
Range :  3,3
1280
12.
5.
8.
Domain :  ,  
Range :  0,  
309
47.420
9.
13.
mA  60
15.
mB  80
17.
16.
sec   1, csc   und
613012
10.
18.
3
1
, cos  
2
2
tan    3, cot   
sec   2, csc   
20.
3
3
2 3
3
15
8
, cos   
17
17
15
8
tan    , cot   
8
15
17
17
sec    , csc  
8
15
sin  
72
58,58
14.
mZ  32, XZ  40
sin   
sin   0, cos   1
tan   0, cot   und
186
my  107  mU
sec    2, csc    2
19.
Range :  0,  
mV  41, ZY  28
mC  40
2
2
sin   
, cos   
2
2
tan   1, cot   1
Domain :  1,  
6.
sin   1, cos   0
21.
tan   und , cot   0
22.
sec   und , csc   1
5 34
3 34
, cos  
34
34
5
3
tan   , cot  
3
5
2
2
, cos  
2
2
tan   1, cot   1
sin  
sec   2, csc   2
sin  
23.
24.
6
25.
4
34
34
, csc  
3
5
sec  
possible
26.
1  sin   1
csc   1
27.
impossible
sec  1or sec  1
reciprocals
28.
possible
  tan   
sin  
30.
32.
29.
3
22
, cos   
5
5
tan   
66
66
, cot   
22
3
sec   
5 22
5 3
, csc  
22
3
3
4
sin    , cos  
5
5
3
4
tan    , cot   
4
3
5
5
sec   , csc   
4
3
impossible
not reciprocals
sin   
31.
tan  
39
5
, cos   
8
8
39
5 39
, cot  
5
39
8
8 39
sec    , csc   
5
39