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Trigonometric Ratios
Trigonometry means “ triangle measurement”
Trigonometry was originally used in surveying. Now it is used in such fields as astronomy, navigation and architecture.
hypotenuse
Opposite
side
A
Adjacent
side
Sine ratio
sin
A
= opposite side
hypotenuse
=
o
h
Cosine ratio
cos
A
= adjacent side =
hypotenuse
a
h
Tangent ratio
tan
A
= opposite side =
adjacent side
o
a
Angle of depression
Angle of elevation
Line of sight
Remember the old Indian “Sohcahtoa”
Tangent
1. tan 15o = _________
2.
tan 30o = _________
3.
tan 65o = ____
4. tan ____ = 0.364
5.
tan ___ = 1.732
6.
tan ___ = 0.087
Use the definition of tangent to find the value of tan A for each right triangle. Express answer in lowest terms.
7. __________
8.
__________
B
9.
__________
B
5
5
A
C
A
10
6
C
A
8
C
8
B
o
o
o
10. What do you know about the legs of a 45 -45 -90 triangle? Without using a
table give the value of tan 45o. __________
11. Find the range of tan by giving these values:
tan 0o = _______
tan 30o = ______
tan 45o = ______
tan 75o = ______
tan 89o = ______
tan 90o = ______
tan 91o = ______
tan 100o = ______
tan 180o= ______ tan 270o = _____
tan 360o = _____
tan 390o = ______
--------------------------------------------------------------12. tan 5o = _________
13. tan 25o = _________
14.
tan 70o = ______
15. tan ____ = 3.732
16.
17.
tan ___ = 0.176
tan ___ = 0.577
Use the definition of tangent to find the value of tan A for each right triangle. Express answer in lowest terms.
18. tan A =_____
19. tan A = _____
B
13
A
C
C
4
A
4 2
A
21. tan A = ____
C
C
4
6
6 3
A
9
12
5
12
20. tan A = _____
15
B
12
B
B
Find the value of x to the nearest thousandth: Use a calculator.
22.
=
x
x
= _________
_____
23.
15
24.
3
40
35
10
25. x = _______
______
x = _________
x
2
26.
x = _______
x
27.
x=
x
65
25
70
x
8
7
x
5
80
20
10
x
28.
x = _______
29.
x = _______
30.
x = _______
x
x
58
63
32
12
27
44
50
22
31.
x = _______
32.
x = _______
33.
x
61
1.2
x
x = _______
x
50
7.1
100
Find x correct to the
34.
25
nearest degree.
x = _______
35.
x = _______
36.
x = _______
n 5
8
n
15
6.1
2n
17
4
37.
x
x
x
x = _______
38.
x = _______
39.
x
3
x
13
2
5n
4n
x = _______
x
34
Find w,
40.
then z, correct to the nearest hundredth:
3n
w = _______
z = _______
41.
w = _______
z = _______
35
z
z
40
w
42
150
w
w
w = _______
z = _______
30
200
z
42.
120
43. w = _______
z = _______
w
44.
45
900
28
Draw a
45.
w = _______
z = _______
z
82
60
w
diagram, write a trig
The diagram shows the path of an
airplane after take-off. Find x, the
x
15
80
42
z
equation, solve and label your answer.
altitude of the plane to the nearest
hundredth.
46.
x
The shadow of a building is 40 m
long. The angle between the ground
and the line to the sun is 35o.
Find x, the height of the
building to the nearest
hundredth.
35
40
47.
The grade of the road is 7%. What
angle does the road make with the
horizontal?
48.
A road climbs at an 8o angle with the
horizontal. What is the grade of the
road? Give your answer as a
percent.
49.
The base of an isosceles triangle is
70 cm long. The altitude to the base
is 75 cm long. Find, to the nearest
degree, the base angles of the
triangle.
50.
A rhombus has diagonals of lengths
4 and 10. Find the angles of the
rhombus to the nearest degree.
51.
The shorter diagonal of a rhombus
with a 70o is 122 long. How long, to
the nearest centimeter, is the longer
diagonal?
52. A rectangle is 80” long and 20” wide.
Find, to nearest degree, the acute
angle formed at the intersection of the
diagonals.
53. A rectangular box has lengths 4,
width 3 and height 2. Find BD and
 GBD to the nearest degree.
57. A surveyor is standing 118 feet from
the base of the Washington
Monument. The surveyor measures
the angle between the ground and the
top of the monument to be 78o. Find
the height of the Washington
Monument to the nearest foot.
G
2
D
C
3
A
4
B
54. Find the vertex angle of an isosceles
triangle with a base of 20 and a height
of 50.
x
32
48
y
7
58.
55. Find the angles of a rhombus with
diagonals of 10 and 24
28
40
x
12
56.
A rectangle is 10 wide and 15 long.
Find, to the nearest degree, the acute
angle formed at the intersection of the
diagonals.
59.
Q
Sine and Cosine
R
1.
sin P = _______
5
3
Refer to  PQR. Find each ratio:
2.
cos P = _______
P
4
3.
tan P = _______
4.
sin Q = _______
5.
cos Q = _______
6.
tan Q = _______
Express sin A, cos A and tanA as a fraction
C
b
B
B
17
7
A
8
C
A
7.
a
25
24
B
c
A
C
15
sin A = _____
8. sin A = _______
9.
sin A = ______
cos A = _____
cos A = _______
cos A = ______
tan A = _____
tan A = _______
tan A = ______
Find the following to the nearest thousandth:
10. sin 25o = ______
11.
cos 40o = ______
12.
sin 50o = ______
13. sin 5o = ______
14.
cos 15o = ______
15.
cos 80o = ______
Find the measure to the nearest degree:
16.
sin A = 0.259
 A = ______
17.
cos P = 0.643
 P = ______
18. sin S = 0.350
 S = ______
19.
sin A = 0.966
 A = _____
20.
cos A = 0.574
 A = _____
21.
cos A = 0.490
 A = ______
State an equation you could use to find the value of x and solve.
22.
24.
10
50
x
23.
25
12
5
x
x
20
_____________
______________
___________
State 2 different equations you could use to find the value of x:
100
x
41
12
50
x
49
35
55
8
40
x
The word cosine is related to the phrase “complement’s sine.”
Explain the relationship by using the diagram to express the cosine of  A
and the sine of its complement -  B
B
c
a
A
C
b
Find the value of y to the nearest hundredth:
8
10
y
40
y
35
y
6
20
5
y
y
7
50
65
35
3
y
Find the values of the variables to the nearest hundredth:
x
y
30
x
30
x
x
58
y
70
120
65
y
24
20
10
30
10
w
75
y
x
x
x
x
16
y
9
Applications: Sketch each problem. Solve each problem.
Round measures of segments to the
nearest hundredth and measures of angles to the nearest degree.
1.
A 20 ft ladder leans against a wall so
that the base of the ladder is 8 ft from the base
of the building. What angle does the ladder
make with the ground?
6.
A ladder leaning against a house makes
an angle of 60o with the ground. The foot of
the ladder is 7 feet from the foot of the house.
How long is the ladder?
2.
A 50-meter vertical tower is braced with
a cable secured at the top of the tower and tied
30 meters from the base. What angle does the
cable form with the vertical tower?
7.
A balloon on a 40 foot string makes an
angle of 50o with the ground. How high above
the ground is the balloon if the hand of the
person holding the balloon is 6 feet above the
ground?
3.
At a point on the ground 50 ft from the
foot of a tree, the angle of elevation to the top
of the tree is 53o. Find the height of the tree.
8.
From the top of a lighthouse 210 feet
high, the angle of depression of a boat is 27o.
Find the distance from the boat to the foot of
the lighthouse. The lighthouse was built at sea
level.
4.
From the top of a tower, the angle of
depression to a stake on the ground is 72o.
The top of the tower is 80 ft above the ground.
How far is the stake from the foot of the tower?
9.
A person is flying a kite. The kite string
makes an angle of 57o with the ground. If the
person is standing 100 feet from the point on
the ground directly below the kite, find the
length of the kite string.
5.
A tree 40 feet high casts a shadow 58 ft
long. Find the measure of the angle of
elevation of the sun.
10.
An airplane rises vertically 1000 feet
over a horizontal distance of 1 mile. What is
the angle of elevation of the airplane’s path?
Law of Sines
C
a
b
h
A
h
c
B
sin A =
C
sin B =
a
b
= h
h
A
= h
B
c
C
sin A =
a
sin C =
b
h
A
= h
= h
B
c
 ________________________________
Use in  s when given: 2  s and 1 side
2 sides and the  opposite one of the given sides
Solve each triangle: (find the missing sides- to nearest hundredth- and angles- to nearest whole no.)
1.
a = 20, m  A = 30, m  B = 45
3.
c = 8, b = 11, m  B = 87
2.
a = 3.5,  B = 35, m  A = 25
4.
b = 20, c = 9.2, m  B = 103
Law of
Cosines
C
a
b
I
 base into
II
A
B
c
x and ____________
I
– find altitude in terms of x and b
 II – use the Pythagorean Theorem to find a
a2 = ________________________________
*cos A =
square
a2 = ________________________________
combine
a2 = ________________________________
sub for x*
a2 = ________________________________
rearrange
Use in  s when given :
a2 = ________________________________
2 sides and included 
3 sides – start with largest side opposite the largest angle
a = 19, b = 20, m  C = 50o , c = ___
Sketch  . Find the measure rounded to nearest tenth.
1.
a = 5, b = 6, c = 8, m  A = _____
 . Find the measure
Sketch
2.
rounded
C
2.1
3.5
B
to nearest tenth
3.9
A
Solve the triangles to the nearest tenth:
3.
4
C
b
28
43
A
32
B
Find the measure indicated to the nearest tenth.
Show ALL your work!!
1.
a = 15, b = 12, c = 10,
 A = _______
2.
a = 2.2, b= 4.3,
 C = 52o
c=
_______
5.
a = 27, b = 41, c =15
 B _______
3.
 A = 23o,  B = 87o, a = 16
b = ______ c = _______
6.
 A = 110, a = 12, b = 5
 B = ______ c = ______
Law of Sines and Cosines
4.
a = 12,
 B = 70o,  C = 15
b = _______ c = _______
Laws of Sine and Cosine
nearest tenth
1.
In parallelogram ABCD, AB = 6,
AD = 3 and  A = 80o. Find
the length of the diagonals.
- answer to
2.
Find the base of an isosceles triangle
if each leg is 35 and the base angles
are 24o.
3.
Two angles of a triangle are 20o and
65o. If the longest side is 34, find
the length of the shortest side.
3
6.
Find the
area:
115
120
x
equilateral
4.
The diagonals of a parallelogram are
12 and 20. They meet at a 60o
angle. Find the perimeter of the
parallelogram.
7.
5.
Find x:
Find the area:
Solving
49.7
56.1
Triangles
114.6
The Ambiguous Case
SSS / SAS
ASS
ASA / AAS
4
Law of Cosines
Law of Sines
? A
acute/obtuse
obtuse
acute
compare
a to b
opp to adj
compare
a to h
h = b sinA
a<h
a=h
a<b
a>h
a>b
compare
a to b
a>b
a<b h < a < b
2
a>h and a>b
Indicate whether a solution exists and if so the
number of solutions for each set of data.
DO NOT SOLVE  .
1.
a = 22, b = 12,  A = 42o
2.
a = 15, b = 25,  A = 85o
8.
a = 4, b = 5,  A = 16o
9.
a = 7 , b = 2 ,  A = 106o
10.
a = 500, b = 330,  A = 40o
a = 15.2, b = 20,  A = 110o
3.
a = 12, b = 31,  A = 21o
4.
a = 4.5, b = 12.8,  A = 58o
5.
a
b
h
A
6.
a = 4.5, b = 5,  A = 58o
7.
a = 125, b = 200,  A = 110o
Two sides and an angle are given. Determine
whether the given information results in one
triangle, two triangles, or no triangles. Solve
any triangle(s) that results.
11.
 A = 50o, a = 3, b = 2
12.
 B = 20o, b = 4, c = 6
16.
13.
 C = 100o, a = 2, c = 1
 A = 60o, a = 4, b = 5
If 2 solutions exist find both.
a
b
h
h
A
1.
 D = 58o d= 4.5 e = 12.8
2.
 F = 58o f= 11.4 g = 12.8
3.
 J = 76o j= 18 k = 20
A
14.
15.
 C = 25o, a = 2, c = 1
 B = 100o, b = 5, c = 3
a
b
Use info to solve  (if possible)
No 
4.
 P = 76o p= 34 j = 21
5.
 T = 110o t= 125 v = 200
6.
 M = 110o m= 125 n = 100
a < h and h = b sinA
a < b sin A
a
a
< b therefore b >
sin A
sin A
2s
a
and b > a
sin A
a
a<b<
sin A
7.
 A = 36o a= 5
8.
 A = 60o a= 10
9.
 A = 88o a= 315.
b<
Find b such that  has
A)
1 solution
B)
No solutions
C)
2 solutions
1
a=h
h
sin A =
b
h
sin A
a
b=
sin A
b=
Example of Angle of
Elevation
Examples of Angle of
Depression
A searchlight located 200m from a tower
is turned on. If the angle of elevation to the
spot of light on the clouds is 35o, how high is
the cloud ceiling?
1.
2.
A fire is sighted from a tower. The
ranger found that the angle of depression to
the fire is 22o. If the tower is 75m tall, how far
is the fire from the base of the tower?
3.
From a spacecraft a crater is seen. The
angles of depression to the far and near edges
of the crater are 25o and 18o. If the spacecraft
is 3 miles above the crater, how wide is the
crater?
depression was 24o. How far is the bench from
the foot of the building?
8.
An observer at the top of a 50m
lighthouse sights 2 ships approaching, one
behind the other. The angles of depression of
the ships are 36o and 25o. Find the distance
between the ships.
9.
A surveyor could not measure the
distance ending directly under the top of a
mountain. So he marked 2 locations A and B
1000 m apart. He then measured the angle of
elevation to the top of the mountain from each
of these locations – 21o and 30o and drew the
diagram. Find the height of the mountain.
4.
A monument casts a shadow 215’ long
when the angle of elevation of the sun is 52o.
Find the height of the monument.
5.
The Chrysler Bldg. in NY is 1046’ tall. A
person stands half a mile away and views the
top of the building. Find the angle of elevation
to the top of the building.
6.
A person 1320’ from a TV tower sights
its top. The angle of elevation is 24o. How tall
is the tower?
Using the Law of Sines
1.
a = 12, m  B = 70o, m  C = 15o
 A = _____
b = _____
7.
A person on a building 180’ high looked
at a bench in a park below. The angle of
c = _____
 C = _____
a = _____
c = _____
2.
a = 12, b = 5, m  A =
110o
 B = _____
 C = _____
c = _____
6.
a = 7, m  A = 37o, m  B = 76o
 C = _____
b = _____
c = _____
3.
a = 8, m  A = 60o, m  C = 40o
 B = _____
b = _____
c = _____
7.
a = 9, b = 9, m  C = 20o
 B = _____
 A = _____
4.
a = 5, c = 4, m  A =
c = _____
65o
 C = _____
 B = _____
b = ____
8.
Solve each  rounding to tenths for lengths and
degree for s
5.
A ship is sighted from2 radar stations 43
km apart. The angle between the line segment
joining the two stations and the radar beam of
the first station is 37o. The angle between the
line segment joining the 2 stations and the beam
from the 2nd station is 113o. How far is the ship
from the 2nd station?
b = 6, m  A = 44o, m  B = 68o
Law of Cosines
c
(sin A )2 + (cos A)2 = (
)2 + (
)2
sin2A + cos2A = ________________
B
_________________
a
_________________
A
C
b
sin2A + cos2A =
C (x,y)
b
Distance from B to C is:
a
y
x
c
A (0,0)
______
B (c,0)
sinA =
y=
cosA =
x=
a=
_______________________
Remove
a2 =
________________________
Sub for x and y
a2 =
________________________
Foil and square
a2 = ____________________
Rearrange
a2 =
_________________________
Factor out b2
a2 =
__________________________
Substitute
a2 =
________________________
(Rearrange
a2 =
_______________________ )
 ________________________________
Can also be done for b and c
b2 = _________________________________
C2 = ____________________________________
Using the Law of Cosines
Solve each triangle  ABC. Round measures to the nearest tenth.
1.
 A =30o, b= 15, c= 30
a = _____
 B = _____
 C = _____
6.
 A = 43, b = 23, c = 26
a = _____
2.
 C = _____
a = 10, b = 15, c = 12
 B = _____
 A = _____
 B = _____
 C = _____
7.
a = 11, b = 14, c= 20
 A = _____
3.
a = 42, c = 60,  B = 58
 B = _____
 C = _____
b = _____
 A = _____
 C = _____
8.
a = 12.9, b = 18.4, c= 15.6
 A = _____
4.
 A = 115o, b = 10, c = 15
 B = _____
 C = _____
a = _____
 B = _____
 C = _____
5.
a = 7, b = 12, c = 15
 A = _____
Write in:
 B = _____
Cover: rtangle and angle A
 C = _____
p.1
#20 4 sq rt 2
#21 6 sq rt3
#24 change x to vertical
p. 2
#33 7.1 to adj side
#29 put in x
#38 sq rt 34
#40 and 41 squiggle for whole side
#36n sq rt 5
p.7 Rt angle signs
p. 9 Rt angle signs
p. 11
move # 4 down!!!
Vertical line separating cos A from
problem
p. 14
#6 Rt angle