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NAME:
FMP1O
Trigonometry Preview Assignment
Part 1: Right Triangles
A) Find the measure of each unknown angle (variable) in the triangle. Remember that all
angles in a triangle add up to 1800.
B) Label all the sides of each right triangle (Hypotenuse, Adjacent, Opposite).
2.
1.
x
=
3.
4.
11
5.
=
N
33
I
x=
Part 2: Pythagorus
A) Label the triangles (legs are a & b, hypotenuse is c).
B) Use Pythagorean relation to solve for the missing side length.
2 a
)
2
) or (b
2
2 b
2 c
) or (a
2
2+b
2=c
2=a
(c
—.
1.
16
63
2.
33
56
3.
37
4.
40
2$
5.
48
55
—
FMPIO
Unit 3: Trigonometry
Lesson 1: Primary Trig Ratios (Finding Lengths)
Trigonometry is the study of triangles.
For the right triangles below measure all the side lengths and all the angles.
Without measuring find the missing side lengths on the triangle below.
4 cm
x
y
Three Primary Trigonometric Ratios
Opposite
A
SOH
sin ZA
=
oppostie
hypotenus
Adjacent
CAH
cos LA
Example 1: Find then exp’ain the resu’t.
a) sin3O°
b) cos75°
c) tan65°
=
adjacent
hypotenus
TOA
tan LA
=
opposite
adjacent
Example 2: Use the sine, cosine or tangent ratios to find x andy.
b)
a)
62°
x
y
c)
2Zi
x
HW:
1) Pg 82: 3-5,9
2) Pg 101:3-5,12
FMPIO
Unit 3: Trigonometry
Lesson 2: Finding Angles
Three Primary Trigonometric Ratios
Opposite
A
Adjacent
sinZA=
TOA
CAH
SOH
oppostie
hypotenus
cosL4=
adjacent
hypotenus
tanL4=
opposite
adjacent
Example 1: Calculate sin A, cos A, tan A, sin C, cos C, tan C
C
A
Inverse Trig Ratios
For any angle sine, cosine and tangent give the corresponding ratio of sides in a right
triangle.
. These are often written or
1
, tan
1
, cos
1
To find angles we use the inverse trig ratios sin
with reciprocals. The
confusion
avoid
“arctangent”
to
or
“arccosine”
called “arcsine”,
inverse trig ratios do the opposite operation of sine, cosine and tangent (they take the any
ratio of sides and give the corresponding angle).
Example 2:
a) For the triangle below find tan LA and LA.
11
A
b) For the triangle below find cos LA and LA.
C)
For the triangle below find sin LA and LA.
JA
Example 3: Find all missing angles.
16cm
HW:
1) Pg. 75:3-5,8
2) Pg. 95: 4-8,10
FMPIO
Unit 3: Trigonometry
Lesson 2: Finding Angles
Three Primary Trigonometric Ratios
Opposite
A
Adjacent
sinZA=
TOA
CAH
SOH
opposite
cosLA=
hypotenus
adjacent
hypotenus
tan ZA=
opposite
adjacent
Example 1: Calculate sin A, cos A, tan A, sin C, cos C, tan C
C
A
4
B
Inverse Trig Ratios
For any angle sine, cosine and tangent give the corresponding ratio of sides in a right
triangle.
. These are often written or
1
, tan
1
To find angles we use the inverse trig ratios sin-, cos
called “arcsine”, “arccosine” or “arctangent” to avoid confusion with reciprocals. The
inverse trig ratios do the opposite operation of sine, cosine and tangent (they take the any
ratio of sides and give the corresponding angle).
Example 2:
a) For the triangle below find tan LA and ZA.
11
A
b) For the triangle below find cos LA and LA.
C)
For the triangle below find sin LA and LA.
Example 3: Find all missing angles.
16 cm
A
HW:
1) Pg. 75:3-5,8
2) Pg. 95: 4-8,10
FMPIO
Unit 3: Trigonometry
Lesson 3: Solving Right Triangles
Solving a Triangle
Solving a right triangle means finding all missing lengths and angles. When solving
a right triangle try to use only the original numbers to find each missing value.
The Angle sum of a Triangle = 1800
2
2 =a
2 +b
Pythagoras’ Theorem: c
Example 1: Solve the following triangles:
a)
C
25cm
16cm
B
A
b) Given that ang’e C
=
C
A
52°.
32.0 in
B
c)
23.0 cm
K
J
9.0 cm
L
HW:
Pg. 82: 3-5,9
Pg. 101:3-5
Pg.111:3-6
FMPIO
Unit 3: Trigonometry
Lesson 4: Trig Word Problems
Anale of Inclination (Elevation): upward angle from the horizontal
/1
Pnç$ect&evatiai
/
—
Iiorinta
Angle of Declination (Depression): downward angle from the horizontal
[tori,ntaf
\Angie of Depresson
Example 1: A guy wire for a flag pole is 10 m long. The foot of the guy wire is 7 m to
the foot of the flag pole. What is the angle of inclination of the guy wire?
Example 2: A 10-ft ladder leans against the side of a building with its base 4-ft from
the walL What angle, to the nearest degree, does the ladder make with the
wall?
Example 3: The angle of elevation of the sun is 68° when the tree casts a shadow
14.3 m long. How tall is the tree?
Example 4: A surveyor, 31 m from a building, uses a transit to measure the angle of
elevation to the top of the building to be 37°. The transit is set at a height
of 1.5 m.
a) Calculate the distance from the transit to the top of the building.
b) Calculate the height of the building.
HW:
1) Pg 76: 12,14,17-19
2) Pg 82: 6-8
3) Pg 96: 11-13
4) Pg 101: 6,7,9
FMPIO
Unit 3: Trigonometry
Lesson 5: Problems Involving More than One Right Triangle
Example 1: Calculate the length of x.
a)
x
8 cm
b)
12cm
x
Example 2: Cacu late the measure of angle ABC
a)
A
8 cm
B
5 cm
C
b)
6 cm
C
5 cm
B
cm
Example 3: Two TV towers are 40.5 m apart. From the top of shorter tower the angle
of elevaUon to the top of the taller tower is 31.2°. The angle of depression
to the base of the taller tower is 46.7°. Ca’culate the height of each tower.
HW:
Pg 118: 3(a,c), 4(a,c), 6,8,9,14
Name:
Unit 3: Trigonometry Review
1. Find tanL4
4
A
A
11
C.±
11
B
D.
4
2. What is the measure LC?
C
21
13
A
A. 38°
B
C. 62°
B. 52°
D. 74°
3. What is the measure of Lx to the nearest degree?
D
C
A
A. 13
B. 17
B
C. 23
D. 28
4. A window on the fourth floor of a building is 20 m above the ground. From the
window, the angle of depression to the base of a nearby building is 31° and the angle
of elevation to the top of the building is 40°. How tall is the nearby building to the
nearest metre.
A.24
B. 48
C.56
D.72
5. Determine tan A and tan C.
C
8
A
a. tan A =
b.tanA =
c.tanA =
d.tanA =
B
10
1.25; tan C = 0.8
0.8; tan C = 0.7809...
0.8; tan C = 1.25
0.6247...; tan C = 1.25
6. Determine the angle of inclination of the line to the nearest tenth of a degree.
a. 63.3°
d. 26.7°
c. 65.8°
b. 24.2°
7. Determine the measure of angle ABD to the nearest tenth of a degree.
C
D
8cm
A
a. 65.1°
b. 67.2°
19cm
c. 22.8°
B
d. 24.9°
8. Determine the tangent ratio for angle K.
M
L
12
K
12
a. 35
37
c. 12
12
37
b.
35
d. 12
9. Determine the length of side z to the nearest tenth of a centimetre.
4Jc7
a. 9.7 cm
d. 8.5 cm
c. 5.4 cm
b. 2.6 cm
10. Determine sin A and cos A to the nearest tenth.
C
20
A
a. sinA= 1.7;
b. sin A = 0.8;
c.sinA=0.6;
d.sinA=0.6;
cosA=0.8
cos A = 0.6
cosA=1.3
cosA=0.8
16
12
B
11. Determine the measure of angle D to the nearest tenth of a degree.
E
D
F
a. 67.6°
d. 20.9°
c. 22.4°
b. 69.1°
12. Determine the measure of angle Q to the nearest tenth of a degree.
P
7
Q
R
a. 68.4°
b. 69.8°
c. 21.6°
d. 20.2°
13. A helicopter is hovering 200 m above a road. A car stopped on the side of the road
is 300 m from the helicopter. What is the angle of elevation of the helicopter
measured from the car, to the nearest degree?
a. 56°
b. 48°
c. 42°
d. 34°
14. A rope that anchors a hot air balloon to the ground is 136 m long. The balloon is
72 m above the ground. What is the angle of inclination of the rope to the nearest
tenth of a degree?
a. 58.0°
b.62.1°
c.32.0°
d. 27.9°
15. Two guy wires are attached to the top of a radio tower. The wires are 75 ft. and 52 ft.
long. The longer wire is anchored to the ground at a point 58 ft. from the base of the
tower. The shorter wire is anchored to the ground at a point between the base of the
tower and the longer wire. Calculate the angle of inclination of the shorter guy wire
to the nearest tenth of a degree.
a. 66.10
d. 42.4°
c. 39.3°
b. 23.9°
16. Determine the perimeter of an equilateral triangle with height 11.9 cm. Give the
measure to the nearest tenth of a centimetre.
a. 81.8 cm
d. 71.4 cm
c. 30.9 cm
b. 41.2 cm
17. Determine the Oength of RS to the nearest tenth of a centimetre.
R
Q
S
T
cm
7
.
a. 6
b. 9.3 cm
c. 11.4 cm
d. 3.3 cm
18. Two trees are 55 yd. apart. From a point halfway between the trees, the angles of
elevation of the tops of the trees are measured. What is the height of each tree to
the nearest yard?
tree
tree
1/
55 yd.
a. 33 yd.; 31 yd.
b. 19 yd.; 15 yd.
c. 41 yd.; 50 yd.
d. 40 yd.; 49 yd.
19. From the top of an 80-ft. building, the angle of elevation of the top of a taller building
is 49° and the angle of depression of the base of this building is 62°. Determine the
height of the taller building to the nearest foot.
a.211ft.
d.276ft.
c.129ft.
b.112ft.
20. Calculate the measure of angle ABC to the nearest tenth of a degree.
A
7 cm
B’[ D
4 cm
C
a. 47.7°
c. 77.5°
b. 102.5°
d. 52.6°
21. Determine the length of AC to the nearest tenth of a centimetre.
B
C
43.3 cm
38.9 cm
D
a. 70.4cm
b. 141.6cm
c. 39.9 cm
d. 41.9cm
22. From the top of a cliff 60 m above a river, angles are measured as shown in the
diagram below. (this is a very nasty question...)
horizontal
/
---p
60
\
\
Calculate the width, w, of the river. (Answer to the nearest metre.)
A. 45 m
B. 53 m
C. 62 m
D. 71 m
Written Responses
1. From the top of a lighthouse, 40m above the sea, the angle of depression to a boat is
200. How far is the boat from the base of the lighthouse?
2. At a certain time of day, the rays of the sun strike the ground at an angle of 25°.
Calculate the length of the shadow cast by a building 40m high.
3. From a point 14.5m from the base of a flagpole, the angle of elevation to the top of
the flagpole is I 5°. If the person making the observations is I .5m tall, how high is the
flagpole?