unit 4 trigonometry - wordpress.com · unit 4 trigonometry 9 c ___2 = ___2 + ___2 – 2(___)(___)...
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Unit 4 Trigonometry 1
Solving Triangles by Law of Sines
(I) Review of Right Triangle Trigonometry
Labeling Sides of a Right Triangle
State the primary Trigonometric Ratios
soh cah toa
Applying primary trig ratios to determine an unknown side or unknown angle
Example Solve for x
1 A lighthouse is 168 feet up
from the ocean A fishing ship
is 360 feet from the base of
the cliff Determine the angle
of inclination
Reference angle
360 ft
xdeg
168 ft
Objectives
Review of Right Triangle Trigonometry
Applying the Law of Sines to determine an unknown side length or
angle measure
Unit 4 Trigonometry 2
2 A cable 200 m in length is attached to a telephone pole and forms a 65deg angle on the
pole Determine the distance from the base of the pole to where the cable attaches to
the ground Include a sketch
(II) Solving an unknown side length or angle measure in an acute triangle
by applying the law of sines
Law of Sines
sin 119860
119886=
sin 119861
119887 =
sin 119862
119888
or
119886
sin 119860=
119887
119904119894119899 119861 =
119888
119904119894119899119862
A
a
B
b C
c
Sine of the angle
NOTE
bullThe law of sines is applied when
(i) an angle is known
and
(ii) the side opposite that angle is known
Unit 4 Trigonometry 3
Example 1 Determining an unknown side
Determine which ship is closer
to the ship in distress
Example 2 Determining an unknown angle
Determine the angle at the air traffic control tower
20 km
70deg
60deg
x
y
2 km
6 km 10deg
x
Unit 4 Trigonometry 4
Example 3 Applying the law of sines based on a verbal description
A pulley is suspended from the ceiling by two chains One chain 62 m in length
forms an angle of 55deg with the ceiling Determine the length of the other chain
which forms an angle 30deg with the other ceiling
P139 ndash 141 3a c d 4b 5 7 10 13
P143 4 5a 7
Unit 4 Trigonometry 5
Solving Triangles by Law of Cosines
(I) Law of Cosines
Law of Cosines
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
Or rearranging for cos of the angle cos 119860 = 1198872+ 1198882minus 1198862
2119887119888
Applied when there isnrsquot an angle and side opposite known
A
a
B
b C
c
Side Opposite this angle
Other Two sides
Objectives
Introduction of Law of Cosines
Applying the Law of Cosines to Determine an Unknown Side
Applying the Law of Cosines to Determine an Unknown Angle
Unit 4 Trigonometry 6
(II) Applying the Law of Cosines to determine an unknown side
Example Determine the distance between the two satellites
(II) Applying the Law of Cosines to Determine an Unknown Angle
Example
Two flights depart from Deer Lake
One flight heads to St Anthony and
the other towards St Johnrsquos
Determine the angle formed from
the departing flights at Deer Lake
36 km 40 km
25deg
x
374 km
241 km
489 km
x
Unit 4 Trigonometry 7
(III) Applying the Law of Cosines to solve a word problem
Example A ship passing an island establishes by sonar a distance of 35 km from
the ship to one end of the island and 51 km to the other end The angle
from the ship contained between the tips of the island is 115deg Determine
the length of the island
P151 ndash 153 2 3 4 5 8 9 12
Unit 4 Trigonometry 8
Applying the Law of SinesCosines to Solve Problems
(I) Remember
Primary Trig Ratios
Consider a reference right angle
triangle
The three primary trigonometry ratios are
Law of Sines
sin 119860
119886=
sin 119861
119887 =
sin 119862
119888
or
119886
sin 119860=
119887
119904119894119899 119861 =
119888
119904119894119899119862
Law of Cosines
NOTE
The law of sines is applied when
(i) an angle is known
and
(ii) the side opposite that angle is known
A
a
B
b C
c Side opposite the angle
Sine of the angle
Objectives
Review of Trig Ratios Law of Sines amp Law of Cosines
Applying Trig to Solve Triangle Problems
Unit 4 Trigonometry 9
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in
more than one triangle to solve a problem
Example A ship navigating the coast knows from a map that the height of the cliff is
150 m high From the ships position the angle of elevation to the base of
the lighthouse is 10deg and the angle of elevation to the top of the lighthouse
is 15deg Determine the height of the lighthouse
A
a
B
b C
c
Applied when there isnrsquot an angle AND side opposite known
Side Opposite this angle
Other Two sides
150 m
15deg 10deg
h
Unit 4 Trigonometry 10
Example Determine the value of x
Example Two ships 2 km apart using sonar have located a sunken ship One ship
emits its sonar at an angle of depression of 30deg and the other ship at 80deg
Determine the distance of each ship from the sunken ship
x 32ordm
55ordm
25ordm
10
9
P161 4 5b 6 7 9 11a 13
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 2
2 A cable 200 m in length is attached to a telephone pole and forms a 65deg angle on the
pole Determine the distance from the base of the pole to where the cable attaches to
the ground Include a sketch
(II) Solving an unknown side length or angle measure in an acute triangle
by applying the law of sines
Law of Sines
sin 119860
119886=
sin 119861
119887 =
sin 119862
119888
or
119886
sin 119860=
119887
119904119894119899 119861 =
119888
119904119894119899119862
A
a
B
b C
c
Sine of the angle
NOTE
bullThe law of sines is applied when
(i) an angle is known
and
(ii) the side opposite that angle is known
Unit 4 Trigonometry 3
Example 1 Determining an unknown side
Determine which ship is closer
to the ship in distress
Example 2 Determining an unknown angle
Determine the angle at the air traffic control tower
20 km
70deg
60deg
x
y
2 km
6 km 10deg
x
Unit 4 Trigonometry 4
Example 3 Applying the law of sines based on a verbal description
A pulley is suspended from the ceiling by two chains One chain 62 m in length
forms an angle of 55deg with the ceiling Determine the length of the other chain
which forms an angle 30deg with the other ceiling
P139 ndash 141 3a c d 4b 5 7 10 13
P143 4 5a 7
Unit 4 Trigonometry 5
Solving Triangles by Law of Cosines
(I) Law of Cosines
Law of Cosines
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
Or rearranging for cos of the angle cos 119860 = 1198872+ 1198882minus 1198862
2119887119888
Applied when there isnrsquot an angle and side opposite known
A
a
B
b C
c
Side Opposite this angle
Other Two sides
Objectives
Introduction of Law of Cosines
Applying the Law of Cosines to Determine an Unknown Side
Applying the Law of Cosines to Determine an Unknown Angle
Unit 4 Trigonometry 6
(II) Applying the Law of Cosines to determine an unknown side
Example Determine the distance between the two satellites
(II) Applying the Law of Cosines to Determine an Unknown Angle
Example
Two flights depart from Deer Lake
One flight heads to St Anthony and
the other towards St Johnrsquos
Determine the angle formed from
the departing flights at Deer Lake
36 km 40 km
25deg
x
374 km
241 km
489 km
x
Unit 4 Trigonometry 7
(III) Applying the Law of Cosines to solve a word problem
Example A ship passing an island establishes by sonar a distance of 35 km from
the ship to one end of the island and 51 km to the other end The angle
from the ship contained between the tips of the island is 115deg Determine
the length of the island
P151 ndash 153 2 3 4 5 8 9 12
Unit 4 Trigonometry 8
Applying the Law of SinesCosines to Solve Problems
(I) Remember
Primary Trig Ratios
Consider a reference right angle
triangle
The three primary trigonometry ratios are
Law of Sines
sin 119860
119886=
sin 119861
119887 =
sin 119862
119888
or
119886
sin 119860=
119887
119904119894119899 119861 =
119888
119904119894119899119862
Law of Cosines
NOTE
The law of sines is applied when
(i) an angle is known
and
(ii) the side opposite that angle is known
A
a
B
b C
c Side opposite the angle
Sine of the angle
Objectives
Review of Trig Ratios Law of Sines amp Law of Cosines
Applying Trig to Solve Triangle Problems
Unit 4 Trigonometry 9
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in
more than one triangle to solve a problem
Example A ship navigating the coast knows from a map that the height of the cliff is
150 m high From the ships position the angle of elevation to the base of
the lighthouse is 10deg and the angle of elevation to the top of the lighthouse
is 15deg Determine the height of the lighthouse
A
a
B
b C
c
Applied when there isnrsquot an angle AND side opposite known
Side Opposite this angle
Other Two sides
150 m
15deg 10deg
h
Unit 4 Trigonometry 10
Example Determine the value of x
Example Two ships 2 km apart using sonar have located a sunken ship One ship
emits its sonar at an angle of depression of 30deg and the other ship at 80deg
Determine the distance of each ship from the sunken ship
x 32ordm
55ordm
25ordm
10
9
P161 4 5b 6 7 9 11a 13
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 3
Example 1 Determining an unknown side
Determine which ship is closer
to the ship in distress
Example 2 Determining an unknown angle
Determine the angle at the air traffic control tower
20 km
70deg
60deg
x
y
2 km
6 km 10deg
x
Unit 4 Trigonometry 4
Example 3 Applying the law of sines based on a verbal description
A pulley is suspended from the ceiling by two chains One chain 62 m in length
forms an angle of 55deg with the ceiling Determine the length of the other chain
which forms an angle 30deg with the other ceiling
P139 ndash 141 3a c d 4b 5 7 10 13
P143 4 5a 7
Unit 4 Trigonometry 5
Solving Triangles by Law of Cosines
(I) Law of Cosines
Law of Cosines
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
Or rearranging for cos of the angle cos 119860 = 1198872+ 1198882minus 1198862
2119887119888
Applied when there isnrsquot an angle and side opposite known
A
a
B
b C
c
Side Opposite this angle
Other Two sides
Objectives
Introduction of Law of Cosines
Applying the Law of Cosines to Determine an Unknown Side
Applying the Law of Cosines to Determine an Unknown Angle
Unit 4 Trigonometry 6
(II) Applying the Law of Cosines to determine an unknown side
Example Determine the distance between the two satellites
(II) Applying the Law of Cosines to Determine an Unknown Angle
Example
Two flights depart from Deer Lake
One flight heads to St Anthony and
the other towards St Johnrsquos
Determine the angle formed from
the departing flights at Deer Lake
36 km 40 km
25deg
x
374 km
241 km
489 km
x
Unit 4 Trigonometry 7
(III) Applying the Law of Cosines to solve a word problem
Example A ship passing an island establishes by sonar a distance of 35 km from
the ship to one end of the island and 51 km to the other end The angle
from the ship contained between the tips of the island is 115deg Determine
the length of the island
P151 ndash 153 2 3 4 5 8 9 12
Unit 4 Trigonometry 8
Applying the Law of SinesCosines to Solve Problems
(I) Remember
Primary Trig Ratios
Consider a reference right angle
triangle
The three primary trigonometry ratios are
Law of Sines
sin 119860
119886=
sin 119861
119887 =
sin 119862
119888
or
119886
sin 119860=
119887
119904119894119899 119861 =
119888
119904119894119899119862
Law of Cosines
NOTE
The law of sines is applied when
(i) an angle is known
and
(ii) the side opposite that angle is known
A
a
B
b C
c Side opposite the angle
Sine of the angle
Objectives
Review of Trig Ratios Law of Sines amp Law of Cosines
Applying Trig to Solve Triangle Problems
Unit 4 Trigonometry 9
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in
more than one triangle to solve a problem
Example A ship navigating the coast knows from a map that the height of the cliff is
150 m high From the ships position the angle of elevation to the base of
the lighthouse is 10deg and the angle of elevation to the top of the lighthouse
is 15deg Determine the height of the lighthouse
A
a
B
b C
c
Applied when there isnrsquot an angle AND side opposite known
Side Opposite this angle
Other Two sides
150 m
15deg 10deg
h
Unit 4 Trigonometry 10
Example Determine the value of x
Example Two ships 2 km apart using sonar have located a sunken ship One ship
emits its sonar at an angle of depression of 30deg and the other ship at 80deg
Determine the distance of each ship from the sunken ship
x 32ordm
55ordm
25ordm
10
9
P161 4 5b 6 7 9 11a 13
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 4
Example 3 Applying the law of sines based on a verbal description
A pulley is suspended from the ceiling by two chains One chain 62 m in length
forms an angle of 55deg with the ceiling Determine the length of the other chain
which forms an angle 30deg with the other ceiling
P139 ndash 141 3a c d 4b 5 7 10 13
P143 4 5a 7
Unit 4 Trigonometry 5
Solving Triangles by Law of Cosines
(I) Law of Cosines
Law of Cosines
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
Or rearranging for cos of the angle cos 119860 = 1198872+ 1198882minus 1198862
2119887119888
Applied when there isnrsquot an angle and side opposite known
A
a
B
b C
c
Side Opposite this angle
Other Two sides
Objectives
Introduction of Law of Cosines
Applying the Law of Cosines to Determine an Unknown Side
Applying the Law of Cosines to Determine an Unknown Angle
Unit 4 Trigonometry 6
(II) Applying the Law of Cosines to determine an unknown side
Example Determine the distance between the two satellites
(II) Applying the Law of Cosines to Determine an Unknown Angle
Example
Two flights depart from Deer Lake
One flight heads to St Anthony and
the other towards St Johnrsquos
Determine the angle formed from
the departing flights at Deer Lake
36 km 40 km
25deg
x
374 km
241 km
489 km
x
Unit 4 Trigonometry 7
(III) Applying the Law of Cosines to solve a word problem
Example A ship passing an island establishes by sonar a distance of 35 km from
the ship to one end of the island and 51 km to the other end The angle
from the ship contained between the tips of the island is 115deg Determine
the length of the island
P151 ndash 153 2 3 4 5 8 9 12
Unit 4 Trigonometry 8
Applying the Law of SinesCosines to Solve Problems
(I) Remember
Primary Trig Ratios
Consider a reference right angle
triangle
The three primary trigonometry ratios are
Law of Sines
sin 119860
119886=
sin 119861
119887 =
sin 119862
119888
or
119886
sin 119860=
119887
119904119894119899 119861 =
119888
119904119894119899119862
Law of Cosines
NOTE
The law of sines is applied when
(i) an angle is known
and
(ii) the side opposite that angle is known
A
a
B
b C
c Side opposite the angle
Sine of the angle
Objectives
Review of Trig Ratios Law of Sines amp Law of Cosines
Applying Trig to Solve Triangle Problems
Unit 4 Trigonometry 9
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in
more than one triangle to solve a problem
Example A ship navigating the coast knows from a map that the height of the cliff is
150 m high From the ships position the angle of elevation to the base of
the lighthouse is 10deg and the angle of elevation to the top of the lighthouse
is 15deg Determine the height of the lighthouse
A
a
B
b C
c
Applied when there isnrsquot an angle AND side opposite known
Side Opposite this angle
Other Two sides
150 m
15deg 10deg
h
Unit 4 Trigonometry 10
Example Determine the value of x
Example Two ships 2 km apart using sonar have located a sunken ship One ship
emits its sonar at an angle of depression of 30deg and the other ship at 80deg
Determine the distance of each ship from the sunken ship
x 32ordm
55ordm
25ordm
10
9
P161 4 5b 6 7 9 11a 13
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 5
Solving Triangles by Law of Cosines
(I) Law of Cosines
Law of Cosines
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
Or rearranging for cos of the angle cos 119860 = 1198872+ 1198882minus 1198862
2119887119888
Applied when there isnrsquot an angle and side opposite known
A
a
B
b C
c
Side Opposite this angle
Other Two sides
Objectives
Introduction of Law of Cosines
Applying the Law of Cosines to Determine an Unknown Side
Applying the Law of Cosines to Determine an Unknown Angle
Unit 4 Trigonometry 6
(II) Applying the Law of Cosines to determine an unknown side
Example Determine the distance between the two satellites
(II) Applying the Law of Cosines to Determine an Unknown Angle
Example
Two flights depart from Deer Lake
One flight heads to St Anthony and
the other towards St Johnrsquos
Determine the angle formed from
the departing flights at Deer Lake
36 km 40 km
25deg
x
374 km
241 km
489 km
x
Unit 4 Trigonometry 7
(III) Applying the Law of Cosines to solve a word problem
Example A ship passing an island establishes by sonar a distance of 35 km from
the ship to one end of the island and 51 km to the other end The angle
from the ship contained between the tips of the island is 115deg Determine
the length of the island
P151 ndash 153 2 3 4 5 8 9 12
Unit 4 Trigonometry 8
Applying the Law of SinesCosines to Solve Problems
(I) Remember
Primary Trig Ratios
Consider a reference right angle
triangle
The three primary trigonometry ratios are
Law of Sines
sin 119860
119886=
sin 119861
119887 =
sin 119862
119888
or
119886
sin 119860=
119887
119904119894119899 119861 =
119888
119904119894119899119862
Law of Cosines
NOTE
The law of sines is applied when
(i) an angle is known
and
(ii) the side opposite that angle is known
A
a
B
b C
c Side opposite the angle
Sine of the angle
Objectives
Review of Trig Ratios Law of Sines amp Law of Cosines
Applying Trig to Solve Triangle Problems
Unit 4 Trigonometry 9
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in
more than one triangle to solve a problem
Example A ship navigating the coast knows from a map that the height of the cliff is
150 m high From the ships position the angle of elevation to the base of
the lighthouse is 10deg and the angle of elevation to the top of the lighthouse
is 15deg Determine the height of the lighthouse
A
a
B
b C
c
Applied when there isnrsquot an angle AND side opposite known
Side Opposite this angle
Other Two sides
150 m
15deg 10deg
h
Unit 4 Trigonometry 10
Example Determine the value of x
Example Two ships 2 km apart using sonar have located a sunken ship One ship
emits its sonar at an angle of depression of 30deg and the other ship at 80deg
Determine the distance of each ship from the sunken ship
x 32ordm
55ordm
25ordm
10
9
P161 4 5b 6 7 9 11a 13
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 6
(II) Applying the Law of Cosines to determine an unknown side
Example Determine the distance between the two satellites
(II) Applying the Law of Cosines to Determine an Unknown Angle
Example
Two flights depart from Deer Lake
One flight heads to St Anthony and
the other towards St Johnrsquos
Determine the angle formed from
the departing flights at Deer Lake
36 km 40 km
25deg
x
374 km
241 km
489 km
x
Unit 4 Trigonometry 7
(III) Applying the Law of Cosines to solve a word problem
Example A ship passing an island establishes by sonar a distance of 35 km from
the ship to one end of the island and 51 km to the other end The angle
from the ship contained between the tips of the island is 115deg Determine
the length of the island
P151 ndash 153 2 3 4 5 8 9 12
Unit 4 Trigonometry 8
Applying the Law of SinesCosines to Solve Problems
(I) Remember
Primary Trig Ratios
Consider a reference right angle
triangle
The three primary trigonometry ratios are
Law of Sines
sin 119860
119886=
sin 119861
119887 =
sin 119862
119888
or
119886
sin 119860=
119887
119904119894119899 119861 =
119888
119904119894119899119862
Law of Cosines
NOTE
The law of sines is applied when
(i) an angle is known
and
(ii) the side opposite that angle is known
A
a
B
b C
c Side opposite the angle
Sine of the angle
Objectives
Review of Trig Ratios Law of Sines amp Law of Cosines
Applying Trig to Solve Triangle Problems
Unit 4 Trigonometry 9
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in
more than one triangle to solve a problem
Example A ship navigating the coast knows from a map that the height of the cliff is
150 m high From the ships position the angle of elevation to the base of
the lighthouse is 10deg and the angle of elevation to the top of the lighthouse
is 15deg Determine the height of the lighthouse
A
a
B
b C
c
Applied when there isnrsquot an angle AND side opposite known
Side Opposite this angle
Other Two sides
150 m
15deg 10deg
h
Unit 4 Trigonometry 10
Example Determine the value of x
Example Two ships 2 km apart using sonar have located a sunken ship One ship
emits its sonar at an angle of depression of 30deg and the other ship at 80deg
Determine the distance of each ship from the sunken ship
x 32ordm
55ordm
25ordm
10
9
P161 4 5b 6 7 9 11a 13
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 7
(III) Applying the Law of Cosines to solve a word problem
Example A ship passing an island establishes by sonar a distance of 35 km from
the ship to one end of the island and 51 km to the other end The angle
from the ship contained between the tips of the island is 115deg Determine
the length of the island
P151 ndash 153 2 3 4 5 8 9 12
Unit 4 Trigonometry 8
Applying the Law of SinesCosines to Solve Problems
(I) Remember
Primary Trig Ratios
Consider a reference right angle
triangle
The three primary trigonometry ratios are
Law of Sines
sin 119860
119886=
sin 119861
119887 =
sin 119862
119888
or
119886
sin 119860=
119887
119904119894119899 119861 =
119888
119904119894119899119862
Law of Cosines
NOTE
The law of sines is applied when
(i) an angle is known
and
(ii) the side opposite that angle is known
A
a
B
b C
c Side opposite the angle
Sine of the angle
Objectives
Review of Trig Ratios Law of Sines amp Law of Cosines
Applying Trig to Solve Triangle Problems
Unit 4 Trigonometry 9
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in
more than one triangle to solve a problem
Example A ship navigating the coast knows from a map that the height of the cliff is
150 m high From the ships position the angle of elevation to the base of
the lighthouse is 10deg and the angle of elevation to the top of the lighthouse
is 15deg Determine the height of the lighthouse
A
a
B
b C
c
Applied when there isnrsquot an angle AND side opposite known
Side Opposite this angle
Other Two sides
150 m
15deg 10deg
h
Unit 4 Trigonometry 10
Example Determine the value of x
Example Two ships 2 km apart using sonar have located a sunken ship One ship
emits its sonar at an angle of depression of 30deg and the other ship at 80deg
Determine the distance of each ship from the sunken ship
x 32ordm
55ordm
25ordm
10
9
P161 4 5b 6 7 9 11a 13
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 8
Applying the Law of SinesCosines to Solve Problems
(I) Remember
Primary Trig Ratios
Consider a reference right angle
triangle
The three primary trigonometry ratios are
Law of Sines
sin 119860
119886=
sin 119861
119887 =
sin 119862
119888
or
119886
sin 119860=
119887
119904119894119899 119861 =
119888
119904119894119899119862
Law of Cosines
NOTE
The law of sines is applied when
(i) an angle is known
and
(ii) the side opposite that angle is known
A
a
B
b C
c Side opposite the angle
Sine of the angle
Objectives
Review of Trig Ratios Law of Sines amp Law of Cosines
Applying Trig to Solve Triangle Problems
Unit 4 Trigonometry 9
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in
more than one triangle to solve a problem
Example A ship navigating the coast knows from a map that the height of the cliff is
150 m high From the ships position the angle of elevation to the base of
the lighthouse is 10deg and the angle of elevation to the top of the lighthouse
is 15deg Determine the height of the lighthouse
A
a
B
b C
c
Applied when there isnrsquot an angle AND side opposite known
Side Opposite this angle
Other Two sides
150 m
15deg 10deg
h
Unit 4 Trigonometry 10
Example Determine the value of x
Example Two ships 2 km apart using sonar have located a sunken ship One ship
emits its sonar at an angle of depression of 30deg and the other ship at 80deg
Determine the distance of each ship from the sunken ship
x 32ordm
55ordm
25ordm
10
9
P161 4 5b 6 7 9 11a 13
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 9
___2 = ___2 + ___2 ndash 2(___)(___) cos ___
(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in
more than one triangle to solve a problem
Example A ship navigating the coast knows from a map that the height of the cliff is
150 m high From the ships position the angle of elevation to the base of
the lighthouse is 10deg and the angle of elevation to the top of the lighthouse
is 15deg Determine the height of the lighthouse
A
a
B
b C
c
Applied when there isnrsquot an angle AND side opposite known
Side Opposite this angle
Other Two sides
150 m
15deg 10deg
h
Unit 4 Trigonometry 10
Example Determine the value of x
Example Two ships 2 km apart using sonar have located a sunken ship One ship
emits its sonar at an angle of depression of 30deg and the other ship at 80deg
Determine the distance of each ship from the sunken ship
x 32ordm
55ordm
25ordm
10
9
P161 4 5b 6 7 9 11a 13
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 10
Example Determine the value of x
Example Two ships 2 km apart using sonar have located a sunken ship One ship
emits its sonar at an angle of depression of 30deg and the other ship at 80deg
Determine the distance of each ship from the sunken ship
x 32ordm
55ordm
25ordm
10
9
P161 4 5b 6 7 9 11a 13
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 11
TRIGONOMETRY TEST REVIEW SHEET
Formulae
CSin
c
BSin
b
ASin
a
c
CSin
b
BSin
a
ASin
AbcCoscba 2222 bc
acbACos
2
222
1 Which represents the correct trigonometric equation for the diagram below
(A) 17
25cosx
(B) x
1725cos
(C) 17
25sinx
(D) x
1725sin
2 For which triangle would you use the Law of Sines to determine the missing side of x
(A) (B)
(C) (D)
105ordm
35ordm
7
x
105ordm 7
x
10
25
35ordm
x
10
x
7
20
15
x
25ordm
17
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 12
3 Solve for x
10
20
30 SinSin
x
(A) 1000 (B) 1012 (C) 5926 (D) 6392
4 Solve for x
40
2510
SinxSin
(A) 588deg (B) 1611deg (C) 1511deg (D) 7902
5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm
(A) 624 (B) 436 (C) 342 (D) 450
6 Solve for x 60
162536 xCos
(A) 46deg (B) 54deg (C) 41deg (D) 60deg
7 Which represents the appropriate equation to solve for x
(A)
55sin
28
45sin
x
(B)
45sin
28
55sin
x
(C)
45sin
28
80sin
x
(D)
55sin
28
80sin
x
x
28
55deg 45deg
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 13
8 Which represents the appropriate equation to solve for x
(A) 144
496481 xCos (B)
112
814964 xCos
(C) 126
644981 xCos (D)
126
644981 xCos
9 Determine the value of x
(a) (b)
(c) (d)
x
9
8 7
x
12
x
65ordm 45deg
x 8
34ordm 42ordm 12
10
x 30ordm
55ordm
25ordm
12
9
11 50deg
9 32ordm
9
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 14
10 Find the distance between the two police officers
11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an
angle of 62deg with the ceiling Determine the angle the second chain makes with the
ceiling if it has a length of 10 m
12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away
from the store If G = 63deg and H = 67deg then how far apart are George and Henry
before traveling to the store
S
G H
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km
Unit 4 Trigonometry 15
13 Determine the distance d between the two satellites
14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one
end of the island and 9 km to the other end of the island The angle formed at the ship
from the sonar is 74deg Determine the length of the island
15 What is the angle between the two planes as recorded at the radar
d
40 km 42 km
35deg
80 km
100 km
120 km
SOLUTIONS
1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C
9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km
13 2473 km 14 1026 km 15 8282 km