6.30 Finding Angles Using the Cosine Law.notebook
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May 30, 2020
Solutions
cos(A) = 2bc
b2 + c2 a2
cos(A) = 2(9)(10)
92 + 102 82
cos(A) = 180117
cos(A) = 0.65
A = cos1(0.65)A = 49.458...
A = 49Ο
6.30 Finding Angles Using the Cosine Law.notebook
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May 30, 2020
cos(B) = 2ac
a2 + c2 b2
cos(B) = 2(13)(10)
132 + 102 122
cos(B) = 260125
cos(B) = 0.4807...
B = cos1(0.4807...)B = 61.264...
B = 61Ο
cos(C) = 2ab
a2 + b2 c2
cos(C) = 2(16)(17)
162 + 172 142
cos(C) = 544349
cos(C) = 0.6415...
C = cos1(0.6415...)C = 50.092...
C = 50Ο
6.30 Finding Angles Using the Cosine Law.notebook
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May 30, 2020
cos(K) = 2mn
m2 + n2 k2
cos(K) = 2(14)(16)
142 + 162 132
cos(K) = 448283
cos(K) = 0.6316...
K = cos1(0.6316...)K = 50.824...
K = 51Ο
cos(U) = 2tv
t2 + v2 u2
cos(U) = 2(1.8)(2.5)
1.82 + 2.52 2.42
cos(U) = 9
3.73
cos(U) = 0.4144...
U = cos1(0.4144...)U = 65.515...
U = 66Ο
6.30 Finding Angles Using the Cosine Law.notebook
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May 30, 2020
cos(G) = 2fh
f2 + h2 g2
cos(G) = 2(5.1)(6.2)
5.12 + 6.22 4.82
cos(G) = 63.2441.41
cos(G) = 0.6548...
G = cos1(0.6548...)G = 49.094...
G = 49Ο
A
RD
H
WN
190 mm
170 mm
210 mm
1.4 km
1.7 km 1.2 km
cos(D) = 2ar
a2 + r2 d2
cos(D) = 2(170)(190)
1702 + 1902 2102
cos(D) = 6460020900
cos(D) = 0.3235...
D = cos1(0.3235...)D = 71.123... D = 71Ο
cos(W) = 2hnh2 + n2 w2
cos(W) = 2(1.4)(1.2)
1.42 + 1.22 1.72
cos(W) = 3.360.51
cos(W) = 0.1517...
W = cos1(0.1517...)W = 81.269... W = 81Ο
6.30 Finding Angles Using the Cosine Law.notebook
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May 30, 2020
cos(J) = 2mv
m2 + v2 j2
cos(J) = 2(8.3)(12.0)
8.32 + 12.02 10.02
cos(J) = 199.2112.89
cos(J) = 0.5667...
J = cos1(0.5667...)J = 55.478...
J = 55.5Ο
cos(V) = 2jm
j2 + m2 v2
cos(V) = 2(10.0)(8.3)
10.02 + 8.32 12.02
cos(V) = 16624.89
cos(V) = 0.1499...
V = cos1(0.1499...)V = 81.376...
cos(M) = 2jv
j2 + v2 m2
cos(M) = 2(10.0)(12.0)
10.02 + 12.02 8.32
cos(M) = 240175.11
cos(M) = 0.7296...
M = cos1(0.7296...)M = 43.145...
V = 81.4Ο M = 43.1Ο
cos(J) = 2mv
m2 + v2 j2
cos(J) = 2(8.3)(12.0)
8.32 + 12.02 10.02
cos(J) = 199.2112.89
cos(J) = 0.5667...
J = cos1(0.5667...)J = 55.478...
J = 55.5Ο
b) sin(V) v =
sin(J)j
sin(V) 12.0 =
sin(55.5)10.0
sin(V) = 12.0sin(55.5)10.0sin(V) = 0.9889...
V = sin1(0.9889...)
V = 81.475...
V = 81.5Ο<M = 180 55.5 81.5
M = 43.0Ο
c) The answers are virtually identical (differences are due to rounding). The second method is preferred because it less calculating.
6.30 Finding Angles Using the Cosine Law.notebook
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May 30, 2020
cos(T) = 2uv
u2 + v2 t2
cos(T) = 2(5)(7)
52 + 72 62
cos(T) = 7038
cos(T) = 0.5428...
T = cos1(0.5428...)T = 57.121...
T = 57.1Ο
sin(V) v =
sin(T)t
sin(V) 7 =
sin(57.1)6
sin(V) = 7sin(57.1)
6sin(V) = 0.9795...
V = sin1(0.9795...)
V = 78.394...
V = 78.4Ο<U = 180 57.1 78.4
U = 44.5Ο
cos(P) = 2my
m2 + y2 p2
cos(P) = 2(5.4)(4.4)
5.42 + 4.42 4.92
cos(P) = 47.5224.51
cos(P) = 0.5157...
P = cos1(0.5157...)P = 58.950...
P = 59.0Ο
sin(Y) y =
sin(P)p
sin(Y) 4.4 =
sin(59.0)4.9
sin(Y) = 4.4sin(59.0)4.9
sin(Y) = 0.7697...
Y = sin1(0.7697...)
Y = 50.327...
Y = 50.3Ο<M = 180 59.0 50.3
M = 70.7Ο
6.30 Finding Angles Using the Cosine Law.notebook
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May 30, 2020
cos(B) = 2gn
g2 + n2 b2
cos(B) = 2(12)(15)
122 + 152 142
cos(B) = 360173
cos(B) = 0.4805...
B = cos1(0.4805...)B = 61.278...
B = 61.3Ο
sin(G) g =
sin(B)b
sin(G) 12 =
sin(61.3)14
sin(G) = 12sin(61.3)14
sin(G) = 0.7518...
G = sin1(0.7518...)
G = 48.749...
G = 48.7Ο<N = 180 61.3 48.7
N = 70.0Ο
N B
G
15 m14 m
12 m
cos(D) = 2rt
r2 + t2 d2
cos(D) = 2(3.8)(4.6)
3.82 + 4.62 5.02
cos(D) = 34.9610.6
cos(D) = 0.3032...
D = cos1(0.3032...)D = 72.349...
D = 72.3Ο
sin(R) r =
sin(D)d
sin(R) 3.8 =
sin(72.3)5.0
sin(R) = 3.8sin(72.3)5.0
sin(R) = 0.7240...
R = sin1(0.7240...)
R = 46.387...
R = 46.4Ο<T = 180 72.3 46.4
T = 61.3Ο
D T
R
5.0 km4.6 km
3.8 km
6.30 Finding Angles Using the Cosine Law.notebook
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May 30, 2020
cos(A) = 2bc
b2 + c2 a2
cos(A) = 2(3.8)(3.8)
3.82 + 3.82 4.82
cos(A) = 28.885.84
cos(A) = 0.2022...
A = cos1(0.2022...)A = 78.333...
A = 78Ο
B = 51Ο C = 51Ο
AB
C
The triangle is isosceles where <A is the nonequal angle.
<B = <C = (180 78) ÷ 2
= 51Ο
a)
b) 3.8 m
A
h
sin(A) = opphyp
sin(78) = h3.8
3.8sin(78) = h
3.7169... = h
Area = base x height ÷ 2
Area = 3.8 x 3.7169... ÷ 2
Area = 7.06...
Total area = 2(7.06...)
= 14 metres2
cos(W) = 2es
e2 + s2 w2
cos(W) = 2(17)(24)
172 + 242 212
cos(W) = 816424
cos(W) = 0.5196...
W = cos1(0.5196...)
W = 58.694...
W = 59Ο<? = 90 59
? = 31Ο
W E
S
21 km17 km
24 km
?
The ship should head at an angle of 31Ο to the western shore to
dock at the western port.
6.30 Finding Angles Using the Cosine Law.notebook
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May 30, 2020
cos(B) = 2ac
a2 + c2 b2
cos(B) = 2(5.0)(4.1)
5.02 + 4.12 6.02
cos(B) = 41.05.81
cos(B) = 0.1417...
B = cos1(0.1417...)B = 81.853...
B = 82Ο
sin(C) c =
sin(B)b
sin(C) 4.1 =
sin(82)6.0
sin(C) = 4.1sin(82)6.0
sin(C) = 0.6766...
C = sin1(0.6766...)
C = 42.584...
C = 43Ο<A = 180 82 43
A = 55Ο
C B
A
6.0 m
5.0 m
4.1 m
c)
a) b)
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