Trigonometric functions are not additive, that is
cos
(
A
+
B
)
≠
cos
A
+
cos
B
.
cosine open eh plus b close not equal to cosine eh plus cosine b . It is also true that
cos
(
A
−
B
)
≠
cos
A
−
cos
B
.
cosine open eh minus b close not equal to cosine eh minus cosine b .
Here's Why It Works In the figure, angles A, B, and
A
−
B
eh minus b are shown.
Image Long Description
First, use the distance formula to find the square of the distance between P and Q.
(
P
Q
)
2
=
(
x
1
−
x
2
)
2
+
(
y
1
−
y
2
)
2
=
(
cos
A
−
cos
B
)
2
+
(
sin
A
−
sin
B
)
2
=
cos
2
A
−
2
cos
A
cos
B
+
cos
2
B
+
sin
2
A
−
2
sin
A
sin
B
+
sin
2
B
=
2
−
2
cos
A
cos
B
−
2
sin
A
sin
B
Use the Pythagorean identity
sin
2
θ
+
cos
2
θ
=
1.
table with 4 rows and 2 columns , row1 column 1 , open , p q , close squared , column 2 equals . open . x sub 1 , minus , x sub 2 . close squared . plus . open . y sub 1 , minus , y sub 2 . close squared , row2 column 1 , , column 2 equals . open . cosine eh minus cosine b . close squared . plus . open . sine eh minus sine b . close squared , row3 column 1 , , column 2 equals , cosine squared eh minus 2 cosine eh cosine b plus , cosine squared b plus , sine squared eh minus 2 sine eh sine b plus , sine squared b , row4 column 1 , , column 2 equals 2 minus 2 cosine eh cosine b minus 2 sine eh sine b . cap usethecap pythagoreanidentity . sine squared , theta plus , cosine squared , theta equals 1. , end table
Now use the Law of Cosines to find
(
P
Q
)
2
open p q close squared in
Δ
POQ
cap delta .
(
P
Q
)
2
=
(
P
Q
)
2
+
(
Q
O
)
2
−
2
(
P
O
)
(
Q
O
)
cos
(
A
−
B
)
=
1
2
+
1
2
−
2
(
1
)
(
1
)
cos
(
A
−
B
)
=
2
−
2
cos
(
A
−
B
)
table with 3 rows and 2 columns , row1 column 1 , open , p q , close squared , column 2 equals . open , p q , close squared . plus . open , q o , close squared . minus 2 . open , p o , close . open , q o , close cosine . open , eh minus b , close , row2 column 1 , , column 2 equals , 1 squared , plus , 1 squared , minus 2 , open 1 close . open 1 close cosine . open , eh minus b , close , row3 column 1 , , column 2 equals 2 minus 2 cosine . open , eh minus b , close , end table
The Transitive Property for Equality tells you that the two expressions for
(
P
Q
)
2
open p q close squared are equal.
2
−
2
cos
(
A
−
B
)
=
2
−
2
cos
A
cos
B
−
2
sin
A
sin
B
−
2
cos
(
A
−
B
)
=
−
2
cos
A
cos
B
−
2
sin
A
sin
B
Subtract
2
from each side
.
cos
(
A
−
B
)
=
cos
A
cos
B
+
sin
A
sin
B
Divide each side by
−
2.
table with 3 rows and 3 columns , row1 column 1 , 2 minus 2 cosine . open , eh minus b , close , column 2 equals 2 minus 2 cosine eh cosine b minus 2 sine eh sine b , column 3 , row2 column 1 , negative 2 cosine . open , eh minus b , close , column 2 equals negative 2 cosine eh cosine b minus 2 sine eh sine b , column 3 cap subtract . 2 . fromeachside . . , row3 column 1 , cosine . open , eh minus b , close , column 2 equals cosine eh cosine b plus sine eh sine b , column 3 cap divideeachsideby . minus 2. , end table
