Download cos(x+y) = cos(x) cos(y) – sin(x) sin(y)

Proof of the cosine of a sum formula
To prove: cos(x+y) = cos(x) cos(y) – sin(x) sin(y)
D
∆ACB is a right triangle. ∠CAB = x. ∠B = 90o. AB = 1
E
C
P
∠y
A
1
2
3
4
5
6
∠x
F
∆ACD is a right triangle, with ∠DAC =y, and ACD=90o.
Segment DF is perpendicular to AB.
Segment EC is perpendicular to CB.
Figure FECB is a rectangle.
P is the intersection of DF and AC.
B
Statement
Construct ∆ACB with ∠CAB = x, ∠B = 90o, AB= 1
AB / AC = cos(x)
AC = 1/cos(x)
Construct ∆ACD, with D on the opposite side of segment AC
from B, with ∠DAC =y, and ACD=90o.
∠ BAD = ∠ (x+y)
7
8
9
10
11
12
From D, drop a perpendicular to A
B, meeting AB in point F
AC/AD = cos(y)
(1/cos(x) ) / AD = cos(y)
AD = 1/ [ cos(x) cos(y) ]
AF/AD = cos(x+y)
AF = AD cos(x+y) = cos(x+y) / [ cos(x) cos(y) ]
Segment AC and segment DF intersect in a point P
13
14
15
16
∠ APF = ∠ DPC
∠ FDC is complementary to ∠ DPC
∠ APF is complementary to ∠ x
∠ FDC = ∠ BAC = x
17
Construct CE perpendicular to DF from the point C
18
19
20
21
22
23
24
25
Figure FECB is a rectangle
CE = FB
DC = sin(y) AD = sin(y) / [ cos(x) cos(y) ]
CE =DC sin(x) = sin(x) sin(y) / [ cos(x) cos(y) ]
AF+FB = AB = 1
cos(x+y) / [ cos(x) cos(y) ] + sin(x)sin(y)/[cos(x)cos(y)] = 1
cos(x+y) + sin(x)sin(y) = cos(x)cos(y)
cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Which was to be proved.
Reason
Angle construction postulate.
cos = adjacent/hypotenuse
AB=1 (#1), and #2
Angle construction postulate.
By construction, and the Angle Addition
Theorem (segment AC is interior to ∠
BAD); #4.
From a point not on a line, there exists a
perpendicular to the line.
cos = adjacent/hypotenuse
Substitution, #3, #7
#8, multiply both sides by AD/cos(y)
cos = adjacent/hypotenuse
#9, #10
Since ∠B is a right angle, ∠x is acute, and
AC and DF are not parallel.
Vertical angles are congruent
#4, #12
#1, #6
Complements of the same ∠ are
congruent, #14, #15
From a point not on a line, there exists a
perpendicular to the line.
#1,#4,#17, so it has 3 right angles
Opposite sides of a rectangle
sin = opposite/hypotenuse in ∆CED
sin = opposite/hypotenuse in ∆CED
Segment addition postulate
#22, substitution, #11,#19,#21,
#23, multiply by cos(x)cos(y)
#24, subtraction