Proceedings of the International Conference on Theory and Applications of
Mathematics and Informatics - ICTAMI 2004, Thessaloniki, Greece
AN ORIGINAL METHOD OF PROVING THE FORMULA
OF A TRIGONOMETRIC FUNCTION OF A SUM OF
ANGLES
by
Emil Olteanu
Abstract: The formulas for calculating the trigonometric functions whose angle sum or
differences can be deduced from one another. However, we have to prove one of these
formulas. Considering this problem with multiple implications in trigonometry, we searched
for various methods of solving this formula and we searched for relations with different other
mathematics chapters (linear algebra, analysis in complex etc.). We searched for several proofs
for the initial formula, of which the method of the fictitious scalar product is an original one.
Proving the trigonometric functions of the angle sums and differences is
a problem starting from an initial formula, then the rest of the formulas
(including for double and triple angles and many other formulas) are deduced
from these. I give below a series of such proofs (the fictitious scalar product
method is original), considering that they make a connection between different
chapters of mathematics.
1. Geometric proof. We start from the following drawing:
N
β
α
P'
α
O
M
337
P
A
Figure 1.
Emil Olteanu - An original method of proving the formula of a trigonometric
function of a sum of angles
OA = r = 1
OP' = cos(β)
OP = OP' cos(α) = cos(β)cos(α)
OM = cos(α+β)
NP' = sin(β)
MP = NP' sin(α) = sin(β)sin(α)
but OM = OP - MP, results the formula
cos(α+β) = cos(α)cos(β) - sin(α)sin(β)
2. Using relations from analytical geometry
N
P
b
M
a
a-b
O(0,0)
Figure 2.
The points O, A, M, N and P have the following coordinates:
O (0,0);
A (1,0);
M (cos(a), sin(a));
N (cos(b), sin(b));
338
A(1,0)
Emil Olteanu - An original method of proving the formula of a trigonometric
function of a sum of angles
P (cos(a-b), sin(a-b)).
|MN|2 = (cos(a) - cos(b))2 + (sin(a) - sin(b))2 =
cos (a) - 2cos(a)cos(b) + cos2(b) + sin2(a) - 2sin(a)sin(b) + sin2(b) =
2 - 2(cos(a)cos(b) + sin(a)sin(b))
2
|AP|2 = (cos(a-b) - 1)2 + sin2(a-b) =
cos (a-b) - 2cos(a-b) + 1 + sin2(a-b) = 2 - 2cos(a-b)
2
but |MN| = |AP| , results
cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
3. We use relations from vectorial analysis.
We start from the scalar product formula.
( )
a • b = ab cos a, b
for the vectors i, j şi k , the module of which is one, the scalar product
table is the following:
•
i
j
i
1
0
j
0
1
k
0
0
k
0
0
1
Table 1.
i = j = k = 1;
cos(0o) = 1;
cos(90o) = 0.
In plane we have the vectorial relations:
(
)
a = a x i + a y j = i cos(α ) + j sin (α ) × a ;
339
Emil Olteanu - An original method of proving the formula of a trigonometric
function of a sum of angles
(
)
b = bx i + b y j = i cos(β ) + j sin (β ) × b .
a
α-β
α
b
β
Figure 3.
The angle between the vectors a and b is:
(a, b) = α − β
( )
a • b = ab cos a, b = ab cos(α − β )
(
) (
)
a • b = a i cos(α ) + j sin (α ) • b i cos(β ) + j sin (β ) =
(
)
= ab i • i cos(α )cos(β ) + i • j cos(α )sin (β ) + j • i sin (α )cos(β ) + j • j sin (α )sin (β ) =
= ab(cos(α ) cos(β ) + sin (α )sin (β ))
results
cos(α − β ) = cos(α )cos(β ) + sin (α )sin (β )
4. Using a product created for this proof (which in this material we shall call
fictitious scalar product).
We introduce a new type of scalar product, for instance:
a ⊗ b = ab sin( a, b)
340
Emil Olteanu - An original method of proving the formula of a trigonometric
function of a sum of angles
We have the following properties:
i = j = 1;
( )
j ⊗ j = sin (0 ) = 0 ;
i ⊗ j = sin (− 90 ) = − 1 ;
j ⊗ i = sin (90 ) = 1
i ⊗ i = sin 0 0 = 0 ;
0
0
0
For the vectors i and j , the module of which is one, the scalar product
table is the following:
⊗
j
-1
0
i
0
1
i
j
Table 2.
According to the prior proof results:
a ⊗ b = ab sin (α − β )
From the relations:
(
(
)
)
a = a x i + a y j = i cos(α ) + j sin (α ) a

b = bx i + b y j = i cos(β ) + j sin (β ) b
Results:
(
(
) (
)
a ⊗ b = a i cos(α ) + j sin (α ) ⊗ b i cos(β ) + j sin (β ) =
)
= ab i ⊗ i cos (α )cos (β ) + i ⊗ j cos (α )sin (β ) + j ⊗ i sin (α )cos (β ) + j ⊗ j sin (α )sin (β ) =
341
Emil Olteanu - An original method of proving the formula of a trigonometric
function of a sum of angles
= ab(− cos(α )sin (β ) + sin (α )cos(β ))
From these relations results:
sin (α − β ) = − cos(α )sin (β ) + sin (α )cos(β ) = sin (α )cos(β ) − cos(α )sin (β )
There are also other methods to prove a trigonometric formula of a sum
or difference/differences of angles. For instance, it can be solved with the help
of Euler’s formulas:
e ix = cos( x ) + i sin ( x )
.
 −ix
e = cos( x ) − i sin ( x )
References:
[1] W. Kecs - Complemente de matematici cu aplicaţii în tehnică (Mathematics
complements with technical applications) , Editura Tehnică, Bucureşti, 1981.
[2] I. Gh. Şabac - Matematici speciale (Special Mathematics) , Editura
Didactică şi Pedagogică, Bucureşti, 1981.
[3] Crstici Borislav (coordonator), ş.a. - Matematici speciale (Special
Mathematics), Editura Didactică şi Pedagogică, Bucureşti, 1981.
Author:
Emil Olteanu - „1 Decembrie 1918” University of Alba Iulia, Romania, E-mail
address: eolteanu@uab.ro
342