When we add two angles and calculate a trigonometric function of them we realize that we won't get the same result if before we add these angles we apply the addition property in some cases, that is, we cannot always apply the following property cos (x + y) = cos x + cos y. See some examples:
Example 1:
cos (π + π) = cos (2π + π) = cos (3π) = cos 270º = 0
2 2 2
cos (π + π) = cos π + cos π = cos 180° + cos 90° = -1. 0 = 0
2 2
In this example it was possible to obtain the same result, but see the example below:
Example 2:
cos (π + π) = cos (2π) = cos 120º = 0
3 3 3
cos (π + π) = cos π + cos π = cos 60th + cos 60th = 1 + 1 = 1
3 3 3 3 2 2
We verify that the equality cos(x + y) = cos x + cos y is not true for any value that x and y take, so we conclude that the equalities:
sin (x + y) = sin x + sin y
sin (x - y) = sin x - sin y
cos (x + y) = cos x + cos y
cos (x - y) = cos x + cos y
tg (x + y) = tg x + tg y
tg (x - y) = tg x + tg y
These are equals that are not true for any value that x and y take, so look at the true equalities for calculating the addition or difference of sine, cosine, and tangent arcs.
• sin (x + y) = sin x. cos y + sin y. cos x
• sin (x - y) = sin x. cos y – sin y. cos x
• cos (x + y) = cos x. cos y – sin x. if you
• cos (x – y) = cos x. cos y + sin x. if you
• tg (x + y) = tg x + tg y
1 - tg x. yy
• tg (x - y) = tg x - tg y
1 + tg x. yy
by Danielle de Miranda
Graduated in Mathematics
Brazil School Team
Trigonometry - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/formulas-adicao-arcos.htm
