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It is fun to let vectors beat up trigonometry. The image shows two unit vectors depicted implicitly in Cartesian coordinates.
We will write these vectors in math notation then show that their scalar multiplication produces a common trigonometric relation. Vectors and their operations contain trigonometry.
The vectors (of length "1") of the sketch are written as:
e0A = cosαI + sinαJ
e0B = cosβ I - sinβ J
The scalar product can be accomplished in two manners. We write both manners then equate them to obtain relationships. First we use the idea, "the dot product of two vectors equals the length of the first times the length of the second times the cosine of the smaller angle between them. Or symbolically:
e0A • e0B = |1|·|1|cos(α + β)
Next we scalar multiply the right sides of the first expression we wrote (above):
OA • OB
(cosα I + sinα J) • (cosβ I - sinβ J) = cosα cosβ - sinα sinβ
Equating the right sides of the two above equations yields:
cos(α + β) = cosα cosβ - sinα sinβ
Obviously the unit vectors can be cast to obtain results for 2θ, (α - β) and so on. Vector multiplication makes quick work of trigonometric identities. Use the vector product for sine-type identities.