Trigonometria

Dati:

TAN(α + β) = 5

TAN(α) = 2/3

con 0 < α < pi/2 ; 0 < β < pi/2

Incognite:

TAN(β) = ?

TAN(α - β) = ?

COS(α + β) = ?

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TAN(α + β) = (TAN(α) + TAN(β))/(1 - TAN(α)·TAN(β))

(2/3 + TAN(β))/(1 - 2/3·TAN(β)) = 5

TAN(β) = t

(2/3 + t)/(1 - 2/3·t) = 5---> t = 1

TAN(β) = 1

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TAN(α - β) = (TAN(α) - TAN(β))/(1 + TAN(α)·TAN(β))

TAN(α - β) = (2/3 - 1)/(1 + 2/3·1)

TAN(α - β) = - 1/5

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COS(α) = 1/√(1 + TAN(α)^2)

SIN(α) = TAN(α)/√(1 + TAN(α)^2)

COS(α) = 1/√(1 + (2/3)^2)----> COS(α) = 3·√13/13

SIN(α) = 2/3/√(1 + (2/3)^2)----> SIN(α) = 2·√13/13

(verifica: (3·√13/13)^2 + (2·√13/13)^2 = 1)

Analogamente:

COS(β) = 1/√(1 + TAN(β)^2)

SIN(β) = TAN(β)/√(1 + TAN(β)^2)

COS(β) = 1/√(1 + 1^2)---> COS(β) = √2/2

SIN(β) = 1/√(1 + 1^2)---> SIN(β) = √2/2

COS(α + β) = COS(α)·COS(β) - SIN(α)·SIN(β)

COS(α + β) = 3·√13/13·(√2/2) - 2·√13/13·(√2/2)

COS(α + β) = 3·√26/26 - √26/13

COS(α + β) = √26/26