Espressioni goniometriche con angoli associati

$-cos\left( -\theta-\dfrac{3}{2}\pi \right) =-cos\left( -\left( \theta+\dfrac{3}{2}\pi \right) \right) =-cos\left( \dfrac{3}{2}\pi +\theta \right) = -sen(\theta)$

$-sen( \theta-5\pi) =sen(5\pi-\theta) =sen( \pi-\theta+4\pi )=sen( \pi-\theta) = sen(\theta)$

$sen\left( -\theta-\dfrac{7}{2}\pi \right) =sen\left( -\left( \theta+\dfrac{7}{2}\pi \right) \right) =-sen\left( \theta +\dfrac{7}{2}\pi \right) = -sen\left( \theta+\dfrac{3}{2}\pi+2\pi \right) = -sen\left( \theta+\dfrac{3}{2}\pi \right) = -(-cos(\theta)) = cos(\theta)$

Da ciò, concludiamo che

$-cos\left( -\theta-\dfrac{3}{2}\pi \right) -sen( \theta-5\pi)+sen\left( -\theta-\dfrac{7}{2}\pi \right) = -sen(\theta) + sen(\theta) + cos(\theta) = cos(\theta)$

COS(-α - 3/2·pi) = COS(α + 3/2·pi)=COS(α)·COS(3/2·pi) - SIN(α)·SIN(3/2·pi)=

=COS(α)·0 - SIN(α)·(-1)=SIN(α)

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SIN(α - 5·pi) = SIN(α)·COS(5·pi) - SIN(5·pi)·COS(α) =

=SIN(α)·(-1) - 0·COS(α)=- SIN(α)

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SIN(-α - 7/2·pi) = - SIN(α + 7/2·pi)=

=- (SIN(α)·COS(7/2·pi) + SIN(7/2·pi)·COS(α))=

=- (SIN(α)·0 + (-1)·COS(α))=COS(α)

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Quindi:

- COS(-α - 3/2·pi) - SIN(α - 5·pi) + SIN(-α - 7/2·pi)=

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