Formule trigonometriche
Vediamo in questo ultimo capitolo un riassunto di tutte le formule trigonometriche che vi saranno utili nel futuro
Da memorizzare!
Formule di addizione e sottrazione
$$ cos(\alpha+\beta)=cos\alpha cos\beta-sin\alpha sin\beta $$ $$ cos(\alpha+\beta)=cos\alpha cos\beta-sin\alpha sin\beta $$ $$ cos(\alpha+\beta)=cos\alpha cos\beta-sin\alpha sin\beta $$ $$ sin(\alpha+\beta)=sin\alpha cos\beta+con\alpha sin\beta $$ $$ sin(\alpha+\beta)=sin\alpha cos\beta+con\alpha sin\beta $$ $$ sin(\alpha+\beta)=sin\alpha cos\beta+con\alpha sin\beta $$ $$ cos(\alpha-\beta)=cos\alpha cos\beta+sin\alpha sin\beta $$ $$ cos(\alpha-\beta)=cos\alpha cos\beta+sin\alpha sin\beta $$ $$ cos(\alpha-\beta)=cos\alpha cos\beta+sin\alpha sin\beta $$ $$ sin(\alpha-\beta)=sin\alpha cos\beta-con\alpha sin\beta $$ $$ sin(\alpha-\beta)=sin\alpha cos\beta-con\alpha sin\beta $$ $$ sin(\alpha-\beta)=sin\alpha cos\beta-con\alpha sin\beta $$
Da memorizzare!
Formule di duplicazione
$$ sin2\alpha=2sin\alpha cos\alpha $$ $$ cos2\alpha=cos^2\alpha-sin^2\alpha=2cos^2\alpha-1=1-2sin^2\alpha $$ $$ cos2\alpha=cos^2\alpha-sin^2\alpha$$ $$ \Downarrow $$ $$ 2cos^2\alpha-1=1-2sin^2\alpha $$ $$ cos2\alpha=cos^2\alpha-sin^2\alpha $$ $$ \Downarrow $$ $$ 2cos^2\alpha-1=1-2sin^2\alpha $$ $$ tg2\alpha=\frac{2tg\alpha}{1-tg^2\alpha} $$
Da memorizzare!
Formule di bisezione
$$ sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{2}} $$ $$ cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+cos\alpha}{2}} $$ $$ tg\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{1+cos\alpha}} $$ $$ sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{2}} $$ $$ cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+cos\alpha}{2}} $$ $$ tg\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{1+cos\alpha}} $$ $$ sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{2}} $$ $$ cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+cos\alpha}{2}} $$ $$ tg\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{1+cos\alpha}} $$
Da memorizzare!
Formule di prostaferesi
$$ sinp+sinq=2sin\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ sinp+sinq=2sin\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ sinp+sinq=2sin\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ sinp-sinq=2cos\frac{p+q}{2}sin\frac{p-q}{2} $$ $$ sinp-sinq=2cos\frac{p+q}{2}sin\frac{p-q}{2} $$ $$ sinp-sinq=2cos\frac{p+q}{2}sin\frac{p-q}{2} $$ $$ cosp+cosq=2cos\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ cosp+cosq=2cos\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ cosp+cosq=2cos\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ cosp-cosq=-2sin\frac{p+q}{2}sin\frac{p-q}{2} $$ $$ cosp-cosq=-2sin\frac{p+q}{2}sin\frac{p-q}{2} $$ $$ cosp-cosq=-2sin\frac{p+q}{2}sin\frac{p-q}{2} $$
Da memorizzare!Formule di addizione e sottrazione
$$ cos(\alpha+\beta)=cos\alpha cos\beta-sin\alpha sin\beta $$ $$ cos(\alpha+\beta)=cos\alpha cos\beta-sin\alpha sin\beta $$ $$ cos(\alpha+\beta)=cos\alpha cos\beta-sin\alpha sin\beta $$ $$ sin(\alpha+\beta)=sin\alpha cos\beta+con\alpha sin\beta $$ $$ sin(\alpha+\beta)=sin\alpha cos\beta+con\alpha sin\beta $$ $$ sin(\alpha+\beta)=sin\alpha cos\beta+con\alpha sin\beta $$ $$ cos(\alpha-\beta)=cos\alpha cos\beta+sin\alpha sin\beta $$ $$ cos(\alpha-\beta)=cos\alpha cos\beta+sin\alpha sin\beta $$ $$ cos(\alpha-\beta)=cos\alpha cos\beta+sin\alpha sin\beta $$ $$ sin(\alpha-\beta)=sin\alpha cos\beta-con\alpha sin\beta $$ $$ sin(\alpha-\beta)=sin\alpha cos\beta-con\alpha sin\beta $$ $$ sin(\alpha-\beta)=sin\alpha cos\beta-con\alpha sin\beta $$
Da memorizzare!Formule di duplicazione
$$ sin2\alpha=2sin\alpha cos\alpha $$ $$ cos2\alpha=cos^2\alpha-sin^2\alpha=2cos^2\alpha-1=1-2sin^2\alpha $$ $$ cos2\alpha=cos^2\alpha-sin^2\alpha$$ $$ \Downarrow $$ $$ 2cos^2\alpha-1=1-2sin^2\alpha $$ $$ cos2\alpha=cos^2\alpha-sin^2\alpha $$ $$ \Downarrow $$ $$ 2cos^2\alpha-1=1-2sin^2\alpha $$ $$ tg2\alpha=\frac{2tg\alpha}{1-tg^2\alpha} $$
Da memorizzare!Formule di bisezione
$$ sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{2}} $$ $$ cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+cos\alpha}{2}} $$ $$ tg\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{1+cos\alpha}} $$ $$ sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{2}} $$ $$ cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+cos\alpha}{2}} $$ $$ tg\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{1+cos\alpha}} $$ $$ sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{2}} $$ $$ cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+cos\alpha}{2}} $$ $$ tg\frac{\alpha}{2}=\pm\sqrt{\frac{1-cos\alpha}{1+cos\alpha}} $$
Da memorizzare!Formule di prostaferesi
$$ sinp+sinq=2sin\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ sinp+sinq=2sin\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ sinp+sinq=2sin\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ sinp-sinq=2cos\frac{p+q}{2}sin\frac{p-q}{2} $$ $$ sinp-sinq=2cos\frac{p+q}{2}sin\frac{p-q}{2} $$ $$ sinp-sinq=2cos\frac{p+q}{2}sin\frac{p-q}{2} $$ $$ cosp+cosq=2cos\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ cosp+cosq=2cos\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ cosp+cosq=2cos\frac{p+q}{2}cos\frac{p-q}{2} $$ $$ cosp-cosq=-2sin\frac{p+q}{2}sin\frac{p-q}{2} $$ $$ cosp-cosq=-2sin\frac{p+q}{2}sin\frac{p-q}{2} $$ $$ cosp-cosq=-2sin\frac{p+q}{2}sin\frac{p-q}{2} $$
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