Довідникові матеріали


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Department of Theoretical Physics, Odessa National I.I.Mechnikov University

Тригонометрия

sin2(x)+cos2(x) = 1   tg(x) · ctg(x) = 1
tg(x) = sin(x)/cos(x)   ctg(x) = cos(x)/sin(x)
1+tg2(x) = 1/cos2(x)   1+ctg2(x) = 1/sin2(x)

2 sin2(x) = 1 − cos(2x)   2 cos2(x) = 1 + cos(2x)
4 sin3(x) = 3 sin(x) − sin(3x)   4 cos3(x) = 3cos(x) + cos(3x)
8 sin4(x) = 3 − 4cos(2x) + cos(4x)   8 cos4(x) = 3 + 4cos(2x) + cos(4x)

sin(a+b) = sin(a)·cos(b) + cos(a)·sin(b)   cos(a+b) = cos(a)·cos(b) − sin(a)·sin(b)
sin(a−b) = sin(a)·cos(b) − cos(a)·sin(b)   cos(a−b) = cos(a)·cos(b) + sin(a)·sin(b)
tg(a+b) = ( tg(a) + tg(b) ) / ( 1 − tg(a)·tg(b) )   ctg(a+b) = ( ctg(a)·ctg(b) − 1 ) / ( ctg(a) + ctg(b) )
tg(a−b) = ( tg(a) − tg(b) ) / ( 1 + tg(a)·tg(b) )   ctg(a−b) = ( ctg(a)·ctg(b) + 1 ) / ( ctg(a) − ctg(b) )

sin(2x) = 2 sin(x)·cos(x) = 2 tg(x) / ( 1 + tg2(x) )   cos(2x) = cos2(x) − sin2(x) = 2 cos2(x) − 1 = 1 − 2 sin2(x) =
            = ( 1 − tg2(x) ) / ( 1 + tg2(x) )
tg(2x) = 2 tg(x) / ( 1 − tg2(x) ) = 2 / ( ctg(x) − tg(x) )   ctg(2x) = ( ctg2(x) − 1 ) / ( 2 ctg(x) ) = ( ctg(x) − tg(x) ) / 2
sin(3x) = 3 sin(x) − 4 sin3(x)   cos(3x) = 4 cos3(x) − 3 cos(x)
tg(3x) = ( 3 tg(x) − tg3(x) ) / ( 1 − 3 tg2(x) )   ctg(3x) = ( ctg3(x) − 3 ctg(x) ) / ( 3 ctg2(x) − 1 )

sin(a)·sin(b) = 0.5 ( cos(a−b) − cos(a+b) )   sin(a)·cos(b) = 0.5 ( sin(a−b) + sin(a+b) )
cos(a)·cos(b) = 0.5 ( cos(a−b) + cos(a+b) )   cos(a)·sin(b) = 0.5 ( sin(a+b) − sin(a−b) )
tg(a)·tg(b) = ( tg(a) + tg(b) ) / (ctg(a) + ctg(b) )   ctg(a)·ctg(b) = (ctg(a) + ctg(b) ) / ( tg(a) + tg(b) )
sin(a+b)·sin(a−b) = cos2(b) − cos2(a)   cos(a+b)·cos(a−b) = cos2(b) − sin2(a)

sin(a) + sin(b) = 2 sin[(a+b)/2]·cos[(a−b)/2]   cos(a) + cos(b) = 2 cos[(a+b)/2]·cos[(a−b)/2]
sin(a) − sin(b) = 2 cos[(a+b)/2]·sin[(a−b)/2]   cos(a) − cos(b) = − 2 sin[(a+b)/2]·sin[(a−b)/2]
cos(a) + sin(a) = Sqrt(2)·sin(π/4+a) = Sqrt(2)·cos(π/4−a) *   cos(a) − sin(a) = Sqrt(2)·sin(π/4−a) = Sqrt(2)·cos(π/4+a) *
tg(a) ± tg(b) = sin(a±b) / ( cos(a)·cos(b) )   ctg(a) ± ctg(b) = ±sin(a±b) / ( sin(a)·sin(b) )
tg(a) + ctg(b) = cos(a−b) / ( cos(a)·sin(b) )   ctg(a) − tg(b) = cos(a+b) / ( sin(a)·cos(b) )

$$\sin^4(x) - \cos^4(x) = \sin^2(x) - \cos^2(x) = \cos(2x)$$   $$ \left( \sin(x) + \cos(x) \right)^2 = 1 + \sin(2x) $$
$$\sin(x) \pm \cos(x) = \sqrt{2} \pm \sin\left(x \pm \frac{\pi}{4}\right) $$   $$ \cos^6(x) + \sin^6(x) = \frac{5 + 3 \cos(4x)}{8} = \frac{1 + 3 \cos^2(2x)}{4} $$
$$\sin(x)\pm \sqrt{3} \cos(x) = 2 \sin\left(x \pm \frac{\pi}{3}\right) $$   $$ \cos^6(x) - \sin^6(x) = \frac{1}{16} ( 15 \cos(2x) + \cos(6x) ) $$
$$\cos(x)\pm \sqrt{3} \sin(x) = 2 \sin\left(x \pm \frac{\pi}{6}\right) $$   $$ \cos^8(x) - \sin^8(x) = \frac{1}{4} \cos(2x) \left( 3 + \cos(4x) \right) $$

* − здесь Sqrt(x) означает квадратный корень из числа x