sin 2 α + cos2 α = 1 - основное тригонометрическое тождество sin (α + β ) = sin α cos β + cos α sin β cos (α + β ) = cos α cos β − sin α sin β
2 sin α sin β = cos (α − β ) − cos (α + β ) 2 cos α cos β = cos (α − β ) + cos (α + β ) 2 sin α cos β = sin (α − β ) + sin (α + β ) sin α + sin β = 2 sin sin α − sin β = 2 cos
α +β 2
cos
α +β
cos α + cos β = 2 cos
2
α +β
sin
α −β
tg x ± tg y =
2
α −β
cos
ctg x ± ctg y =
2
α −β
π⎞ π⎞ ⎛ ⎛ sin x − cos x = 2 ⋅ sin ⎜ x − ⎟ = 2 ⋅ cos ⎜ x + ⎟ ⎝ 4⎠ ⎝ 4⎠
cos 2α = cos2 α − sin2 α
2 cos2 α = 1 + cos 2α
sin 3α = 3 sin α − 4 sin 3 α
2 sin 2 α = 1 − cos 2α
cos 3α = 4 cos3 α − 3 cos α cos α =
1 − tg2 1 + tg
tg (α + β ) =
tg α + tg β 1 − tg α tg β
ctg (α + β ) = tg 2α =
ctg α ctg β − 1 ctg α + ctg β
2 tg α 1 − tg2 α ctg2 α − 1 2 ctg α 1 sec α = cos α 1 cosec α = sin α
α 2
sin α =
2α
2 tg 1 + tg
2 tg α tg β =
sin ( y ± x ) sin x sin y
tg x ± ctg y = ±
2 2 α +β α −β cos α − cos β = −2 sin sin 2 2 π⎞ π⎞ ⎛ ⎛ sin x + cos x = 2 ⋅ sin ⎜ x + ⎟ = 2 ⋅ cos ⎜ x − ⎟ ⎝ 4⎠ ⎝ 4⎠
sin 2α = 2 sin α cos α
sin ( x ± y ) cos x cos y
2
2α
tg α + ctg β ctg α + tg β
ctg α ctg β =
ctg α + ctg β tg α + tg β
ctg 2α =
ctg x − tg x = 2 ctg 2 x
α
tg α + tg β ctg α + ctg β
tg α ctg β =
cos ( x ∓ y ) cos x sin y
tg2
α
=
2
2 tg α ± tg β =
1 − cos α 1 + cos α
sin (α ± β ) cos α cos β
ctg α ± ctg β =
sin ( β ± α ) sin α sin β
tg α + ctg β =
cos (α − β ) cos α sin β
ctg α − tg β =
cos (α + β ) sin α cos β
sec2 x = 1 + tg2 x cosec2 x = 1 + ctg2 x
sin x = a x = ( −1) k arcsin a + π k cos x = a x = ± arccos a + 2π k - решения простейших тригонометрических уравнений tg x = a x = arctg a + π k ctg x = a x = arcctg a + π k 1