Простейшие формулы

Page 1

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 =

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

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


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