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TaissaFarmiga
@TaissaFarmiga
July 2022
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докажите торжество
arctg(1/2)+arcctg(1/3)=pi/4
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oganesbagoyan
Verified answer
task/26323516
--------------------
Докажите
тождество
:
arctg(1/2) + arcctg(1/3) =
π
/4
---------------------
Пусть arctg(1/2) + arcctg(1/3)= φ
* * * tg(α + β) =
(
tgα + tgβ
)
/
(
1 -tgα *tgβ
)
* * *
tgφ
= tg( arctg1/2 + arcctg1/3 ) =tg( arctg1/2 + arctg3 )
(
tg( arctg1/2) +tg( arctg3)
) /
(
1 - tg( arctg1/2) *tg( arcctg3 )
) =
= (1/2 +3) /(1 -(1/2)*3) =3,5 /(-0,5)
= - 7 .
φ
≠ π/4 ⇒ пример неверно
.
========================================
Верное тождество :
arctg(1/2) + arc
t
g(1/3) = π /4
.
φ = arctg1/2 + arc
t
g1/3
tgφ
= tg
(
arctg1/2 + arc
t
g1/3
)
=
(
tg( arctg1/2 +tg( arctg1/3 ) /
(
1 -tg( arctg1/2) *tg( arctg1/3 )
) =
=
(
1/2 +1/3
)
/
(
1 -(1/2)*(1/3)
)
= (5 /6 )
/
(5/6)
= 1 .
⇒
φ = π/4
.
7 votes
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Answers & Comments
Verified answer
task/26323516--------------------
Докажите тождество :
arctg(1/2) + arcctg(1/3) = π /4
---------------------
Пусть arctg(1/2) + arcctg(1/3)= φ
* * * tg(α + β) = ( tgα + tgβ) / (1 -tgα *tgβ ) * * *
tgφ = tg( arctg1/2 + arcctg1/3 ) =tg( arctg1/2 + arctg3 )
( tg( arctg1/2) +tg( arctg3) ) / (1 - tg( arctg1/2) *tg( arcctg3 ) ) =
= (1/2 +3) /(1 -(1/2)*3) =3,5 /(-0,5) = - 7 . φ ≠ π/4 ⇒ пример неверно .
========================================
Верное тождество : arctg(1/2) + arctg(1/3) = π /4 .
φ = arctg1/2 + arctg1/3
tgφ = tg( arctg1/2 + arctg1/3 ) =
( tg( arctg1/2 +tg( arctg1/3 ) / (1 -tg( arctg1/2) *tg( arctg1/3 ) ) =
= (1/2 +1/3) / ( 1 -(1/2)*(1/3) ) = (5 /6 ) / (5/6) = 1 . ⇒ φ = π/4 .