ЛЕКЦІЇ ² ВПРАВИ

14.3.11. Ïàðí³ñòü ³ íåïàðí³ñòü òðèãîíîìåòðè÷íèõ
ôóíêö³é:
sin (–x) = –sin x, sin — íåïàðíà ôóíêö³ÿ;
cos (–x) = cos x, cos — ïàðíà ôóíêö³ÿ;
tg (–x) = –tg x, tg — íåïàðíà ôóíêö³ÿ;
ctg (–x) = –ctg x, ctg — íåïàðíà ôóíêö³ÿ.
3.12. Ïåð³îäè÷í³ñòü òðèãîíîìåòðè÷íèõ ôóíêö³é:
sin x, cos x — ïåð³îäè÷í³ ôóíêö³¿ (íàéìåíøèé äîäàòíèé
ïåð³îä 2π):
( x k ) x ( x k ) x ( k )
sin + 2 π = sin , cos + 2 π = cos ∈Z .
tg x, ctg x — ïåð³îäè÷í³ ôóíêö³¿ (íàéìåíøèé äîäàòíèé
ïåð³îä π).
( x k ) x ( x k ) x ( k )
tg + π = tg , ctg + π = ctg ∈Z .
14.3.13. Âëàñòèâîñò³ îáåðíåíèõ òðèãîíîìåòðè÷íèõ
ôóíêö³é:
arcsin
arctg
( − a) = −arcsin
a ( a ≤1 ); arccos( a) arccos a ( a )
− = π− ≤1 ;
( − a) =−arctg
a ( a∈R ); arcctg( − ) =π−arcctg
( ∈ )
a a a R ;
14.3.15. Çíà÷åííÿ îáåðíåíèõ òðèãîíîìåòðè÷íèõ
ôóíêö³é
14.3.16. Òåîðåìà ñèíóñ³â (ðèñ. 14.6):
a b c
= =
sin α sin β sin γ .
π
arcsin a+ arccos a = a ≤1
2
π
a a a R .
2
( ); arctg + arcctg = ( ∈ )
14.3.14. Ðîçâ’ÿçàííÿ íàéïðîñò³øèõ òðèãîíîìåòðè÷íèõ
ð³âíÿíü:
k
( ) ( ) ( )
sin x = a a ≤ 1 , x = − 1 arcsin a +πk k∈Z ;
( ) ( )
cos x = a a ≤ 1 , x = ± arccos a + 2πk k∈Z ;
( ),
arctg ( )
tg x = a a∈ R x = a + πk k∈Z ;
( ),
arcctg ( )
ctg x = a a∈ R x = a+ πk k∈Z .
Ðèñ. 14.6
14.3.17. Òåîðåìà êîñèíóñ³â:
2 2 2
a = b + c − bc α ;
2 cos
2 2 2
b = a + c − ac β ;
2 cos
2 2 2
c a c 2ab cos .
= + − γ
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