22.3–22.4. Арктангенс ва арккотангенс - Алгебра — синфи 10
Арктангенс
Тангенс дар фосилаи ( − π 2 ; π 2 ) \left(-\dfrac{\pi}{2};\ \dfrac{\pi}{2}\right) ( − 2 π ; 2 π ) ҳамаи қиматҳои ҳақиқиро мегирад. Аз теоремаи реша барои ҳар як адади ҳақиқии a a
навбатӣ 22.5. Алоқаи функсияҳои роста ва чаппа. Сохтани кунҷ аз рӯйи қимати функсия 1 соат
a
tg b = a \operatorname{tg} b = a tg b = a Мисоли 1. arctg 1 = π 4 , \operatorname{arctg} 1 = \dfrac{\pi}{4}, arctg 1 = 4 π , tg π 4 = 1 \operatorname{tg}\dfrac{\pi}{4} = 1 tg 4 π = 1 π 4 ∈ ( − π 2 ; π 2 ) . \dfrac{\pi}{4} \in \left(-\dfrac{\pi}{2};\ \dfrac{\pi}{2}\right). 4 π ∈ ( − 2 π ;
Мисоли 2. arctg ( − 3 ) = − π 3 , \operatorname{arctg}\left(-\sqrt{3}\right) = -\dfrac{\pi}{3}, arctg ( − 3 ) = − 3 π , tg ( − π 3 ) = − 3 \operatorname{tg}\left(-\dfrac{\pi}{3}\right) = -\sqrt{3} tg ( − 3 π ) = − 3 − π 3 ∈ ( − π 2 ; π 2 ) . -\dfrac{\pi}{3} \in \left(-\dfrac{\pi}{2};\ \dfrac{\pi}{2}\right). − 3 π ∈ ( − 2 π ;
Мисоли 3. sin ( arctg ( − 3 ) + arcsin ( − 1 ) + arctg 0 ) \sin\left(\operatorname{arctg}\left(-\sqrt{3}\right) + \arcsin(-1) + \operatorname{arctg} 0\right) sin ( arctg ( − 3 ) + arcsin ( − 1 ) + arctg 0 )
Ҳал. Ҳар якро алоҳида меёбем. arctg ( − 3 ) = − π 3 ; \operatorname{arctg}\left(-\sqrt{3}\right) = -\dfrac{\pi}{3}; arctg ( − 3 ) = − 3 π ; arcsin ( − 1 ) = − π 2 , \arcsin(-1) = -\dfrac{\pi}{2}, arcsin ( − 1 ) = − 2 π , sin ( − π 2 ) = − 1 ; \sin\left(-\dfrac{\pi}{2}\right) = -1; sin ( − 2 π ) = − 1 ; arctg 0 = 0 , \operatorname{arctg} 0 = 0, arctg 0 = 0 , tg 0 = 0. \operatorname{tg} 0 = 0. tg 0 = 0. sin ( − π 3 − π 2 + 0 ) = sin ( − 5 π 6 ) = − sin 5 π 6 = − 1 2 . \sin\left(-\frac{\pi}{3} - \frac{\pi}{2} + 0\right) = \sin\left(-\frac{5\pi}{6}\right) = -\sin\frac{5\pi}{6} = -\frac{1}{2}. sin ( − 3 π − 2
Масалан, arctg ( tg π 4 ) = π 4 , \operatorname{arctg}\left(\operatorname{tg}\dfrac{\pi}{4}\right) = \dfrac{\pi}{4}, arctg ( tg 4 π ) = 4 π , tg ( arctg 1 ) = 1. \operatorname{tg}(\operatorname{arctg} 1) = 1. tg ( arctg 1 ) = 1. − a -a − a ( − π 2 ; π 2 ) \left(-\dfrac{\pi}{2};\ \dfrac{\pi}{2}\right) ( − 2 π ; 2 π )
Арккотангенс Котангенс дар фосилаи ( 0 ; π ) (0;\ \pi) ( 0 ; π ) a a a ( 0 ; π ) (0;\ \pi) ( 0 ; π ) b b b ctg b = a \operatorname{ctg} b = a ctg b = a
Мисоли 4. arcctg 1 3 = π 3 , \operatorname{arcctg}\dfrac{1}{\sqrt{3}} = \dfrac{\pi}{3}, arcctg 3 1 = 3 π , ctg π 3 = 1 3 \operatorname{ctg}\dfrac{\pi}{3} = \dfrac{1}{\sqrt{3}} ctg 3 π = 3 π 3 ∈ ( 0 ; π ) . \dfrac{\pi}{3} \in (0;\ \pi). 3 π ∈ ( 0 ; π ) .
Мисоли 5. arcctg ( − 3 ) = 5 π 6 , \operatorname{arcctg}\left(-\sqrt{3}\right) = \dfrac{5\pi}{6}, arcctg ( − 3 ) = 6 5 π , ctg 5 π 6 = − 3 \operatorname{ctg}\dfrac{5\pi}{6} = -\sqrt{3} ctg 6 5 π = − 3 5 π 6 ∈ ( 0 ; π ) . \dfrac{5\pi}{6} \in (0;\ \pi). 6 5 π ∈ ( 0 ; π ) .
Мисоли 6. tg ( arctg ( − 1 ) + 2 arcctg ( − 1 ) + arcctg 1 3 ) \operatorname{tg}\left(\operatorname{arctg}(-1) + 2\operatorname{arcctg}(-1) + \operatorname{arcctg}\dfrac{1}{\sqrt{3}}\right) tg ( arctg ( − 1 ) + 2 arcctg ( − 1 ) + arcctg 3 1 )
Ҳал. arctg ( − 1 ) = − π 4 ; \operatorname{arctg}(-1) = -\dfrac{\pi}{4}; arctg ( − 1 ) = − 4 π ; arcctg ( − 1 ) = 3 π 4 ; \operatorname{arcctg}(-1) = \dfrac{3\pi}{4}; arcctg ( − 1 ) = 4 3 π ; arcctg 1 3 = π 3 . \operatorname{arcctg}\dfrac{1}{\sqrt{3}} = \dfrac{\pi}{3}. arcctg 3 1 = − π 4 + 3 π 2 + π 3 = − 3 π + 18 π + 4 π 12 = 19 π 12 . -\dfrac{\pi}{4} + \dfrac{3\pi}{2} + \dfrac{\pi}{3} = \dfrac{-3\pi + 18\pi + 4\pi}{12} = \dfrac{19\pi}{12}. − 4 π + 2 3 π π \pi π 19 π 12 − π = 7 π 12 \dfrac{19\pi}{12} - \pi = \dfrac{7\pi}{12} 12 19 π − π = 12 7 π tg 7 π 12 = tg 105 ∘ = − ( 2 + 3 ) . \operatorname{tg}\dfrac{7\pi}{12} = \operatorname{tg}105^\circ = -(2 + \sqrt{3}). tg 12 7 π = tg 10 5 ∘ = − ( 2 +
Хосияти 3 дар машқҳо зарур аст. Исбот: котангенси ду тараф ба − a -a − a π − arcctg a ∈ ( 0 ; π ) \pi - \operatorname{arcctg} a \in (0;\ \pi) π − arcctg a ∈ ( 0 ; π ) arcctg a ∈ ( 0 ; π ) \operatorname{arcctg} a \in (0;\ \pi) arcctg a ∈ ( 0 ; π ) arcctg ( − 1 ) = π − π 4 = 3 π 4 . \operatorname{arcctg}(-1) = \pi - \dfrac{\pi}{4} = \dfrac{3\pi}{4}. arcctg ( − 1 ) = π − 4 π = 4
Хати тангенсҳо ва котангенсҳоро дар давра бубинед:
Арктангенс ва арккотангенс санҷида бин! Аз нав
Кор дар синф
Машқҳо
Дар синф
а) arctg 3 ; \operatorname{arctg}\sqrt{3}; arctg 3 ;
б) arctg ( − 1 ) ; \operatorname{arctg}(-1); arctg ( − 1 ) ;
в) arctg 0 ; \operatorname{arctg} 0; arctg 0 ;
г) arctg 3 3 ; \operatorname{arctg}\dfrac{\sqrt{3}}{3}; arctg 3 3 ;
ғ) arctg ( − 3 3 ) ; \operatorname{arctg}\left(\dfrac{-\sqrt{3}}{3}\right); arctg ( 3 − 3 ) ;
д) arctg 1 3 ; \operatorname{arctg}\dfrac{1}{\sqrt{3}}; arctg 3 1 ;
е) arctg ( tg ( − π 3 ) ) . \operatorname{arctg}\left(\operatorname{tg}\left(-\dfrac{\pi}{3}\right)\right). arctg ( tg ( − 3 π ) ) .
а) tg ( arctg 2 ) ; \operatorname{tg}(\operatorname{arctg} 2); tg ( arctg 2 ) ;
б) tg ( arctg ( − 1 3 ) ) ; \operatorname{tg}\left(\operatorname{arctg}\left(-\dfrac{1}{3}\right)\right); tg ( arctg ( − 3 1 ) ) ;
в) arctg ( tg 5 6 π ) ; \operatorname{arctg}\left(\operatorname{tg}\dfrac{5}{6}\pi\right); arctg ( tg 6 5 π ) ;
г) arctg ( tg π 10 ) ; \operatorname{arctg}\left(\operatorname{tg}\dfrac{\pi}{10}\right); arctg ( tg 10 π ) ;
д) arctg ( tg 3 ) ; \operatorname{arctg}(\operatorname{tg} 3); arctg ( tg 3 ) ;
е) tg ( 3 arctg 4 3 ) . \operatorname{tg}\left(3\operatorname{arctg}\dfrac{4}{3}\right). tg ( 3 arctg 3 4 ) .
а) tg ( arctg 1 + arctg ( − 1 ) ) ; \operatorname{tg}(\operatorname{arctg} 1 + \operatorname{arctg}(-1)); tg ( arctg 1 + arctg ( − 1 )) ;
б) tg ( arctg 1 3 − arctg 3 ) ; \operatorname{tg}\left(\operatorname{arctg}\dfrac{1}{\sqrt{3}} - \operatorname{arctg}\sqrt{3}\right); tg ( arctg 3 1 − arctg 3 ) ;
в) tg ( arctg 0 + arctg ( − 3 ) ) ; \operatorname{tg}(\operatorname{arctg} 0 + \operatorname{arctg}(-\sqrt{3})); tg ( arctg 0 + arctg ( − 3 )) ;
г) tg ( arctg 0 + arctg ( − 1 3 ) ) . \operatorname{tg}\left(\operatorname{arctg} 0 + \operatorname{arctg}\left(-\dfrac{1}{\sqrt{3}}\right)\right). tg ( arctg 0 + arctg ( − 3
а) arcctg 3 ; \operatorname{arcctg}\sqrt{3}; arcctg 3 ;
б) arcctg 0 ; \operatorname{arcctg} 0; arcctg 0 ;
в) arcctg 1 ; \operatorname{arcctg} 1; arcctg 1 ;
г) arcctg ( − 3 ) ; \operatorname{arcctg}(-\sqrt{3}); arcctg ( − 3 ) ;
д) arcctg ( − 1 3 ) ; \operatorname{arcctg}\left(-\dfrac{1}{\sqrt{3}}\right); arcctg ( − 3 1 ) ;
е) arcctg ( − 1 ) . \operatorname{arcctg}(-1). arcctg ( − 1 ) .
а) arctg ( − 3 ) + arccos ( 3 2 ) + arcsin 1 ; \operatorname{arctg}(-\sqrt{3}) + \arccos\left(\dfrac{\sqrt{3}}{2}\right) + \arcsin 1; arctg ( − 3 ) + arccos ( 2 3 ) + arcsin 1 ;
б) arcsin ( − 1 ) − 3 2 arccos 1 2 + 3 arctg ( − 1 3 ) . \arcsin(-1) - \dfrac{3}{2}\arccos\dfrac{1}{2} + 3\operatorname{arctg}\left(-\dfrac{1}{\sqrt{3}}\right). arcsin ( − 1 ) − 2 3 arccos 2 1
Дар хона
а) tg ( arctg ( − 3 5 ) ) ; \operatorname{tg}\left(\operatorname{arctg}\left(-\dfrac{3}{5}\right)\right); tg ( arctg ( − 5 3 ) ) ;
б) ctg ( arctg 5 6 ) ; \operatorname{ctg}\left(\operatorname{arctg}\dfrac{5}{6}\right); ctg ( arctg 6 5 ) ;
в) tg ( arctg 1 2 + arctg 1 4 ) ; \operatorname{tg}\left(\operatorname{arctg}\dfrac{1}{2} + \operatorname{arctg}\dfrac{1}{4}\right); tg ( arctg 2 1 + arctg 4 1
г) arcctg ( ctg π 2 ) ; \operatorname{arcctg}\left(\operatorname{ctg}\dfrac{\pi}{2}\right); arcctg ( ctg 2 π ) ;
д) ctg ( arcctg 3 ) ; \operatorname{ctg}\left(\operatorname{arcctg}\sqrt{3}\right); ctg ( arcctg 3 ) ;
е) arcctg ( ctg π 4 ) . \operatorname{arcctg}\left(\operatorname{ctg}\dfrac{\pi}{4}\right). arcctg ( ctg 4 π ) .
а) cos π 4 cos π 6 − sin π 4 sin π 6 ; \cos\dfrac{\pi}{4}\cos\dfrac{\pi}{6} - \sin\dfrac{\pi}{4}\sin\dfrac{\pi}{6}; cos 4 π cos 6 π − sin 4 π sin 6 π ;
б) sin π 3 cos π 4 − cos π 3 sin π 4 . \sin\dfrac{\pi}{3}\cos\dfrac{\pi}{4} - \cos\dfrac{\pi}{3}\sin\dfrac{\pi}{4}. sin 3 π cos 4 π − cos
а) arccos 1 + 2 arcsin 1 2 ; \arccos 1 + 2\arcsin\dfrac{1}{2}; arccos 1 + 2 arcsin 2 1 ;
б) arcsin 1 2 + arccos 1 2 . \arcsin\dfrac{1}{2} + \arccos\dfrac{1}{2}. arcsin 2 1 + arccos 2 1 .
9. Нобаробариро ҳал кунед:
а) ( x + 5 ) ≤ 2 x 2 + 4 ; (x + 5) \le 2x^2 + 4; ( x + 5 ) ≤ 2 x 2 + 4 ;
б) 10 − ( 2 x − 1 ) ≥ 1 − 7 x . 10 - (2x - 1) \ge 1 - 7x. 10 − ( 2 x − 1 ) ≥ 1 − 7 x .
10. Қатора мебоист масофаи байни шаҳрҳои A A A B B B 4 4 4 30 30 30 A A A 30 30 30 B B B 10 10 10 A A A B B B
11. Системаи муодилаҳоро ҳал кунед:
а) { x 2 + y 2 + x + y = 18 , x 2 − y 2 + x − y = 6 ; \begin{cases} x^2 + y^2 + x + y = 18, \\ x^2 - y^2 + x - y = 6; \end{cases} { x 2 + y 2 + x + y = 18 , x 2 − y 2 + x − y = 6 ;
б) { x 2 y 2 + x y = 72 , x + y = 6. \begin{cases} x^2y^2 + xy = 72, \\ x + y = 6. \end{cases} { x 2 y 2 + x y = 72 , x + y = 6.
12. Айниятро исбот кунед:
а) sin ( α + β ) − sin ( α − β ) = 2 cos α sin β ; \sin(\alpha + \beta) - \sin(\alpha - \beta) = 2\cos\alpha\sin\beta; sin ( α + β ) − sin ( α − β ) = 2 cos α sin β ;
б) cos ( α + β ) cos ( α − β ) = cos 2 α − sin 2 β . \cos(\alpha + \beta)\cos(\alpha - \beta) = \cos^2\alpha - \sin^2\beta. cos ( α + β ) cos ( α − β ) = cos 2 α − sin 2 β .
13. Завод мебоист дар муҳлати муайян 800 800 800 25 % 25\% 25% 10 10 10 2 2 2
Вазифаи хонагӣ
Машқҳои 6–13 ва қисми иҷронашудаи машқҳои «Дар синф».
Вазифаи хонагии онлайн: «Алгебра 10 · Вазифаи хонагӣ: 22.3–22.4. Арктангенс ва арккотангенс».
Хулоса
arctg a ∈ ( − π 2 ; π 2 ) , \operatorname{arctg} a \in \left(-\dfrac{\pi}{2};\ \dfrac{\pi}{2}\right), arctg a ∈ ( − 2 π ; 2 π ) , arcctg a ∈ ( 0 ; π ) ; \operatorname{arcctg} a \in (0;\ \pi); arcctg a ∈ ( 0 ; π ) ; a a a tg ( arctg x ) = x ; \operatorname{tg}(\operatorname{arctg} x) = x; tg ( arctg x ) = x ; arctg ( tg x ) = x \operatorname{arctg}(\operatorname{tg} x) = x arctg ( tg x ) = x ( − π 2 ; π 2 ) ; \left(-\dfrac{\pi}{2};\ \dfrac{\pi}{2}\right); ctg ( arcctg x ) = x ; \operatorname{ctg}(\operatorname{arcctg} x) = x; ctg ( arcctg x ) = x ; arcctg ( ctg x ) = x \operatorname{arcctg}(\operatorname{ctg} x) = x arcctg ( ctg x ) = x ( 0 ; π ) ; (0;\ \pi); ( 0 ; π Дар ин дарс
Ман медонам таърифи арктангенс (дар ( − π 2 ; π 2 ) \left(-\dfrac{\pi}{2};\ \dfrac{\pi}{2}\right) ( − 2 π ; 2 π ) ( 0 ; π ) (0;\ \pi) ( 0 ; π ) Ман метавонам arctg a \operatorname{arctg} a arctg a arcctg a \operatorname{arcctg} a arcctg a Ман медонам айниятҳои tg ( arctg x ) = x , \operatorname{tg}(\operatorname{arctg} x) = x, tg ( arctg x ) = x , arctg ( tg x ) = x , \operatorname{arctg}(\operatorname{tg} x) = x, arctg ( tg x ) = x , arctg ( − a ) = − arctg a , \operatorname{arctg}(-a) = -\operatorname{arctg} a, Такрор (2 дақиқа)
tg π 6 , \operatorname{tg}\dfrac{\pi}{6}, tg 6 π , tg π 4 , \operatorname{tg}\dfrac{\pi}{4}, tg 4 π , tg π 3 \operatorname{tg}\dfrac{\pi}{3} tg 3 π Дар кадом фосила тангенс меафзояд ва ҳамаи қиматҳоро мегирад?
Дар кадом фосила котангенс кам мешавад?
Ҷавобҳо
3 3 ; \dfrac{\sqrt{3}}{3}; 3 3 ; 1 ; 1; 1 ; Таъриф Арктангенси адади a a a b b b ( − π 2 ; π 2 ) \left(-\dfrac{\pi}{2};\ \dfrac{\pi}{2}\right) ( − 2 π
Хосиятҳо
arctg ( tg x ) = x \operatorname{arctg}(\operatorname{tg} x) = x arctg ( tg x ) = x − π 2 < x < π 2 . -\dfrac{\pi}{2} < x < \dfrac{\pi}{2}. − 2 π < Таъриф Арккотангенси адади a a a b b b ( 0 ; π ) (0;\ \pi) ( 0 ; π ) ctg b = a . \operatorname{ctg} b = a. ctg b = a . b = arcctg a . b = \operatorname{arcctg} a.
Расми 1. Чап: arctg √3 = π/3 — камони асосӣ дар нимдавраи рост, нуқтаи T аз хати тангенсҳо. Рост: arcctg(−1) = 3π/4 — камони асосӣ дар нимдавраи боло, нуқтаи N аз хати котангенсҳо. Хосиятҳо
arcctg ( ctg x ) = x \operatorname{arcctg}(\operatorname{ctg} x) = x arcctg ( ctg x ) = x 0 < x < π . 0 < x < \pi. 0 < x < π . ctg ( arcctg x ) = x \operatorname{ctg}(\operatorname{arcctg} x) = x Калимаҳои калидӣ арктангенс арккотангенс хати тангенсҳо камони асосӣ функсияи чаппа
Ман дар ин дарс чӣ омӯхтам? Ман метавонам:
✓ / ✗ арктангенс ва арккотангенси ададҳои махсусро ёбам;
✓ / ✗ нависам, дар кадом фосила arctg \operatorname{arctg} arctg arcctg \operatorname{arcctg} arcctg
✓ / ✗ arcctg ( − a ) \operatorname{arcctg}(-a) arcctg ( − a ) arcctg a \operatorname{arcctg} a
arcctg ( − a ) = − arcctg a . \operatorname{arcctg}(-a) = -\operatorname{arcctg} a. arcctg ( − a ) = − arcctg a .
arctg ( − 1 ) \operatorname{arctg}(-1) arctg ( − 1 )
Як ҷавобро интихоб кунед A − 3 π 4 -\dfrac{3\pi}{4} − 4 3 π
B − π 4 -\dfrac{\pi}{4} − 4
C D arcctg ( − 1 ) \operatorname{arcctg}(-1) arcctg ( − 1 )
Як ҷавобро интихоб кунед A 3 π 4 \dfrac{3\pi}{4} 4 3 π
B − π 4 -\dfrac{\pi}{4} − 4
C D arctg 3 + arcctg 3 \operatorname{arctg}\sqrt{3} + \operatorname{arcctg}\sqrt{3} arctg 3 + arcctg 3
Як ҷавобро интихоб кунед A B C D
Арк-функсияро бо маҷмӯи қиматҳояш мувофиқ кунед.
Ҳар як унсури чапро бо яке аз рост мувофиқ кунед arcsin a \arcsin a arcsin a
→ … [−π/2; π/2] (0; π) (−π; π) (−π/2; π/2) [0; π] [0; 2π] arccos a \arccos a arccos a
→ … [−π/2; π/2] (0; π) (−π; π) (−π/2; π/2) [0; π] [0; 2π] arctg a \operatorname{arctg} a arctg a
→ … [−π/2; π/2] (0; π) (−π; π) (−π/2; π/2) [0; π] [0; 2π] arcctg a \operatorname{arcctg} a arcctg a
→ … [−π/2; π/2] (0; π) (−π; π) (−π/2; π/2) [0; π] [0; 2π] 2
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arcctg ( − a ) = π − arcctg a . \operatorname{arcctg}(-a) = \pi - \operatorname{arcctg} a. arcctg ( − a ) = π − arcctg a .
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