23 uždavinys
1. Apskaičiuokite f(x) reikšmę, kai =
Sprendimas.
sin(x)-cos( 2* x) =
|
sin(x)-cos( 2* x) = 
=
Paaiškinimas: Keitimas 
= 
.
sin(
)-cos(
) = 
=
| π |
| / 2 |
| 2* π |
| / 2 |
sin(
)-cos(π) = 
=
| π |
| / 2 |
) = 1-cos(π) = 
=
) = 1+1 = 
=
2
Atsakymas: 2
2.Parodykite, kad f(x) = (sin(x)+1)(2sin(x)-1)
Sprendimas.
sin(x)-cos( 2* x) =
|
|
sin(x)-cos( 2* x) = =
Paaiškinimas: Kosinuso argumento žeminimo formulė 
= 
sin(x)-( cos(x)^2- sin(x)^2) = 
=
sin(x)- cos(x)^2+ sin(x)^2 = 
=
sin(x)-(1- sin(x)^2)+ sin(x)^2 = 
=
sin(x)-1+ sin(x)^2+ sin(x)^2 = 
=
sin(x)-1+ 2* sin(x)^2 = 
=
2* sin(x)^2+sin(x)-1 = 
=
Paaiškinimas: Kvadratinis trinaris 
, kur
a = 2, b = 1, c = -1.
Diskriminantas
= 
= 
= 9.
x1 =
= 
= 
= 1/2
x2 =
= 
= 
= -1
a = 2, b = 1, c = -1.
Diskriminantas
x1 =
x2 =
2* (sin(x)-
)* (sin(x)+1) = 
=
| 1 |
| / 2 |
( 2* sin(x)-
)* (sin(x)+1) = 
=
| 2* 1 |
| / 2 |
( 2* sin(x)-1)* (sin(x)+1)
3.Išspręskite lygtį f(x) = 0
Sprendimas.
2* sin(x)-1 =
0
0
|
|
2* sin(x)-1 = 0
= 0
2* sin(x) = 0+1
= 
2* sin(x) = 1 = 
sin(x) =
= 
| 1 |
| / 2 |
arcsin(sin(x)) = arcsin(
)
= 
| 1 |
| / 2 |
x = arcsin(
)
=
| 1 |
| / 2 |
x =
= 
| π |
| / 6 |
x = ( (-1)^k* (
)+ π* k) = 
| π |
| / 6 |
* k)x = (-1)^k* (
)+ π* k = 
| π |
| / 6 |
* ksin(x)+1 =
0
0
|
|
sin(x)+1 = 0
= 0
sin(x) = -1+0 = 
sin(x) = -1 = 
arcsin(sin(x)) = arcsin(-1) = 
x = arcsin(-1) =
x = -
= 
| π |
| / 2 |
x = -
+ 2* π* k = 
| π |
| / 2 |
* kAtsakymas: =
ir
=