14 . , 500 , 100 . 35 .

" ", "" " ". .
18.03.2013

LiveJournal Vkontakte Facebook Twitter

.

1. ( f(x) = a).

sin x = a (|a| ≤ 1)⇒x = (-1)n arcsin a + πn, n ∈ Z.

cos x = a (|a| ≤ 1)⇒x = arccos a + 2πn, n ∈ Z.

tg x = a (aR)⇒x = arctg a + πn, n ∈ Z.

ctg x = a (aR)⇒x = arcctg a + πn, n ∈ Z.

2. .

, .

a(sin x + cos x) + b sin 2x = c , sin x + cos x = t. 1 + sin 2x = t2,

at + b(t2 - 1) = c.

3. .

, , - . .

4.

a0(cos x)n + a1(cos x)n - 1sin x + ... + an - 1cos x(sin x)n - 1 + an(sin x)n = 0, nN, a0 ≠ 0.

(sin x)n ≠ 0 (.. sin x, cos x 0). ctg x = z

a0zn + a1zn - 1 + ... + an - 1z + an = 0, nN, a0 ≠ 0.

5. .

(, asinx + bcosx = c, a, b, cR) tg x/2 = z. sin x = 2z/(1 + z2), cos x = (1 - z2)/(1 + z2), tg x = 2z/(1 - z2). tg x/2 x = π + 2πn, nZ, . , x = π + 2πn, nZ .


.

( f(x) > a, f(x) < a)

sin x < a

π(2n - 1) - arcsin a < x < arcsin a + 2πn, a ∈ (-1;1] (nN);

xR, a > 1;

x ∈ ∅, a ≤ -1.

sin x > a

2nπ + arcsin a < x < π(2n + 1) - arcsin a, a ∈ [-1;1) (nN);

xR, a < -1;

x ∈ ∅, a ≥ -1.

cos x < a

n + arccos a < x < 2π(n + 1) - arccos a, a ∈ (-1;1] (nN);

xR, a > 1;

x ∈ ∅, a ≤ -1.

cos x > a

n - arccos a < x < 2πn + arccos a, a ∈ [-1;1) (nN);

xR, a < -1;

x ∈ ∅, a ≥ 1.

tg x < a

πn - π/2 < x < πn + arctg a, aR (nN);

tg x > a

πn + arctg a < x < πn + π/2, aR (nN);

tg x < a

πn + arctg a < x < π(n + 1), aR (nN);

tg x > a

πn < x < πn + arctg a, aR (nN);


2009-2012 . " - !" - , . .
Rambler's Top100