??????? ?????????????????? ?????????.

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

ctg x = a (a ∈ R)   ⇒   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 - 1}sin x + ... + a_{n - 1}cos x(sin x)^{n - 1} + a_n(sin x)ⁿ = 0, n ∈ N, a₀ ≠ 0.

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

a₀zⁿ + a₁z^{n - 1} + ... + a_{n - 1}z + a_n = 0, n ∈ N, a₀ ≠ 0.

5. ????????????? ??????.

??? ??????? ????????? ????????? (????????, asinx + bcosx = c, a, b, c ∈ R) ????? ????? ???????????? ?????? 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, n ∈ Z, ?? ??? ??????????? ????? ???????? ? ?????? ??????. ?????? ?????????? ?????????, ?? ???????? ?? ????? ???? x = π + 2πn, n ∈ Z ??????? ????????? ?????????.

??????? ?????????????????? ??????????.

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

sin x < a   ⇒

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

x ∈ R, ??? a > 1;

x ∈ ∅, ??? a ≤ -1.

sin x > a   ⇒

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

x ∈ R, ??? a < -1;

x ∈ ∅, ??? a ≥ -1.

cos x < a   ⇒

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

x ∈ R, ??? a > 1;

x ∈ ∅, ??? a ≤ -1.

cos x > a   ⇒

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

x ∈ R, ??? a < -1;

x ∈ ∅, ??? a ≥ 1.

tg x < a   ⇒

πn - ^π/₂ < x < πn + arctg a, ??? a ∈ R (n ∈ N);

tg x > a   ⇒

πn + arctg a < x < πn + ^π/₂, ??? a ∈ R (n ∈ N);

?tg x < a   ⇒

πn + ar?ctg a < x < π(n + 1), ??? a ∈ R (n ∈ N);

?tg x > a   ⇒

πn < x < πn + ar?ctg a, ??? a ∈ R (n ∈ N);