Funkcje Cyklometryczne

1. Wyznaczy¢ dziedzin¦ funkcji:

(a) f (x) = arcsin (4x − 2),

(b) f (x) = arccos (2

2x+4

− 3)

,

(d) f (x) =

√

4 arcsin 3x − π

,

(c) f (x) = log

3

2 arccos (1 − 2x) −

π

2



,

(e) f (x) = arcsin (arctg x),

(f) f (x) = p3arcctg (1 − 2x) − 2π.

2. Naszkicowa¢ wykres funkcji:

(a) f (x) = 3 arcsin (|x| − 2),

(b) f (x) = arccos

x

2

− 1

−

π

2

,

(d) f (x) = arctg (|x − 1|) + π,

(c) f (x) = 2arcctg (x + 2) − π,

(e) f (x) = 



arccos (x + 1) −

π

4




,

(f) f (x) = 3 arccos (3x − 6) − π.

(g) f (x) = arcsin (sin x),

(h) f (x) = sin (arcsin x),

(i) f (x) = arccos (cos x),

(j) f (x) = cos (arccos x),

(k) f (x) = arctg (tgx),

(l) f (x) = tg (arctgx)

(ª) f (x) = arcctg (ctgx),

(m) f (x) = ctg (arcctgx).

3. Wykaza¢ to»samo±ci:

(a) ∀x ∈ [−1, 1]

arcsin (−x) = − arcsin x

,

(b) ∀x ∈ [−1, 1]

arcsin x + arccos x =

π

2

,

(c) ∀x ∈ [−1, 1]

arccos (−x) = π − arccos x

,

(d) ∀x ∈ [−1, 1]

arcsin x =

arctg

x

√

1−x

2

,

(e) ∀x ∈ [−1, 1]

arccos x =

arcctg

x

√

1−x

2

,

(f) ∀x ∈ R

arctg (−x) = −arctgx,

(g) ∀x ∈ R

arcctg (−x) = π − arcctgx,

(h) ∀x ∈ R

arctgx + arcctgx =

π

2

,

(i) ∀x ∈ [0, 1]

arcsin x = arccos

√

1 − x

2

,

(j) ∀x ∈ [0, 1]

arcsin x =

arcctg

√

1−x

2

x

,

(k) ∀x ∈ [0, 1]

arccos x = arcsin

√

1 − x

2

,

(l) ∀x ∈ [0, 1]

arccos x =

arctg

√

1−x

2

x

,

(m) ∀x ∈ (0, +∞)

arctgx = arcctg

1
x

,

(n) ∀x ∈ (0, +∞)

arcctgx = arctg

1
x

,

(o) ∀x ∈ (0, +∞)

arctgx = arccos

1

√

1+x

2

,

(p) ∀x ∈ (0, +∞)

arcctgx = arcsin

1

√

1+x

2

.

4. Obliczy¢ warto±¢: arcsin

−

√

2

2

+ arccos

√

3

2

+

arctg tg

137π

6

+

arcctg −

√

3

.

1