A Quick Survey of Trigonometry: Formula Sheet. sin α

A Quick Survey of Trigonometry: Formula Sheet.
opposite leg
A
opposite leg
A
sin α
sin α =
=
tan α =
=
=
sec α =
hypotenuse
C
adjacent leg
B
cos α
B
adjacent leg
B
cos α
adjacent leg
=
cot α =
=
=
csc α =
cos α =
hypotenuse
C
opposite leg
A
sin α
sin2 α + cos2 α = 1
tan2 α + 1 = sec2 α
sin(α + 2nπ) = sin α
cos(−α) = cos α
tan(α + nπ) = tan α, for n = 0, ±1, ±2, . . .
cos(α + 2nπ) = cos α
sin(−α) = − sin α
sin(π/2 − α) = cos α
hypotenuse
C
1
=
=
adjacent leg
B
cos α
hypotenuse
C
1
=
=
opposite leg
A
sin α
cos(π − α) = − cos α
tan(π/2 − α) = cot α
cos(α ± β) = cos α cos β ∓ sin α sin β
sin(π − α) = sin α
sec(π/2 − α) = csc α
sin(α ± β) = sin α cos β ± cos α sin β
tan α ± tan β
tan(α ± β) =
1 ∓ tan α tan β
cos(2α) = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α
2 tan α
sin(2α) = 2 sin α cos α
tan(2α) =
1 − tan2 α
q
q
sin(α/2) = ± 12 (1 − cos α)
cos(α/2) = ± 12 (1 + cos α)
sin α
1 − cos α
=
1 + cos α
sin α
sin x
1 − cos x
sin x ≈ x when |x| is small
lim
=1
lim
=0
x→0 x
x→0
x
d
d
d
sin x = cos x
cos x = − sin x
tan x = sec2 x
dx
dx
dx
d
d
d
cot x = − csc2 x
sec x = sec x tan x
csc x = − csc x cot x
dx
dx
dx
1
1
1
d
d
d
arcsin(x) = √
arccos(x) = − √
arctan(x) =
dx
dx
dx
1 + x2
1 − x2
1 − x2
d
1
d
1
1
d
√
arccot(x) = −
arcsec(x) =
arccsc(x) = − √
2
2
dx
1+x
dx
dx
|x| x − 1
|x| x2 − 1
restricted domain
range
domain
range
tan(α/2) =
sin x
[−π/2, π/2]
[−1, 1]
arcsin x
[−1, 1]
[−π/2, π/2]
cos x
[0, π]
[−1, 1]
arccos x
[−1, 1]
[0, π]
tan x
(−π/2, π/2)
(−∞, +∞)
arctan x
(−∞, +∞)
(−π/2, π/2)
sec x
[0, π/2) ∪ (π/2, π]
(−∞, −1] ∪ [1, +∞)
arcsec x
(−∞, −1] ∪ [1, +∞)
[0, π/2) ∪ (π/2, π]
1
y = sin x
y = tan x
y = sec x
1.0
0.5
Kp K
1
2
p
1
2
p
p
3
2
p
2
p
5
2
p
3
Kp K
p
1
2
p
1
2
p
p
3
2
p
2
p
5
2
p
3
Kp K
p
1
2
1
p
2
p
p
3
2
p
2
p
2
p
5
2
p
3
p
p
3
p
K
0.5
K
1.0
y = cos x
y = cot x
y = csc x
1.0
0.5
Kp K
1
2
p
1
2
p
p
3
2
p
2
p
5
2
p
3
Kp K
p
1
2
p
1
2
p
p
3
2
p
2
p
5
2
p
3
Kp K
p
1
2
1
p
2
p
p
3
2
p
5
2
K
0.5
K
1.0
y = sin x
y = arcsin x
1
2
y = cos x
y = arccos x
p
p
1
1
0.5
0.5
1
K
1
2
1
p
2
K
p
K
1
0.5
0.5
1
1
2
2
p
p
p
K
K
0.5
0.5
K
K
1
1
K
1
2
y = tan x
K
p
1
y = arctan x
K
0.5
y = sec x
2
0.5
1
y = arcsec x
3
1
2
p
2
p
1
1
K
1
2
1
p
2
p
K
2
K
1
1
1
2
2
p
1
p
2
p
K
1
K
1
K
2
K
1
K
2
p
K
2
3
2
K
3
K
2
K
1
1
2
3