Download Trigonometric Functions The trigonometric functions sinx and cosx

Trigonometric Functions
The trigonometric functions sin x and cos x are defined on
(−∞, ∞). They are 2π-periodic, i.e., sin(x + 2π) = sin x and
cos(x + 2π) = cos x for all x ∈ (−∞, ∞). The values of sin x
and cos x for some special angles x are listed as follows.
x
0
π/6
sin x
0
cos x
1
1/2
√
3/2
π/4
√
2/2
√
2/2
π/3
√
3/2
π/2
1/2
0
The following identities are useful.
1 = sin2 x + cos2 x
sin (2x) = 2 sin x cos x
cos (2x) = cos2 x − sin2 x
sin2 x = (1 − cos (2x))/2
cos2 x = (1 + cos(2x))/2.
The functions tan x and cot x are defined by
tan x =
sin x
cos x
and
1
cot x =
cos x
.
sin x
1
The functions sin x and cos x are differentiable on (−∞, ∞):
d
(sin x) = cos x and
dx
d
(cos x) = − sin x.
dx
Consequently,
Z
Z
sin x dx = − cos x + C
and
cos x dx = sin x + C.
The derivatives of tan x and cot x are
d
(tan x) = sec2 x and
dx
Note that sec x =
Z
1
cos x
d
(cot x) = − csc2 x.
dx
and csc x =
sec2 x dx = tan x + C
Z
and
1
sin x .
It follows that
csc2 x dx = − cot x + C.
Moreover, we have
Z
Z
tan x dx =
sin x
dx = − ln(cos x) + C
cos x
and
Z
Z
cot x dx =
cos x
dx = ln(sin x) + C.
sin x
2
If tan y = x and −π/2 < y < π/2, then we define
y = arctan x.
For example,
1
π
=√
6
3
π
tan = 1
4
π √
tan = 3
3
tan
=⇒
=⇒
=⇒
1
π
arctan √ = ;
6
3
π
arctan 1 = ;
4
√
π
arctan 3 = .
3
The function arctan x is differentiable on (−∞, ∞):
d
1
.
(arctan x) =
dx
1 + x2
It follows that
Z
1
dx = arctan x + C.
1 + x2
3