Download Note 2. Trigonometric Functions The trigonometric functions sinx

Note 2. Trigonometric Functions
The trigonometric functions sin x and cos x are defined on (−∞, ∞). They are 2πperiodic, 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
1
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 sin x and cos x are differentiable on (−∞, ∞):
d
d
(sin x) = cos x
and
(cos x) = − sin x.
dx
dx
The functions tan x and cot x are defined by
sin x
cos x
The derivatives of tan x and cot x are
d
(tan x) = sec2 x
dx
It follows that
Z
sec2 x dx = tan x + C
tan x =
Moreover, we have
Z
tan x dx = − ln(cos x) + C
and
and
cot x =
d
(cot x) = − csc2 x.
dx
Z
and
cos x
.
sin x
csc2 x dx = − cot x + C.
Z
and
cot x dx = ln(sin x) + C.
If tan y = x and −π/2 < y < π/2, then we define y = arctan x. The function arctan x
is differentiable on (−∞, ∞):
1
d
(arctan x) =
.
dx
1 + x2
It follows that
Z
1
dx = arctan x + C.
1 + x2