Download Some trigonometric identities Periodicity: sinx, cosx have period 2π

Some trigonometric identities
Periodicity: sin x, cos x have period 2π;
tan x, cot x have period π
Symmetry: cos is an even function, sin, tan, cot are odd.
Turning sin to cos:
sin( π2 − x) = cos x
cos( π2 − x) = sin x
Relating tan and sec: 1 + tan2 x = sec2 x , 1 + cot2 x = csc2 x
Expanding sums:
sin(x ± y) = sin x cos y ± cos x sin y
Products to sums:
cos(x ± y) = cos x cos y ∓ sin x sin y
,
x+y
x−y
1
sin
+ sin
sin x cos y =
2
2
2
1
y+x
y−x
1
y+x
y−x
cos x cos y =
cos
+ cos
, sin x sin y =
cos
− cos
2
2
2
2
2
2
Expand double angle:
cos(2x) = cos2 x − sin2 x = 2 cos2 x − 1 = 1 − 2 sin2 x
sin(2x) = 2 sin x cos x ,
or the other way around, go to double angle by
sin x cos =
1
sin(2x) ,
2
cos2 x =
1 + cos(2x)
2
,
sin2 x =
1 − cos(2x)
2
Some integrals:
Z
sec x dx = ln | sec x + tan x| + C
Z
,
csc x dx = − ln | csc x + cot x| + C
Derivatives of inverse trigonometric functions:
d
1
arcsin x = √
dx
1 − x2
,
d
1
arccos x = − √
dx
1 − x2
,
d
1
arctan x =
dx
1 + x2
Hyperbolic functions
Definitions:
sinh x =
ex − e−x
2
,
cosh x =
ex + e−x
2
Relations:
d
sinh x = cosh x ,
dx
cosh2 x − sinh2 x = 1 ,
sinh−1 x = ln x +
√
1 + x2
,
d
cosh x = sinh x
dx
d
1
sinh−1 x = √
dx
1 + x2