Applied Calculus I
Lecture 15
Trigonometric identities and limits
Let us recall that
sin2 x + cos2 x = 1,
1
sec x =
,
cos x
2
2
1 + cot x = csc x,
sin x
tan x =
,
cos x
cos x
cot x =
sin x
1
csc x =
,
sin x
1 + tan2 x = sec2 x,
sin 2x = 2 sin x cos x,
cos 2x = cos2 x − sin2 x
= 1 − 2 sin2 x
2
= 2 cos x − 1
Trigonometric identities and limits
Let us recall that
sin2 x + cos2 x = 1,
1
sec x =
,
cos x
2
2
1 + cot x = csc x,
sin x
tan x =
,
cos x
cos x
cot x =
sin x
1
csc x =
,
sin x
1 + tan2 x = sec2 x,
sin 2x = 2 sin x cos x,
cos 2x = cos2 x − sin2 x
= 1 − 2 sin2 x
2
= 2 cos x − 1
We also have the following identities
sin (x + y) = sin x cos y + cos x sin y
sin (x − y) = sin x cos y − cos x sin y
cos (x + y) = cos x cos y − sin x sin y
cos (x − y) = cos x cos y + sin x sin y
Trigonometric identities and limits
Let us recall that
sin2 x + cos2 x = 1,
1
sec x =
,
cos x
2
2
1 + cot x = csc x,
sin x
tan x =
,
cos x
cos x
cot x =
sin x
1
csc x =
,
sin x
1 + tan2 x = sec2 x,
sin 2x = 2 sin x cos x,
cos 2x = cos2 x − sin2 x
= 1 − 2 sin2 x
2
= 2 cos x − 1
We also have the following identities
sin (x + y) = sin x cos y + cos x sin y
sin (x − y) = sin x cos y − cos x sin y
cos (x + y) = cos x cos y − sin x sin y
cos (x − y) = cos x cos y + sin x sin y
And it can be shown that
sin x
lim
=1
x→0 x
Many other trigonometric
limits can be found from
this one and the identities.
cos h − 1
Find lim
.
h→0
h
Example
cos h − 1
Find lim
.
h→0
h
Example
cos h − 1
cos h − 1 cos h + 1
cos2 h − 1
= lim
·
= lim
=
lim
h→0
h→0
h
h
cos h + 1 h→0 h(cos h + 1)
2
− sin h
sin h
1
= lim
= lim (− sin h)
=
h→0 h(cos h + 1)
h→0
h
cos h + 1
1
=0·1·
=0
1+1
The derivative of sin x
sin (x + h) − sin x
=
(sin x) = lim
h→0
h
sin x cos h + cos x sin h − sin x
= lim
=
h→0
h
sin x(cos h − 1) + cos x sin h
= lim
=
h→0
h
sin x(cos h − 1)
cos x sin h
= lim
+ lim
=
h→0
h→0
h
h
= (sin x) · 0 + (cos x) · 1 =
= cos x
0
The derivative of sin x
sin (x + h) − sin x
=
(sin x) = lim
h→0
h
sin x cos h + cos x sin h − sin x
= lim
=
h→0
h
sin x(cos h − 1) + cos x sin h
= lim
=
h→0
h
sin x(cos h − 1)
cos x sin h
= lim
+ lim
=
h→0
h→0
h
h
= (sin x) · 0 + (cos x) · 1 =
= cos x
0
Therefore,
0
(sin x) = cos x
The derivative of cos x
Notice that
sin
π
2
− x = sin
π cos x − cos
π sin x =
2
2
= 1 · cos x + 0 · sin x = cos x
The derivative of cos x
Notice that
sin
π
2
− x = sin
π cos x − cos
π sin x =
2
2
= 1 · cos x + 0 · sin x = cos x
Therefore,
π
0
(cos x)0 = sin
−x
= cos
−x ·
−x =
2
2 π
h 2 π π i
= − cos
− x = − cos
cos x + sin
sin x =
2
2
2
= −(0 · cos x + 1 · sin x) = − sin x
h
π
i0
π
The derivative of cos x
Notice that
sin
π
2
− x = sin
π cos x − cos
π sin x =
2
2
= 1 · cos x + 0 · sin x = cos x
Therefore,
π
0
(cos x)0 = sin
−x
= cos
−x ·
−x =
2
2 π
h 2 π π i
= − cos
− x = − cos
cos x + sin
sin x =
2
2
2
= −(0 · cos x + 1 · sin x) = − sin x
h
π
i0
π
Therefore,
0
(cos x) = − sin x
The derivatives of tan x and cot x
0
0
0
sin
x
(sin
x)
cos
x
−
sin
x(cos
x)
(tan x)0 =
=
=
2
cos x
cos x
2
2
cos x + sin x
1
2
=
=
=
sec
x
cos2 x
cos2 x
The derivatives of tan x and cot x
0
0
0
sin
x
(sin
x)
cos
x
−
sin
x(cos
x)
(tan x)0 =
=
=
2
cos x
cos x
2
2
cos x + sin x
1
2
=
=
=
sec
x
cos2 x
cos2 x
Similarly,
0
0
(cos
x)
sin
x
−
cos
x(sin
x)
=
(cot x)0 =
=
2
sin x
sin x
2
1
− sin x − cos2 x
2
=
=
−
=
−
sec
x
2
2
sin x
sin x
h cos x i0
The derivatives of sec x and csc x
−1 0
(sec x)0 = (cos x)
= −(cos x)−2 (cos x)0 = −(cos x)−2 (− sin x) =
sin x
sin x
=
=
= sec x tan x
2
cos x
cos x cos x
The derivatives of sec x and csc x
−1 0
(sec x)0 = (cos x)
= −(cos x)−2 (cos x)0 = −(cos x)−2 (− sin x) =
sin x
sin x
=
=
= sec x tan x
2
cos x
cos x cos x
Similarly,
0
−1 0
(csc x) = (sin x)
= −(sin x)−2 (sin x)0 = −(sin x)−2 cos x =
cos x
cos x
=− 2 =−
= − csc x cot x
sin x sin x
sin x
Example
Let f (x) = ln | tan2 x|. Find f 0 (x).
2
2
First notice that ln | tan x| = ln(| tan x| ) = 2 ln(| tan x|).
Example
Let f (x) = ln | tan2 x|. Find f 0 (x).
2
2
First notice that ln | tan x| = ln(| tan x| ) = 2 ln(| tan x|).
Then,
1
cos x 1
0
0
0
f (x) = 2 ln | tan x| = 2
(tan x) = 2
=
2
tan x
sin x cos x
2
4
4
=
=
=
= 4 csc 2x
sin x cos x
2 sin x cos x
sin 2x