Lecture 5: More Calculus of Trigonometric Functions

Lecture 5: More Calculus of Trigonometric Functions
5.1 Some limit calculations
We may use the limits
lim
x→0
sin(x)
=1
x
and
1 − cos(x)
=0
x→0
x
lim
to help evaluate other limits.
Example
sin(3x)
lim
= lim
x→0
x→0
x
Example
sin(2x)
2
= lim
lim
x→0 sin(3x)
3 x→0
Example
sin2 (x)
lim
= lim
x→0
x→0
x2
Example
tan(t)
lim
= lim
t→0
t→0
t
Example
We have
3 sin(3x)
3x
sin(2x)
2x
sin(3x)
3x
sin(x)
x
sin(t)
t
sin(3x)
= (3)(1) = 3.
x→0
3x
= 3 lim
2
1
2
=
= .
3
1
3
sin(x)
x
1
cos(t)
= (1)(1) = 1.
= (1)(1) = 1.
1 − cos(x)
1 + cos(x)
sin(x)
1 + cos(x)
1 − cos2 (x)
= lim
x→0 sin(x)(1 + cos(x))
1 − cos(x)
lim
= lim
x→0
x→0
sin(x)
sin2 (x)
x→0 sin(x)(1 + cos(x))
sin(x)
= lim
x→0 1 + cos(x)
0
= = 0,
2
= lim
or
1 − cos(x)
= lim
x→0
x→0
sin(x)
lim
5-1
1−cos(x)
x
sin(x)
x
=
0
= 0.
1
Lecture 5: More Calculus of Trigonometric Functions
5-2
5.2 More derivatives
Example
If f (x) = sin2 (x), then f 0 (x) = 2 sin(x) cos(x).
Example
If f (x) = sec2 (x), then
f 0 (x) = 2 sec(x) sec(x) tan(x) = 2 sec2 (x) tan(x).
Example
If g(t) = sin(4t), then g 0 (t) = 4 cos(4t).
Example
If g(t) = sin2 (4t), then g 0 (t) = 2 sin(4t) cos(4t)(4) = 8 sin(4t) cos(4t).
Example
If f (x) = sec3 (4x), then
f 0 (x) = 3 sec2 (4x) sec(4x) tan(4x)(4) = 12 sec3 (4x) tan(4x).
Example
If f (x) = sec2 (x2 + 4) tan(3x), then
f 0 (x) = sec2 (x2 + 4) sec2 (3x)(3) + tan(3x)(2 sec(x2 + 4) sec(x2 + 4) tan(x2 + 4)(2x))
= 3 sec2 (x2 + 4) sec2 (3x) + 4x sec2 (x2 + 4) tan(x2 + 4) tan(3x).