Download Lesson 3.5 - Coweta County Schools

3.5
Derivatives of
Trigonometric Functions
Quick Review
1. Convert 135 degrees to radian. 3  2.356
4
2. Convert 1.7 radian to degrees.
97.403o
3. Find the exact value of sin    without a calculator.
3
4. State the domain and range of the cosine function.
Domain : all reals
range :  1, 1
5. State the domain and range of the tangent function.
k
k odd number 
Domain : x 
2
range : all reals
3
2
Quick Review
6. If sin a = – 1, what is cos a?
0
7. If tan a = – 1, what are two possible values of sin a?
2

2
2
1

cos
h
sin
h
8. Verify the identity

.
h
h1  cos h 
9. Find an equation of the line tangent to the curve y  2 x3  7 x 2  10
at the point (3, 1). y  12 x  35
10.A particle moves along a line with velocity v  2t 3  7t 2  10
for time t > 0. Find the acceleration of the particle at t = 3.
12
What you’ll learn about
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Derivative of the Sine Function
Derivative of the Cosine Function
Simple Harmonic Motion
Jerk
Derivatives of Other Basic Trigonometric
Functions
Essential Questions
What are the derivatives of sines and cosines, and
how do they play a key role in describing
periodic change?
Finding the Derivative of the Sine Function
1. Use the definition of derivative to find the derivative of f (x) = sin x.
sin x  h   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
 lim
h 0
h 0
h
h
sin h

cos h  1
 lim cos x
lim sin x
h 0
h 0
h
h
0  cos x  cos x
sin x
lim
1
x 0
x
cos x  1
lim
0
x 0
x
Derivative of the Sine Function
The derivative of the sine is the cosine.
d
sin x  cos x
dx
Derivative of the Cosine Function
The derivative of the cosine is the negative of the sine.
d
cos x   sin x
dx
Using the Derivative of Sine and Cosine
3. Find the derivative of each:
a) y  x sin x
2
y  x  cos x  sin x 2 x 
2
2

y  x cos x  2 x sin x
cos x
b) u 
1  sin x

1  sin x  sin x   cos x  cos x 
u 
2
1  sin x 
1  sin x
 sin x  sin 2 x  cos 2 x


2
2
1  sin x 

1  sin x 
1

1  sin x
Simple Harmonic Motion
The motion of a weight bobbing up and down on the end of
a string is an example of simple harmonic motion.
Example Simple Harmonic Motion
4. A weight hanging from a spring is stretched 5 units beyond its rest
position (s = 0) and released at time t = 0 to bob up and down. Its
position at any later time t is
s  5 cos t
What are its velocity and acceleration at time t?
vt   5 sin t
at   5 cos t
s  5  sin t  cos t 0
Jerk
Jerk is the derivative of acceleration. If a body's position at
time t is
da d 3 s
j t  
 3.
dt dt
5. Find the jerk caused by the constant acceleration of gravity
(g = – 32 ft/sec 2)
j t   0
6. Find the jerk of the simple harmonic motion of the previous
example s = 5 cos t.
vt   5 sin t
at   5 cos t
j t   5 sin t
Finding the Derivative of the Tan Function
7. Use the derivative of sin and cos to find the derivative of f (x) = tan x.
sin x
f x   tan x 
cos x
cos x  cos x  sin x  sin x 
f x  
2
cos x
2
2
1
cos x  sin x



2
2
cos x
cos x
2
sec x
Derivative of the Other Basic Trig Functions
d
tan x  sec 2 x
dx
d
cot x   csc 2 x
dx
d
sec x  sec x tan x
dx
d
csc x   csc x cot x
dx
Example Derivative of the Other
Basic Trigonometric Functions
dy
cot x
8. Find
for y 
.
dx
1  cot x



dy 1  cot x   csc x  cot x  csc 2 x

2
dx
1  cot x 
 csc 2 x  cot x csc 2 x  cot x csc 2 x

2
1  cot x 
 csc x

2
1  cot x 
2
2

Example Finding Horizontal Tangents of
Trigonometric Functions
9. Show that the graphs of y = sec x and y = cos x have horizontal
tangents at x = 0.
y  sec x
y  sec x tan x  0
sec x  0
tan x  0
x0
y  cos x
y   sin x  0
sin x  0
x0
Example Derivative of Trig Functions
10. Find y if y  sec x.
y  sec x tan x
2
y  sec x sec x  tan x sec x tan x
 sec x  tan x sec x
3
2
Example Derivative of Trig Functions
11. Find y if y   cos  .
y    sin    cos  1
  sin   cos 
y   cos   sin   1  sin 
  cos   2 sin 
Pg. 146, 3.5 #2-48 even