MATHEMATICS 201-NYA-05
Differential Calculus
Martin Huard
Fall 2016
IX –Derivatives of Trigonometric Functions
1. Convert from degrees to radians.
a) 135
b)
900
c)
315
2. Convert from radians to degrees.
a) 83
b)
7
12
c)
3
3. Evaluate exactly (without the use of a calculator).
a) sin  23 
b) tan  4 
c) cos  56 
e) sec  74 
d) sin  52 
h) cot  34 
g) csc  76 
f) cot 
4. Use identities to evaluate exactly.
a) sin 75
b)
cos  12 
c)
5. Find the limit.
sin 4x
x0 2x
sin x  cos x sin x
d) lim
x0
x2
a)
g)
lim
limsin3x csc12x
x0
b)
e)
h)
sin3x
x0 sin7 x
tan 2t
lim
t 0
3t
2x  sin x
lim
x0
x
lim
c)
f)
i)
cos 165
sin  cos 
 0
sec
cot x
lim
x0 csc2x
sin  x 1
lim 2
x1 x  2x  3
lim
6. Differentiate the function.
a)
f  x  2cos x 5sin x
b)
f  x 
sin x
x
c)
f  x  sec x  5tan x
d)
f  x  sec x tan x
e)
f t   t3 csct t cot t
f)
f  x 
g)
f  x 
h)
f  x 
j)
f  x   x sin x 
csc x
tan x
cos x
x
k)
x2  1
cos x 1
3cos
f   
2cos  sin
i)
l)
cot x
1 csc x
1 sin x
f  x 
1 2sin x
sin x sec x
f  x 
1 x tan x
IX – Derivatives of Trig Functions
Math NYA
7. Find the equation for the tangent and normal lines to the graph of each function at the given
point.
a) f  x  2sin x at  6 ,1 .
f  x  3tan x at  34 , 3 .
c) f  x  x  cos x at  , 1 .
b)
d)
f  x  sec x  csc x at

e)
f  x  2cot x at
,2 3 .

5
6

4

,2 2 .

8. For what values of x does the graph of f  x  x  2cos x have a horizontal tangent?
9. A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its
equation of motion is x t   8sin t , where t is in seconds and x in centimeters.
a) Find the velocity at time t.
b) Find the position and velocity of the mass at time t  23 . In what direction is it
moving at that time?
0
x
10. The force F (in Newtons) acting at an angle  with the horizontal that is needed to drag a
crate with weight W along a horizontal surface at a constant velocity is given by
F
W
cos   sin
where  is a constant called the coefficient of sliding friction between the crate and the
surface (see the figure). Suppose that the crate weighs 50 N and that   0.4 . Find dF
d
when   30 .
θ
Fall 2016
Martin Huard
2
IX – Derivatives of Trig Functions
Math NYA
ANSWERS
3
4
1. a)
2. a) 480
3. a)
4. a)
3
2
6 2
4
5. a) 2
6.
b) 5
c)
b) 105
c)
b) –1
c)
6 2
4
3
7
b)
b)
c)
7
4
540


 3
2
d) 1
e)
c) sin1
e) 23
d) 0
e) f  t   3t 2 csct  t3 csct cot t  cot t  t csc2 t
a) yT  3x 
3
3
3
6
x
1
3
18
i)
1
4
5
6
h) f   x 
j) f   x   
l) f   x 
yN  x  8  3
1
6
1
2x cos x  2x  x2 sin x  sin x
cos x 12

sin x
cos x
 x cos x  3
2 x
2x 2
1
1 x tan x2
c) yT  x 1
yN  x   1
e) yT  8x  203  2 3
 2 n n 
9. a) v t   8cost
10. 2.7 Newtons/degrees
Fall 2016
x cos x  sin x
x2
d) f   x  sec x tan2 x  sec3 x
b) yT  6x  92  3
d) yT  2 2
xN  4
8. x  6  2 n,
h) 3
 csc3 x  csc2 x  csc x cot 2 x  csc x

1 csc x
1 csc x2
 csc x  csc x sec2 x
g) f   x 
tan2 x
3cos x
i) f   x 
1 2sin x2
3
k) f    
 2cos  sin 2
yN  
h) –1
b) f   x 
c) f   x  sec x tan x  5sec2 x
7.
g) –2
g) 14
f) 2
a) f   x  2sin x  5cos x
f) f   x 
f) 

2
 6 2
4
yN  81 x  548  2 3
b) Position: 4 3 cm
Velocity: – 4 cm/s, moving towards the wall
Martin Huard
3