Inverse Trigonometric Functions - Integration

. For example, if f(x) = sin x, then
f0(x) = cos x; f00(x) = ??sin x; f(3)(x) = ??cos x; f(4)(x) = sin x; f(5)(x) = cos x; : : :
(Note the derivatives follow a similar pattern for cos(x). )
Example Let f(x) = sin x. What is
f(20)(x)?
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A mass on a spring released at some point other than its equilibrium position will follow a pattern of
simple harmonic motion (x(t) = Asin(Cx + D) or equivalently x(t) = Acos(Cx + D) ), when there is
no friction or other forces to dampen the eect. The values of A;C and D depend on the elasticity of
the spring, the mass and the point at which the mass is released. You will be able to prove this easily
later when you learn about dierential equations.
Example An object at the end of a vertical spring is stretched 5cm beyond its rest position and
released at time t = 0. Its position at time t is given by x(t) with the positive direction as shown in a
downward direction, where
x(t) = 5 cos(t):
(a) Find the velocity and acceleration at time t.
(b) Find the position, velocity and acceleration of the mass at time t = 
4 . In which direction is it
moving at that time?
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The following is a summary of the derivatives of the trigonometric functions. You should be able to
verify all of the formulas easily.
d
dx
sin x = cos x;
d
dx
cos x = ??sin x;
d
dx
tan x = sec2 x
d
dx
csc x = ??csc x cot x;
d
dx
sec x = sec x tan x;
d
dx
cot x = ??csc2 x
Example The graph below shows the variations in day length for various degrees of Lattitude.
At 60o North, at what times of the year is the length of the day changing most rapidly?
Extras
Example (Preparation for Related Rates) A police car is parked 40 feet from the road at the
point P in the diagram below. Your vehicle is approaching on the road as in the diagram below and the
police are pointing a radar gun at your car. Let x denote the distance from your car to the police car
and let  be the angle between the line of sight of the radar gun and the road. How fast is x changing
with respect to  when  = 
4 ? (Please attempt this problem before looking at the solution on the
following page.)
40ft
x
!
P
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