Chapter 8: Trigonometric Equations and Applications

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Chapter 8 Trigonometric Equations and
Applications
L8.2 Sine Cosine Curves Simple Harmonic
Motion
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Simple Harmonic Motion
The periodic nature of the trigonometric
functions is useful for describing motion of a
point on an object that vibrates, oscillates,
rotates or is moved by wave motion.
For ex, consider a ball that is bobbing up and
down on the end of a spring.

• 10cm is the maximum distance that the ball moves
  vertically upward or downward from its
  equilibrium (at rest) position.
• It takes 4 seconds for the ball to move from its
  maximum displacement above zero to its maximum
  displacement below zero and back again.

• With ideal conditions of perfect elasticity and
  no friction or air resistance, the ball would
  continue to move up and down in a uniform manner.
• Motion of this nature can be described by a sine
  or cosine function and is called simple harmonic
  motion.
• For this particular example, the amplitude is
  10cm, the period is 4 seconds, and the frequency
  is ¼ cps (cycles per second).

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Simple Harmonic Motion
A point that moves on a coordinate line is in
simple harmonic motion if its distance d from
the origin at time t is given by either d a
sin ?t or d a cos ?t where
a and ? are real numbers such that ? gt 0. The
motion has amplitude a, period 2p/? and
frequency ?/2p.
Ex 1 Write the equation for simple harmonic
motion of a ball suspended from a spring that
moves vertically 8 cm from rest. It takes 4
seconds to go from its maximum displacement to
its minimum and back. What is the frequency of
the motion?
Since the spring is at equilibrium (d 0) when
t0, we will use the equation d a sin ?t.
The maximum displacement from 0 is 8 cm and the
period is 4 sec so amplitude a 8, period
2p/? 4 ? ? p/2. Consequently the equation of
motion is The frequency ?/2p (p/2)/(2p) ¼
cycle per second.
Note that ? (lower case omega) is just a
stand-in for the coefficient, B.
Since time, unlike an angle, is not measured in
p, ? frequently has p in it for cancelation
purposes.
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Simple Harmonic Motion (cont)
A point that moves on a coordinate line is in
simple harmonic motion if its distance d from
the origin at time t is given by either d a
sin ?t or d a cos ?t where
a and ? are real numbers such that ? gt 0. The
motion has amplitude a, period 2p/? and
frequency ?/2p.
Ex 2 Given the equation for simple harmonic
motion , where d is in cm
and t is in seconds, find (a) the maximum
displacement, (b) the frequency,
(c) the value when t 4, and (d) the
least positive value for t for which d 0.

• Max displacement is amplitude, which is 6 cm

(b) Frequency ?/2p (3p/4) / 2p ? cycle per
second.
(c)
cm
(d) The least positive value
sec.