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1
Unit
Two
Functions
in
Trigonometry
2
Sinusoids
A
sinusoid
is
considered
to
be
the
general
form
of
a
sine
wave.It
is
a
function
that
can
be
written
as
f(x)=
a
sin(bx+c)+d
and
since
the
cosine
function
is
identical
to
the
sine
function
it
can
also
be
f(x)=
a
cos(bx+c)+d.
Here
the
amplitude
is
represented
by
the
absolute
value
of
"a",
the
period
is
found
by
using
2pi/b,
the
frequency
is
found
by
using
the
absolute
value
of
b/2pi.
***Here
is
an
example
of
what
a
sinusoid
looks
like***
3
Amplitude
In
a
sinusoid,
you
can
acquire
the
amplitude
by
getting
the
absolute
value
of
"a".
When
identifying
it
in
the
graph,
an
amplitude
is
half
the
height
of
an
entire
wave.
***Example***
Find
the
amplitude
of
y=
cosx
and
y=
1/2cosx
For
y=
cosx
the
amplitude
would
be
1
and
for
y=
1/2cosx
the
amplitude
would
be
1/2
4
Frequency
In
a
sinusoid,
you
can
find
the
frequency
by
taking
the
absolute
valus
of
b
and
dividing
it
by
2pi.
When
looking
at
the
graph,
the
frequency
is
the
number
of
cycles
completed
within
a
certain
time
interval.
***Example***
Find
the
frequency
of
f(x)=
4sin(2x/3)
To
solve
this
problem
you
get
2/3
and
divide
it
by
2pi.
This
would
equal
1/(3pi).
THE
IMAGE
TO
THE
RIGHT
DOES
NOT
CORRESPOND
TO
THE
EXAMPLE
5
Period
To
find
the
period
in
a
sinusoid,
take
2pi
and
divide
it
by
the
absolute
value
of
"b".
When
looking
at
the
graph,
the
period
is
the
length
of
one
cycle
of
a
wave.
***Example***
Find
the
period
of
the
function
y=
3sin
(-2x)
To
find
the
period
divide
2pi
by
the
absolute
value
of
(-2).
From
this
you
should
get
pi
as
the
answer.
PICTURE
NOT
RELATED
TO
EXAMPLE
6
Phase
Shift
From
the
sinusoid
equation
f(x)=a
sin(bx+c)+d,
we
can
get
the
horizontal
shift
by
using
-c/b.
When
looking
at
the
grapg,
the
phase
shift
can
also
be
seen
as
a
horizontal
shift
of
the
graph
either
to
the
left
or
right.
***Example***
Find
the
phase
shift
of
the
function
y=
sin
(2x+pi)
To
find
the
phase
shift
of
this
function
first
rewrite
it
in
the
form
that
would
look
like
y=
sin
(2
(x+
pi/2)).
Then
you
would
have
a
horizontal
shift
of
pi/2
so
the
phase
shift
would
be
at
(pi/2,0)
7
Taking
a
Given
Angle
and
Computing
its
Trig
Function
***Example***
Imagine
that
there
is
a
right
triangle
with
an
angle
of
35
degrees
and
an
opposite
side
of
4.
Use
your
knowledge
of
functions
with
given
angles
to
find
the
value
of
the
hypotenuse.
To
solve
this
problem
we
first
do
sin
35=
4/h.
Next
we
multiply
both
sides
by
"h".
By
this
point
we
have
h
sin35=
4.
Next
we
compute
sin35
to
find
its
value
and
get
.57.
Now
we
divide
4/.57
and
know
that
the
hypotenuse
equals
7.
Click to add new slide
Taking
a
Given
Angle
and
Computing
its
Trig
Function
***Example***
Imagine
that
there
is
a
right
triangle
with
an
angle
of
35
degrees
and
an
opposite
side
of
4.
Use
your
knowledge
of
functions
with
given
angles
to
find
the
value
of
the
hypotenuse.
To
solve
this
problem
we
first
do
sin
35=
4/h.
Next
we
multiply
both
sides
by
"h".
By
this
point
we
have
h
sin35=
4.
Next
we
compute
sin35
to
find
its
value
and
get
.57.
Now
we
divide
4/.57
and
know
that
the
hypotenuse
equals
7.
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