Chapter 8

Chapter 8 - Chapter
8:

Analytical
Trigonometry
...

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Unformatted text preview: Chapter
8:

Analytical
Trigonometry
 
 The
addition
formulas
for
sine
and
cosine
function
(8.1)
 
 
 Distributive
law
from
algebra
will
not
work
to
add
and
subtract
angels
under
trig
 functions
as
shown
below:
 
 
 1) sin( s + t ) = sin s cos t + cos s sin t 2) sin( s − t ) = sin s cos t − cos s sin t 
 3) cos( s + t ) = cos s cos t − sin s sin t 
 
 
 
 
 
 
 
 4 ) cos( s − t ) = cos s cos t + sin s sin t 5) tan( s + t ) = tan s + tan t 1 − tan s tan t 
 6) tan( s − t ) = tan s − tan t 1 + tan s tan t 
 
 
 sine
and
cosine
functions
follow
the
rules
shown
below
to
add
or
subtract
angels
 under
them.

You
must
memorize
the
above
identities
since
they
will
allow
you
to
 successfully
add
and
subtract
angels
under
sine,
cosine
and
tangent
functions.
 
 
 
 
 
 
 
 
 
 
 
 
 Ex : 1) cos(π + θ ) = 2) sin(θ − 3π ) 2 
 3) cos 750 = 
 
 
 
 
 
 
 
 4 ) sin1050 = 5) sin150 = 
 π 6) sin( ) = 12 7) sin( 
 
 
 
 
 
 
 
 
 7π )= 12 Now

let’s
look
at
the
following
two
identities
under
a
new
light
by
applying
these
 addition
formulas:
 
 
 π cos( − α ) = sinα 2 
 sin( 
 
 
 π − β ) = cos β 2 
 
 
 
 
 
 Remember,
in
the
previous
sections
we
proved
the
above
identities
by
looking
at
it
 from
the
graphical
perspective.
 
 
 Proof
of
one
of
the
tan
identity:
 
 tan( s + t ) = 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Simplify
the
following
quantities
:
 
 
 π 1) tan( ) = 12 2) tan( 750 ) = 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3) Given sinα = 12 , 13 π −3 < α < π and cos β = , 2 5 π <β < 3π 2 Find sin(α + β ) = 
 cos(α + β ) = sin(α - β ) = 
 
 
 
 
 
 
 
 cos(α - β ) = The
Double
Angel
Formulas
(8.2)
 
 By
using
the
summation
of
angels
formulas
from
the
previous
section,
derive
the
 use the fact 2θ = θ + θ 
 double
angel
formulas
for
 sin 2θ , cos2θ , tan2θ ; 
 sin 2θ = 
 cos 2θ = tan 2θ = 
 
 
 
 
 
 
 
 
 Given sinθ = 3π , <θ < π 4 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Find sin2θ = cos2θ = tan2θ = 
 
 
 
 
 
 
 
 
 
 
 
 
 The
Half‐Angel
Formulas:
 
 s 1 − cos s 1) sin( ) = ± 2 2 s 1 + cos s 2) cos( ) = ± 2 2 
 s 1 − cos s sin s 3) tan( ) = ± =± 2 1 + cos s 1 + cos s 
 
 
 Proof
of
1)
just
for
your
amusement:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex : Use sinθ = 3π , < θ < π as before but now find 42 θ sin( ) = 2 

 
 θ cos( ) = 2 
 
 
 
 
 
 
 
 θ tan( ) = 
 2 
 
 
 
 
 
 
 
 
 π π π Ex : Find sin( ), cos( ), and tan( ) 
using
half‐angel
formulas:
 8 8 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex : Use the information x = 5sinθ , 0 < θ < 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 π to express sin(2θ ) and cos(2θ ) interms of x. 2 The
Product
to
Sum
and
Sum
to
Product
Formulas
(8.3)
 
 
 1 1) sin A sin B = [ cos( A − B ) − cos( A + B )] 2 2) sin A cos B = 1 [ sin( A − B ) + sin( A + B )] 
 2 1 [ cos( A − B ) + cos( A + B )] 2 3) cos A cos B = 
 
 Derivation
of
the
first
identity:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 The
proof
of
the
other
two
identities
are
very
similar.
 
 
 
 
 
 
 
 
 Ex:

Simplify
 cos 750 cos150 
into
a
simpler
value
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Convert
the
product
 sin 4 x cos 3x 
to
a
sum
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 There
are
also
formulas
that
convert
sums
to
products
but
we
don’t
need
to
worry
 about
that.
 
 π Ex:

Prove
 sin x + cos x = 2 cos( x − ) 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Use
the
above
result
to
graph
 y = sin x + cos x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Trigonometric
Equations
(8.4)
 
 In
this
section,
we
will
learn
some
techniques
on
how
to
solve
trigonometric
 equations.

The
best
way
to
learn
that
will
be
trough
varieties
of
different
types
of
 examples.

Always
pay
attention
to
the
solution
domain
as
denoted
in
the
original
 problem.
 
 1 Ex:

a)

 Given cosx = , find all x in [ 0,2π ]
by
using
your
knowledge
of
the
unit
circle.
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 b)

Now
use
the
calculator
to
verify
your
solution
from
above.

Let’s
discuss
what
 happens
if
you
completely
depend
on
your
calculator
to
give
you
all
the
solutions.
 
 
 
 
 
 
 
 
 
 
 
 c)

Now

find
all
real
number
solutions
to
the
equation
stated
in
part
a).
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Solve
the
trigonometric
equation
defined
by
 2 cos2 θ + cosθ = 0 in [ 0,2π ]
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Solve
the
trigonometric
equation
defined
by
 2 sin 2 x − 3 sin x + 1 = 0 in (0,2π ) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Solve
the
trigonometric
equation
 3 sin x + tan x = 0 in (0,2π ) 
 
 Note:

In
this
example
you
don’t
have
the
equation
expressed
in
terms
of
a
single
 trigonometric
function
as
it
was
in
the
previous
examples.


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Solve
the
trigonometric
equation
 cos 3x = 1 in [ 0,2π ] 
 
 Note:

you
have
to
go
around
the
unit
circle
three
times
to
fish
out
all
the
solutions.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Solve
 2 sin 2 θ − cos 2θ = 0 in [ 0,2π ] 





 
 Remember
the
double
angel
identity
.

Also
you
want
to
express
the
equation
in
 terms
of
 sinθ only
in
order
to
solve
this
equation
by
using
the
factoring
techniques
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

 2 sin x = 1 − cos x for all real x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 How
can
we
make
the
basic
trigonometric
functions
(y=sinx,
y=cosx,
and
y=tanx)
 one‐
to
one?
 
 Look
at
the
graph
of
y=cosx
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Why
is
this
function
not
one
to
one
as
is?
 
 
 
 
 
 
 
 How
can
we
make
y=cosx
be
one
to
one?
 
 
 
 
 
 
 
 The
graph
of
y=cosx
with
the
restricted
domain
looks
as
follows:
 
 
 
 
 
 
 
 
 
 
 
 
 
 Now
Look
at
the
graph
of
y=sinx
and
y=tanx
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Why
are
these
functions
not
one
to
one
as
is?
 
 
 
 
 
 
 
 How
can
we
make
y=sinx

and
y=tanx
be
one
to
one?
 
 
 
 
 
 
 
 The
graph
of
y=sinx
and
y=tanx
with
the
restricted
domain
looks
as
follows:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Definition
of
the
Inverse
Trig
Functions:
 
 Inverse
cosine
denoted
 cos−1 x = unique number in [ 0,π ] whose cosine is x 
 
 ⎡ π π⎤ Inverse
sine
denoted
 sin −1 x = unique number in ⎢− , ⎥ whose sine is x 
 ⎣ 2 2⎦ ⎡ π π⎤ Inverse
tangent
denoted
 tan −1 x = unique number in ⎢− , ⎥ whose tangent is x 
 ⎣ 2 2⎦ 
 Let’s
look
at
our
calculator
and
how
you
can
use
it
to
find
the
inverse
of
the
basic
 trig
functions.
 
 Ex:

Find
 cos−1(.4 )
by
using
your
calculator.
 
 
 
 
 
 
 1 Ex:

Let
 sin x = .

Solve
for
all
real
x
 2 
 Note
:
If
you
are
not
given
domain
restriction,
you
have
to
solve
for
all
x.

Otherwise,
 stick
to
given
x
boundary.
 
 
 
 
 
 
 
 
 
 
 
 
 
 −2 Ex:

Find
 cos−1( )
 2 
 
 
 
 
 
 
 
 The
Inverse
Trig
Functions

(8.5)
 
 
 In
this
section
we
want
to
study
the
inverse
trigonometric
functions
 y = sin −1( x ), y = cos-1( x ), and y = tan -1( x )
with
more
conceptual
foundations.


 
 Recall:

1)
In
order
for
y
=
sinx
and
y
=
tanx
to
have
an
inverse,
we
have
to
restrict
 ⎡ −π π ⎤ , 
and
for
y
=
cosx
to
have
an
inverse,
we
had
to
restrict
the
 their
domains
to
 ⎢ ⎣ 2 2⎥ ⎦ domain
to
 [ 0,π ].

By
restricting
the
domain
we
force
the
trigonometric
functions

 y
=
sinx,
y
=
cosx
and
y
=
tanx
to
become
one
to
one.
 
 2)

We
will
able
to
obtain
the
graphs
of
the
inverse
functions
by
reflecting
the
graphs
 of
the
basic
trigonometric
functions
across
the
line
y
=
x


(Look
below
for
graphical
 explanation)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Algebraically
to
find
an
inverse
we
will
have
to
perform
the
following
steps:
 
 1) Let
y
=
sinx
 
 Switch
the
x
and
y
variables
 
 
 x=
siny
 
 2) Solve
for
the
new
y
 
 
 
 to
get
y
by
itself,
we
have
to
take
the
inverse
of
both
sides
in
step
1.


 
 Therefore,
 y = sin −1( x )
 
 
 Domain:
 
 
 Range:
 
 
 Definition:
 
 sin(sin −1 x ) = x for all x in [ -1,1] ⎡ -π π ⎤ 
 sin −1(sin x ) = x for all x in ⎢ , ⎥ ⎣ 2 2⎦ 
 
 Ex:

Find
 3 sin −1( )= 2 
 sin −1( −3 )= 2 
 
 
 
 
 
 
 
 Similarly
for
 y = cos−1 x 
 
 
Domain:
 
 
 Range:
 
 
 Definition:
 
 cos( cos−1 x ) = x for all x in [ -1,1] cos−1(cos x ) = x for all x in [ 0,π ] 
 2 Ex : Find cos(cos-1( )) = 3 
 2 Find cos(sin −1( )) = 3 
 
 
 
 
 
 
 
 Ex:

arcos(cos4)
 
 
 
 
 
 
 
 
 
 
 
 For
 y = tan −1( x ) ,
find
 
 Domain:
 
 Range:
 
 tan(tan −1 x ) = x for all x and −π π 
 tan -1(tan x ) = x for all x in ( ,) 22 
 Ex:

Find
arc(tan0)=













=
 
 
 
 
 Ex:
 Find tan -1(1) = 
 arctan(tan( −π )) = 7 
 
 Evaluate
the
following
without
using
your
calculator:
 
 4 tan(sin −1( )) = 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Evaluate
without
using
your
calculator
 
 4π sin −1(sin( )) 
 3 
 Note:

Remember
to
pay
extra
attention
to
the
domain
of
the
inverse
sine
function.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex : Show sin(cos-1x ) = 1 − x 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Graph:

 y = − sin −1 x − 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Graph:

 y = − cos−1(1 − x ) − π 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 π 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ...
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This note was uploaded on 02/08/2011 for the course MATH 3 taught by Professor Staff during the Winter '08 term at UCSC.

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