Chapter 6 -...

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Unformatted text preview: Chapter
6:

Trigonometric
Functions
of
Angles
 
 Trigonometric
Functions
of
Acute
Angles

(6.1)
 
 
 Vocabulary:

An
angle
is
a
figure
formed
by
two
rays
with
a
common
end
point.

The
 common
end
point
is
called
the
vertex.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 < = angel = θ = < ABC 
 
 
 
 
 
 One
measurement
of
angle
is
degrees.

A
full
rotation
is
360
degrees.

The
degree
is
 sub‐divided
into
smaller
units.

The
minute
(1/60th
of
a
degree)

and
second
(1/60th
 of
a
minute)


 
 Acute
angles
have
measure


 00 < θ < 900 

 
 Ex:
 
 
 
 
 
 
 
 
 
 
 Obtuse
angels
have
measure

 900 < θ < 1800 
 
 Ex:


 
 
 
 
 
 
 
 
 
 Definition:

Let
θ 
be
an
acute
angle
placed
in
a
right
triangle.

The
basic
trig
 functions
are
defines
as
follows:
 
 
 
 
 
 
 
 
 
 
 
 
 You
can
remember
the
definition
of
the
basic
trigonometric
functions
by
 remembering
SOHCAHTOA
 
 
 Reciprocal
Identities
can
be
defined
from
the
basic
trig
functions
as
follows:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

 ΔABC 
is
a
right
triangle
with
AC
=
3,

BC
=
2

and
<C
=
900
 
 Find
the
six
trigonometric
functions:
 
 
 
 
 
 
 
 
 
 Calculate

sin
(250)
,
cos
(250)

and
tan
(250)
using
degree
mode
in
your
calculator.
 
 
 
 
 
 
 Two
well
known
triangles
from
Geometry:
 
 
 300‐600
Right
Triangle
 
 Theorem:

In
a
300‐600

right
triangle,
the
length
of
the
opposite
side
to
the
300
angel
 is
half
of
the
length
of
the
hypotenuse.
 
 
 
 
 
 
 
 

 
 
 450‐450
Right
Triangle
 
 Theorem:

In
a
450‐450
right
triangle,
the
length
of
the
opposite
and
the
adjacent
 sides
are
equal.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Let’s
now
make
a
table
with
the
trig
functions
with
all
the
special
angels
where
you
 can
express
them
as
an
exact
answer
(without
calculator).


You
have
to
memorize
 this
table
asap.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Prove:

cos(600)
=
cos2(300)
–
sin2(300)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Algebra
and
the
Trigonometric
Functions
(6.2)
 
 Notation:

 
 
1)

 sin(θ ) = said as sin e of the angel θ = sinθ 
 
 In
this
case
θ is the input and sinθ is the output 
 
 Note:

 sinθ is a value ;
For
example,
sin(300)
=
½
 
 2)

When
you
are
multiplying
with
two
trigonometric
quantities,
parenthesis
is
often
 omitted.
 
 For
example,
 
 (sinθ )(cosθ ) = sinθ cosθ Similarly (sinθ )(sinθ ) = sin 2θ 
 (sinθ )n = sin n θ (n ≠ -1) 
 The
convention
stated
above
applies
to
the
other
five
trig
functions.
 
 
 Useful
Hint:

Treat
 sinθ ,cosθ 
or
any
of
the
other
four
trig
expression
as
you
treat
 variables
(like
x
and
y)
in
Algebra
while
simplifying
trigonometric
quantities.
 
 
 Ex:

Simplify
the
following
expressions:
 
 1)

10 sinθ cosθ + 4 sinθ cosθ − 16 sinθ cosθ = 
 
 
 
 
 
 
 
 
 
 2)

 ( 3 sinθ − 4 cosθ )( 2 sinθ + cosθ ) = 
 
 
 
 
 
 
 Factor:

 tan 2 θ − 2 tanθ − 15 =
 
 
 
 
 
 
 
 
 
 
 
 
 
 Basic
Right
Angle
Identities:


 
 1) sin 2θ + cos 2 θ = 
 
 2)

 tanθ = 
 
 3)

 sin( 90 − θ ) = 
 
 4)

 cos( 90 − θ ) = 
 
 cscθ = 5)

 secθ = 
 
 
 Proof:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 cotθ =
 Ex:

Given
 cos A = functions.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Simplify
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 , and
A
is
an
acute
angel,
find
the
other
five
trigonometric
 5 sin 2 θ − cos 2 θ =
 sinθ − cosθ 1 − 2 cos 2 x Ex:

 cot x + 
 sin x cos x 
 
 
 
 
 Right
Triangle
Applications

(6.3):




 
 We
are
going
to
apply
the
knowledge
obtained
in
the
sections
6.1
and
6.2
in
solving
 some
basic
problems
involving
right
triangles.
 
 
 Ex:

In
a
300‐600
right
triangle,
the
hypotenuse
is
60
cm.

Find
the
measure
of
the
 other
two
sides
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

In
a
450‐450
right
triangle,
the
opposite
side
has
length
9
inches.

Find
the
 measure
of
the
other
two
sides.


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A
trigonometric
formula
for
the
area
of
a
triangle
with
side
lengths
a
and
b
and
the
 angle
in
between
θ 
:
 
 1 1 Formula : Area = ( base )( height ) sinθ = ab sinθ 
 2 2 
 Proof:


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Find
the
area
of
a
triangle
with
side
lengths
a
=
5,
b
=
9
and
the
angle
in
between
 θ 
=
600
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Find
the
area
of
a
rectangular
nonagon
that
is
inscribed
in
a
circle
of
radius
6
 cm.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

A
certain
sprinkler
will
water
ground
out
to
a
distance
of
25
feet.

If
the
 sprinkler
head
is
set
to
rotate
back
and
forth
through
an
angel
of
600,
determine
the
 area
in
square
feet
that
is
being
irrigated.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Definition:

The
angle
of
elevation
is
the
angle
formed
by
the
horizontal
and
the
 line
of
sight
to
an
object
above
the
observer.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 The
angle
of
depression
is
the
angel
formed
by
the
horizontal
and
the
line
of
sight
 to
an
object
below
the
observer.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

From
a
point
on
the
ground
level,
you
measure
the
angle
of
elevation
to
the
top
 of
the
mountain
to
be
380.

Then
you
walk
200m
further
away
from
the
mountain
 and
find
that
the
angle
of
elevation
is
now
200.

Find
the
height
of
the
mountain
to
 the
nearest
meter.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Trigonometric
Function
of
Angles
(6.4)
 
 You
need
to
bring
a
copy
of
a
Unit
Circle
to
lecture
today!

Download
it
from
 my
web
site.
 
 In
this
section
we
will
not
be
restricted
to
acute
angle
 ( 00 < θ < 900 )
trigonometry
 only.

We
will
expand
our
definitions
to
include
angles
of
any
size. 
 Remember
the
definition
of
angle
formed
by
two
rays
(sec
6.1).

Let’s
fix
one
side
of
 the
angle
(the
initial
side)
as
shown
below
and
rotate
through
an
angle
θ 
to
get
to
 the
terminal
side
as
shown
below.
 
 
 
 
 
 
 
 
 
 
 By
convention,
the
counterclockwise
rotation
yields
a
positive
angle
measure
θ ,
and
 the
clockwise
rotation
yields
a
negative
angle
measure
(‐θ )
as
shown
below.
 
 
 
 
 
 
 
 
 
 
 
 Standard
Position:

Initial
side
is
always
aligned
with
the
positive
x‐axis
of
the
Unit
 Circle
as
shown
below:
 
 
 
 
 
 
 
 
 
 After
the
rotation
of
angle
θ 
we
land
on
the
terminal
side
with
the
coordinate
(x,
y).
 
 We
can
define
the
six
trig
functions
in
terms
of
x
and
y
as
follows:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (x,
y)
=
coordinate
of
the
terminal
side
=
(cosθ ,
sinθ )
 
 Since
(x,
y)
lies
on
the
Unit
Circle,
it
must
satisfy
the
equation
of
the
circle
with
 radius
1
which
is

 
 x 2 + y 2 = 1
 
 check:
 
 
 
 
 
 
 
 By
fixing
the
initial
side
in
the
standard
position
and
by
rotating
the
terminal
side
by
 an
angel
measure
00<θ <3600
,
we
are
able
to
find
trigonometric
values
of
any
angle
 θ .

Visualize
the
rotation
to
measure
angle
of
any
size
on
an
Unit
Circle
as
 demonstrated
below:
 
 
 
 
 
 
 
 Now
let’s
fill
out
the
table
below
by
consulting
at
the
Unit
Circle:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Let’s
discuss
how
the
Unit
Circle
works
since
you
have
to
be
an
expert
on
this
very
 soon!!
 
 
 
 What
if
the
angle
measurement
is
not
always
falling
on
the
axis
as
shown
on
the
 table
above.
Now
we
will
find
a
procedure
to
find
trigonometric
values
of
angles
that
 do
not
necessarily
fall
on
the
x
or
y
axis.
 
 Definition:

You
can
remember
the
sign
of
the
trigonometric
function
value
by
 remembering
All
Students
Take
Calculus
as
the
quadrant
changes
as
shown
below:
 
 
 Definition:

The
reference
angle
associated
with
an
angel
θ 
in
standard
position
is
 the
acute
angle
(with
positive
measure)
formed
by
the
x­axis
and
the
terminal
side
 of
the
angle
θ .
 Ex:

Find
the
reference
angle
for
2250
 
 
 
 
 
 
 Once
we
have
established
how
to
find
the
reference
angle,
following
is
the
 procedure
to
calculate
trig
function
value
of
any
angel.
 
 
 
 Ex:

Find
the
value
of
 sin 2250 
after
figuring
out
the
corresponding
reference
angel.
 
 a) Sketch
the
angel
 
 
 
 
 
 
 
 
 
 
 
 
 b) Determine
the
reference
angel
 
 
 
 
 
 c) Evaluate
the
trig
function
of
the
reference
angel.
 
 
 
 
 
 d) Affix
the
appropriate
sign
depending
on
which
quadrant
the
angle
lies
in.
 
 
 
 
 
 
 
 Ex:

Use
the
same
steps
as
described
in
the
example
above
to
find
the
 following
quantities:
 
 
 cos1500
 
 
 
 
 
 
 sin1500
 
 
 
 
 










sin(‐1500)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 













cos(‐1500)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Find
sin(4050)
,
cos(4050)
and
tan
(4050)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:
Find
the
area
of
a
triangle
with
sides
5
cm,
7cm
and
the
angle
between
them
is
 1200.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Trigonometric
Identities
(6.5)
 
 
 Basic
Identities
that
you
must
memorize:
 
 1) sin 2θ + cos 2 θ = 1 Pr oof : 2) tanθ = sinθ cosθ 3) secθ = 1 cosθ cscθ = 1 sinθ 4 ) 1 + tan 2θ = sec2 θ Proof : 5) 1 + cot 2θ = csc2 θ Proof : 
 
 
 
 
 
 
 
 These
identities
are
true
for
all
angles
θ 
 cotθ = 1 
 tanθ Ex:

Given
 cosθ = 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 , 2700 < θ < 3600 ( Q IV) 



Find
all
the
other
five
trig
functions.
 4 
 Ex:

Given
 sinθ = −3u , 1800 < θ < 2700 


Find
all
the
other
five
trig
functions.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Prove
the
following
identities:
 
 Ex:

 tan β sin β = sec β − cos β 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 − tan 2 α Ex:

Show
 cos2 α − sin 2 α = 
 1 + tan 2 α 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Show
 cosθ 1 + sinθ 
 = 1 − sinθ cosθ 
 
 Note:

Since
this
example
is
already
in
terms
of
sinθ 
and
cosθ ,
you
will
have
to
use
 the
algebraic
technique
of
rationalizing
the
denominator.

Then
you
will
be
able
to
 use
the
basic
identities.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

 secθ − cscθ tanθ − 1 
 = secθ + cscθ tanθ + 1 
 Note:

In
this
example,
you
may
have
to
work
with
both
sides
of
the
equation
.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ...
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This note was uploaded on 08/09/2011 for the course MATH 3 taught by Professor Staff during the Spring '08 term at University of California, Santa Cruz.

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