Download Trig Review Part Two - Benjamin N. Cardozo High School

Important Idea r > 0
y
opp
sin  

hyp r
adj
x
cos 

hyp r
opp
y
tan  

adj
x
( x, y )
r
y

x
Try This
Find sin, cos
& tan of the
angle 
whose
terminal side
passes
through the
point (5,-12)

(5,-12)
Solution
12
sin   
13
5
cos 
13
12
tan   
5
5

13
(5,-12)
-12
Important Idea
Trig ratios may be positive
or negative
Definition
Reference Angle: the
acute angle between the
terminal side of an angle
and the x axis.
(Note: x axis; not y axis).
Reference angles are
always positive.
Important Idea
How you find the reference
angle depends on which
quadrant contains the given
angle.
The ability to quickly and
accurately find a reference
angle is important.
Example
Find the reference angle if
the given angle is 20°.
y
In quad. 1,
the given
20° angle & the
x
ref. angle are
the same.
Example
Find the reference angle if
the given angle is 120°.
For
given
y
angles in quad.
120°
2,
the
ref.
?
x angle is 180°
less the given
angle.
Example
Find the reference angle if
7

the given angle is
.
6
7

For given
y
angles in quad.
6
3, the ref.
x angle is the
given angle
less 
Try This
Find the reference angle if
7
the given angle is
4
For given
7
angles in quad.
4,
the
ref.
4

angle is 2 less
4 the given
angle.
Important Idea
The trig ratio of a given
angle is the same as the
trig ratio of its reference
angle except, possibly, for
the sign.
We can use
the unit circle
to find trig
functions of
quadrantal
angles.
Definition
1
-1
1
-1
y
x
The unit
circle
(-1,0)
(0,1)
1
-1
1
-1
(0,-1)
(1,0)
Definition
For the
quadrantal
angles:
(-1,0)
(0,1)
1
-1
1
The x values are (0,-1)
the terminal
sides for the cos
function.
-1
(1,0)
Definition
For the
quadrantal
angles:
(-1,0)
(0,1)
1
-1
1
The y values are (0,-1)
the terminal
sides for the sin
function.
-1
(1,0)
Definition
For the
quadrantal
angles : (-1,0)
(0,1)
1
-1
1
The tan function (0,-1)
is the y divided
by the x
-1
(1,0)
Example
Find the
(0,1)
1
values of
(1,0)
(-1,0)
the 6 trig
functions of0°
(0,-1)
the
quadrantal
sin  csc
angle in
sec
cos

standard
tan  cot 
position:
-1
1
-1
(0,1)
Find the Example
values of
 (1,0)
the 6 trig (-1,0)
functions of
the
quadrantal
(0,-1)
angle in
sin  csc
standard
sec

cos

position:
90°
tan  cot 
1
-1
1
-1
(0,1)
Find the Example
values of
the 6 trig (-1,0)
(1,0)
functions of
the
quadrantal
(0,-1)
angle in
sin  csc
standard
sec

cos

position:
180°
tan  cot 
1
-1
1
-1
(0,1)
Find the Example
values of
the 6 trig (-1,0)
(1,0)
functions of
the
quadrantal
(0,-1)
angle in
sin  csc
standard
sec

cos

position:
270°
tan  cot 
1
-1
1
-1
(0,1)
Find the Try This
values of
the 6 trig (-1,0)
(1,0)
functions of
the
quadrantal
(0,-1)
angle in
sin  csc
standard
sec

cos

position:
360°
tan  cot 
1
-1
1
-1
A trigonometric identity is a
statement of equality
between two expressions.
It means one expression
can be used in place of the
other.
A list of the basic identities
can be found on p.317 of
your text.
Reciprocal Identities:
1
1
csc  
sin  
sin 
csc 
1
cos  
sec 
1
sec  
cos 
1
tan  
cot 
1
cot  
tan 
Quotient Identities:
sin A
 tan A
cosA
cos A
 cot A
sin A
y
sin  
r
x
cos  
r
1
r

but…
x y r
2
2
y
-1
x
2
therefore sin
1
2
  cos   1
-1
2
Pythagorean Identities:
sin   cos   1
2
2
Divide by cos  to get:
2
tan   1  sec 
2
2
Pythagorean Identities:
sin   cos   1
2
2
Divide by sin  to get:
2
1  cot   csc 
2
2
Try This
Use the Identities to
simplify the given
expression:
cot t sin t  sin t
2
2
1
2
Try This
Use the
Identities
to simplify
the given
expression:
sec t  tan t
2
2
2
cos t
2
sec t
Prove that this is an identity
sin 
 1  cos
1  cos
2
Now prove that this
is an identity
sin q
1+ cos q
+
= 2cot q sec q
1+ cos q
sin q
One More
1
1
2
= - 2sec x
sin x - 1 sin x + 1