Transcript Trig _Basics_Quick_Review

Definition y
Radian: The
length of the
arc above the
angle divided
by the radius of
the circle.
1
r

-1
-1
s
  ,  in radians
r
s
1
x
Definition y
Unit Circle:
the circle with
radius of 1
unit
If r=1,  =s
1
1

-1
-1
s
  ,  in radians
1
s
1
x
Definition
The radian measure of an
angle is the distance traveled
around the unit circle. Since
circumference of a circle is
2  r and r=1, the distance
around the unit circle is 2 
Important Idea
If a circle contains 360° or
2 radians, how many
radians are in 180°
Use to change
rads to degrees
Use
to
change
rads

•
180° degrees to rads
180°
•
 rads
Try This
Change 240° to radian
measure in terms of .
4
rads
3
Try This
7

Change
radians to
8
degree measure.
157.5°
Try This
Change 3 radians to
degree measure.
171.89°
Definition
Terminal Side
y
Vertex
A
Initial Side
x
Angle A
is in
standard
position
Definition
y
A
x
If the
terminal
side
moves
counterclockwise,
angle A is
positive
Definition
y
A
x
If the
terminal
side
moves
counterclockwise,
angle A is
positive
Definition
y
A
x
If the
terminal
side
moves
counterclockwise,
angle A is
positive
Definition
y
A
x
If the
terminal
side
moves
clockwise,
angle A is
negative
Definition
y
A
x
If the
terminal
side
moves
clockwise,
angle A is
negative
Definition
y
A
x
If the
terminal
side
moves
clockwise,
angle A is
negative
Definition
y
A
x
If the
terminal
side
moves
clockwise,
angle A is
negative
Definition
y
A
If the
terminal
side is on
x an axis,
angle A is a
quadrantel
angle
Definition
y
A
If the
terminal
side is on
x an axis,
angle A is a
quadrantel
angle
Definition
y
A
If the
terminal
side is on
x an axis,
angle A is a
quadrantel
angle
Definition
y
A
If the
terminal
side is on
x an axis,
angle A is a
quadrantel
angle
Definition 
The
quadrantal
angles in
radians
2

0
2
3
2
Definition 
The
quadrantal
angles in
radians
2

0
2
3
2
Definition 
The
quadrantal
angles in
radians
2

0
2
3
2
Definition 
2
The
quadrantal
0
angles in

2
radians
The terminal side is on an
axis.
Definition
Coterminal Angles: Angles
that have the same terminal
side.
Important Idea
In precal, angles can be
larger than 360° or 2 
radians.
Important Idea
To find coterminal angles,
simply add or subtract
either 360° or 2  radians
to the given angle or any
angle that is already
coterminal to the given
angle.
Analysis
30° and
390° have
the same
terminal
side,
therefore,
the angles
are
coterminal
y
30°
x
y
390°
x
Analysis
30° and
750° have
the same
terminal
side,
therefore,
the angles
are
coterminal
y
30°
x
y
750°
x
Analysis
30° and
1110° have
the same
terminal
side,
therefore,
the angles
are
coterminal
y
30°
x
y
1110°
x
Analysis
30° and
-330° have
the same
terminal
side,
therefore,
the angles
are
coterminal
y
30°
x
y
-330°
x
Try This
Find 3 angles coterminal
with 60°
420°,780° and -300°
Try This
Find two positive angle and
one negative angle
5

coterminal with 
6
radians.
19
7
17

and

,
6
6
6
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
Find the exact 11
value of the sin,
cos and tan of the 6
given angle in
standard position.
Do not use a
calculator.
Solution 11
1
 11 
sin 

2
 6 
3
 11 
cos 


 6  2
 11
tan 
 6
1
3



3
3

6
3
-1
2
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 going to be
important in future lessons.
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.
Example:
sin 130   sin  50

sin  230    sin  50

The unit
circle is a
circle with
radius of 1.
We use the
unit circle to
find trig
functions of
quadrantal
angles.
Definition
1
-1
1
-1
Definition
The unit
circle
(-1,0)
y
x
(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 of
(0,-1)
the
quadrantal
sin  csc 
angle in
sec 
cos 
standard
tan  cot 
position: 0°
-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:
tan  cot 
90°
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:
tan  cot 
180°
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:
tan  cot 
270°
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:
tan  cot 
360°
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
y

but…
x y r
2
2
-1
1
x
2
therefore sin
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
+
= 2 cot q sec q
1 + cos q
sin q
One More
1
1
2
= - 2sec x
sin x - 1 sin x + 1