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CHAPTER 6:
TRIGONOMETRY
Section 6.4: Trigonometric Functions
DAY 1
TRIGONOMETRIC RATIOS

r

P (x.y)
Let θ be an angle in
standard position and let
P(x,y) be any point on the
terminal side of θ. Let r be
the distance from (x,y) to
the origin:
x2  y2
Then the trigonometric
ratios of θ are defined as
follows:
r
θ
y
r
r
csc  
y
sin  
y
x
x
r
r
sec 
x
cos  
y
x
x
cot 
y
tan  
TRIGONOMETRIC RATIOS

Find Sin, Cos, and Tan of
the angle θ, whose terminal
side passes through the point
(-3,-2).
x  -3, y  -2, and
r
( 3) 2  ( 2) 2 
2
,
13
-3
cos 
,
13
2
tan  
3
sin  

Find Sin, Cos, and Tan of
the angle θ, whose terminal
side passes through the point
(-2,3).
x  -2, y  3, and
13
r
( 2) 2  (3) 2 
3
,
13
-2
cos 
,
13
3
tan   
2
sin  
13
TRIGONOMETRIC FUNCTIONS

Trigonometric ratios have been
defined for all angles. But
modern applications of
trigonometry deal with
functions whose domains
consist of real numbers. The
basic idea is quite simple: If t
is a real number then:

sin t is defined to be the sine of
an angle of t radians;

cos t is defined to be the cosine
of an angle of t radians;

and so on. Instead of starting
with angles, as was done up
until now, this new approach
starts with a number and only
then moves to angles
Begin
with a
Number t
Form an
angle
of t
radians
Determine
sint, cost,
tant
TRIGONOMETRIC RATIOS

(x.y)
Let t be a real number.
Choose any point (x,y) on
the terminal side of an
angle t radians in standard
position.
r
r
t
x
x2  y2
y
r
r
csc t 
y
sin t 

y
Then the trigonometric
ratios of t radians are
defined as follows:
x
r
r
sec t 
x
cos t 
y
x
x
cot t 
y
tan t 
TRIGONOMETRIC RATIOS

Find Sin t, Cos t, and Tan t
when the terminal side of t
radians passes through the
point (5,-1).
x  5, y  -1, and
r
(5) 2  ( 1) 2 
sin t  
1
,
26
5
,
26
1
tan t  
5
cos t 

Find Sin t, Cos t, and Tan t
when the terminal side of t
radians passes through the
point (-4,4).
x  -4, y  4, and
26
r
( 4) 2  ( 4) 2 
sin t 
4

1
,
2
4 2
-4
1
cos t 
,  4 2
2
4
tan t    1
4
32  4 2
TRIGONOMETRIC RATIOS

The terminal side of an angle of t
radians lies in quadrant 1 on the
line through the origin parallel to
-2y+5x=12. Find Sin t, Cos t, and
Tan t.
First solve for y!
5
y  x6
2
Since the slope is
y  5, and r 
tan t 
5
2
YOU TRY!! The terminal side of
an angle of t radians lies in
quadrant 1 on the line through
the origin parallel to 3y-4x=12.
Find Sin t, Cos t, and Tan t.
First solve for y!
4
y x4
3
5
, then x  2,
2
( 2) 2  (5) 2 
5
, cos t 
29
sin t 

29
2
,
29
Since the slope is
y  4, and r 
4
, then x  3,
3
(3) 2  ( 4) 2 
4
3
, cos t  ,
5
5
4
tan t 
3
sin t 
25  5
HOMEWORK!!!

Page 452: 1-6

6.4 Worksheet #1
DAY 2
TRIGONOMETRY AND THE UNIT CIRCLE

1
1
-1
y
 y
1
x
cos t 
x
1
y
sin t
tan t 

x
cos t
sin t 
1
-1
In the unit circle, the
radius is always 1. So
if r = 1, then:
1
1

sin t
y
1
1
sec t 

cos t
x
x
cos t
cot t 

y
sin t
csc t 
DOMAIN AND RANGE



sin and cos:
Domain is the set of
all real numbers!
Range is the set of all
real numbers between
-1 and 1.



tan:
Domain is the set of
all real numbers
except ±π/2 + kπ, where
k = 0.±1,±2,…
Range is the set of all
real numbers!
EXACT VALUES OF OUR SPECIAL ANGLES
Square root of finger over palm!
t
30o
45o
60o
Sin t
1/2
√2/2
√3/2
Cos t
√3/2
√2/2
1/2
Tan t
√3/3
1
√3
Csc t
2
√2
2√3/3
Sec t
2√3/3
√2
2
Cot t
√3
1
√3/3
Cos
90o
60o
45o
30o
2
0o
Sin
Flip hand over for Tangent!!
EXACT VALUES OF OUR SPECIAL ANGLES
Square root of finger over palm!

Without using a calculator,
Find the sin, cos, and tan
of 30o.
sin

6

1
2
Cos
90o
60o
45o
30o
2

3
cos

6
2
tan

6

1
3

3
3
0o
Sin
Flip hand over for Tangent!!
EXACT VALUES OF OUR SPECIAL ANGLES
Square root of finger over palm!

Without using a
calculator, Find the
sin, cos, and tan of
45o.
sin
cos
tan

4

4

4



2
2
2
2
2
1
2
Cos
90o
60o
45o
30o
2
0o
Sin
Flip hand over for Tangent!!
TRIGONOMETRY AND THE UNIT CIRCLE

Find sint, cost, and tant
when the terminal side of
an angle of t radians
passes through the given
point on the unit circle.
2
1 

,


5
5


Find sint, cost, and tant
when the terminal side of
an angle of t radians
passes through the given
point on the unit circle.
4
 3
  , 
5
 5
2
r
2
1 



 1
5
5

1
2
, cos t  
,
5
5
1
tan t  
2
sin t 
r
4
 3
   
5
 5
sin t  
tan t 
4
3
2
1
4
3
, cos t   ,
5
5
TRIG FUNCTION SIGNS

t 
2
sin t 
sin t 
cos t 
tan t 

2
sin t 
cos t 
tan t 
cos t 
tan t 
 t 
0t
3
2
3
 t  2
2
sin t 
cos t 
tan t 
HOMEWORK!!!

Page 452:

7-10

Create a poster of Trig Signs
in different quadrants. 50 pts



1) Colorful 10 pts
2) All correct signs 30 pts
3) Neatness 10 pts
DAY 3
REFERENCE ANGLES

Reference
Angle is
the positive
acute angle
formed by
the
terminal
side of θ
and the xaxis.
t
t
t’=t
t’=π-t
t
t
t’=t-π
t’=2π-t
REFERENCE ANGLES

1)
Find the reference
angle to the given
angle:
5
5
5
3
2)
4
6
5

1)


3
3
3
2)
5
4



4
4
4
3)

Now find sin, cos, and tan for
each problem and append the
appropriate sign.
1) sin
6
cos
tan

3

3

3


3
2
1
2
 3
3) sin
3)
6
5



6
6
6
tan

6


1
2
3
6
2

3
tan

3
3
cos

2
4
2

2
cos

4
2
2) sin


4

1
PRACTICE

Find the
exact value
of the sin,
cos, and tan
of the
number
without
7 a
using
a)
6
calculator.
7
b)
3
19
c) 3
7


in the Third
6
6
Quadrant, so :
a)
7
1

6
2
7
3
cos

6
2
7
3
tan

6
3
7


in the first
3
3
Quadrant, so :
b)
7
3

3
2
7
1
cos

3
2
7
tan
 3
3
sin
sin
19


in the fourth
3
3
Quadrant, so :
c) -
3
 19 
sin  
3 
2

 19  1
cos 

3

 2
 19 
tan  
 3
3 

COMPLETE THE CHART
T
Sin
Cos
Tan
Csc
Sec
Cot
30o. π/6
1/2
√3/2
√3/3
2
2√3/3
√3
45o, π/4
√2/2
√2/2
1
√2
√2
1
60o, π/3
√3/2
1/2
√3
2√3/3
2
√3/3
90o, π/2
1
0
Undef
1
Undef
0
HOMEWORK!!!

Pg.452

11-14,

15-23 part a only,

24-35.

Complete the chart!
DAY 4
EVALUATING EXPRESSIONS

Write the expression
as a single real
number.

1) sin 
6

 
 cos 

3
 
 
2) sin   cos 
4
6
 
 
3) sin   cos 
3
4
 
  1  1 1
1) sin   cos     
6
 3  22 4
2 3
6
 
 


2) sin   cos  

2 
4
4
6
 2 

3) sin 
3
3 2
6

 


cos


 

2 
4

4
 2 
EVALUATING EXPRESSIONS

YOU TRY!!! Write
the expression as a
single real number.
 
1) sin   tan  
6
 
 
2) tan   cos 
4
6
 
 
 
 
3) sin   cos   sin   cos 
6
3
2
3
1
 
1) sin   tan    0   0
2
6
 3
3
 
 

2) tan  cos   1
 2 
2
4
6



3) sin 
6
3)

 

 cos   sin 

3
2

 
 cos  

3
11
1
1 1 1
   1     
22
4
2 4 2
HOMEWORK!!

Pg.452: 36-53

6.4 Worksheet #2