Download DAY V 4.4

He who is devoid of the
power to forgive
is devoid of the power to
love.
Dr. Martin Luther King,
Jr.
(1929 – 1968)
an American clergyman, activist, and
prominent leader in the AfricanAmerican Civil Rights Movement
Chapter 4
Trigonometry
Day V
Trigonometric Functions of
Any Angle (4.4)
You can use trigonometric
functions to model and solve
real life problems, like
modeling the average daily
temperature in a city.
Goal 1
To evaluate
trigonometric
functions of
any angle
I. Introduction
Life does not always
give us unit circles,
where r = 1.
Why can’t r ≤ 0?
Since r = any value,
we have generalized
trigonometry
definitions
Standard 2.1
Students know the definition
of sine and cosine as y-and
x-coordinates of points on
the unit circle
Standard 5.1
Students know the
definitions of the
tangent and cotangent
functions.
Standard 6.1
Students know the
definitions of the secant
and cosecant functions.
(x, y)
r = x² + y²
y
r
sin  =
csc  = , y  0
r
y
x
r x0
cos  =
sec  = ,
r
x
y
x
tan  = , cot  = , y  0
x
y
x0
Let’s play “Name that
Quadrant”
Name the Quadrants
in which
the sine function is
positive.
I and II
sin 
cos 
tan 
sec 
csc 
cot 
I
+
__
__
__
__
__
__
II
+
__
__
__
__
__
__
III
__
__
__
__
__
__
IV
__
__
__
__
__
__
Name the Quadrants
in which
the sine function is
negative.
III and IV
sin 
cos 
tan 
sec 
csc 
cot 
I
+
__
__
__
__
__
__
II
+
__
__
__
__
__
__
III
–
__
__
__
__
__
__
IV
–
__
__
__
__
__
__
Name the Quadrants
in which
the cosine function is
positive.
I and IV
sin 
cos 
tan 
sec 
csc 
cot 
I
+
__
+
__
__
__
__
__
II
+
__
__
__
__
__
__
III
–
__
__
__
__
__
__
IV
–
__
+
__
__
__
__
__
Name the Quadrants
in which
the cosine function is
negative.
II and III
sin 
cos 
tan 
sec 
csc 
cot 
I
+
__
+
__
__
__
__
__
II
+
__
–
__
__
__
__
__
III
–
__
–
__
__
__
__
__
IV
–
__
+
__
__
__
__
__
Name the Quadrants
in which
the tangent function
is positive.
I and III
sin 
cos 
tan 
sec 
csc 
cot 
I
+
__
+
__
+
__
__
__
__
II
+
__
–
__
__
__
__
__
III
–
__
–
__
+
__
__
__
__
IV
–
__
+
__
__
__
__
__
Name the Quadrants
in which
the tangent function
is negative.
II and IV
sin 
cos 
tan 
sec 
csc 
cot 
I
+
__
+
__
+
__
__
__
__
II
+
__
–
__
–
__
__
__
__
III
–
__
–
__
+
__
__
__
__
IV
–
__
+
__
–
__
__
__
__
Fill in the rest of the
chart!
sin 
cos 
tan 
sec 
csc 
cot 
I
+
__
+
__
+
__
+
__
+
__
+
__
II
+
__
–
__
–
__
–
__
+
__
–
__
III
–
__
–
__
+
__
–
__
–
__
+
__
IV
–
__
+
__
–
__
+
__
–
__
–
__
.8 over the 2. . .
What’s your sine?
Example 1
Evaluating
Trigonometric
Functions
1. Let (-12, -5) be a point on the
terminal side of . Find the sine,
cosine, and tangent of .
-5
-12
y²
r = x²+
?
= (-12)²+(-5)²
= 169
= 13
1. Let (-12, -5) be a point on the
terminal side of . Find the sine,
cosine, and tangent of .
-5
-12
13
sin  = y/r = -5/13
cos  = x/r =-12/13
tan  = y/x = 5/12
2. Let (3, 1) be a point on the
terminal side of . Find the sine,
cosine, and tangent of .
3
1
y²
r = x²+
?
= 3²+1²
= 10
2. Let (3, 1) be a point on the
terminal side of . Find the sine,
cosine, and tangent of .
10
3
1
sin  = y/r = 1/10
cos  = x/r =3/10
tan  = y/x = 1/3
Your Turn
1. Let (-5, 2) be a point on the
terminal side of . Find the sine,
cosine, and tangent of .
29
2
sin =y/r=2/29
-5
cos =x/r= -5/29
tan  =y/x=2/-5
2. Let (3, -7) be a point on the
terminal side of . Find the sine,
cosine, and tangent of .
3 sin =y/r= -7/58
-7 cos =x/r= 3/58
58
tan  =y/x=-7/3
sin   0 and cos   0
Where am I?
sin   0
cos   0
III
Your Turn
sin   0 and tan   0
Where am I?
sin  > 0
tan   0
II
sec   0 and cot   0
Where am I?
sec  > 0
cot   0
IV
Example 2
Evaluating
Trigonometric
Functions
If sin  = ½ and tan   0, find the
exact value of cos .
Where is sin  0?
Where is tan  0?
 is in Quadrant II.
If sin  = ½ and tan   0, find the
exact value of cos .
1
2
-3
cos  = x/r = -3/2
Your Turn
1. If cos  = -4/5 and  is in
Quadrant II, find the exact value of
sin .
5
3
-4
sin θ = 3/5
2. If csc  = 4 and cot   0, find
the exact value of cos .
Where is csc  0?
Where is cot  0?
 is in Quadrant II.
2. If csc  = 4/1 and cot  < 0, find
the exact value of cos .
1 4
-15
cos θ = -15/4
What starts with T
ends with T and is full
of T?
What starts with T
ends with T and is full
of T?
Teapot
A quadrantal angle
is an angle that lies on
the x- or y-axis.
Example 3
Trigonometric
Functions of
Quadrantal Angles
Find the cos  of the four
quadrantal angles.
Unit circle
cos  = x/r
(1
? , ?0 )
cos 0 = 1/1 = 1
Find the cos  of the four
quadrantal angles.
Unit circle
(0
? , 1? )
cos  = x/r
cos /2 = 0/1 = 0
Find the cos  of the four
quadrantal angles.
Unit circle
cos  = x/r
(-1
?)
? ,0
cos  = -1/1 = -1
Find the cos  of the four
quadrantal angles.
cos  = x/r
Unit circle
cos 3/2 = 0/-1 = 0
( ?0 ,-1
?)
Your Turn
If tan  = undefined and  ≤ θ ≤
2, find the exact value of sin .
tan θ = y/x?
When is tan undefined?
(0, -1)
WhatSin
θ isθ x==-1/1
0 between
= -1
 and 2?
Goal 2
To use
reference angles
to evaluate
trigonometric
functions
II. Reference Angles
Standard 9.1
Students compute, by hand,
the values of the
trigonometric functions at
various standard points.
A reference angle
is the acute angle formed
by the terminal side of 
in standard position and
the HORIZONTAL axis.
Quadrant II


 = 180 - 
 =  - 
Quadrant III


 =  - 180
 =  - 
Quadrant IV
 = 360 - 


 = 2 - 
Example 4
Finding Reference
Angles
 = 210
210

 =210
 - 180
 = 30
 = 4.1
3.14

4.1
 = 4.1
 -
  .96
4.71
Your Turn
θ = 309
360 – 309 = 51
θ = -149
180 – 149 = 31
θ = 7/4
8 – 7 = 1
4
4
4
θ = 11/3
12 – 11 = 1
3
3
3
Example 7
Evaluate
NO calculator - evaluate
2
sec θ = -2/1
-1 60
2
r = 2, x = -1
120 V 240
Where am I?
2/3 V 4/3
Your Turn
NO calculator - evaluate
sin θ = 2/2
22
1
1
sin θ= 1/2
45
y = 1, r = 2
45
V
135
Where am I?
/4 V 3/4