Download 1 radian

Section 2.1
Angles and Their Measure

Vertex
Initial Side
Counterclockwise rotation
Positive Angle
Consider a circle of radius r.
Construct an angle whose vertex is at
the center of this circle, called the
central angle, and whose rays subtend
an arc on the circle whose length is r.
The measure of such an angle is 1
radian.
r

r
1 radian
Theorem Arc Length
For a circle of radius r, a central angle
of  radians subtends an arc whose
length s is
s  r
Relationship between Degrees and Radians
-> 1 revolution = 2 π radians
-> 180o = π radians

1 

180
radian
 180 
1 radian  

  

Announcements
• Test Friday (Jan 30) in lab, ARM
213/219, material through section 2.2
• Sample test posted...link from course
Web site
• Bring picture ID…you will need to
scan your ID upon entering the lab
• You may use a calculator up to TI 86.
• You can’t your the book or notes
Section 2.2
Right Angle Trigonometry
A triangle in which one angle is a right
angle is called a right triangle. The side
opposite the right angle is called the
hypotenuse, and the remaining two sides
are called the legs of the triangle.
c
b

90
a

Initial side
c

a
b
The six ratios of a right triangle are called
trigonometric functions of acute angles
and are defined as follows:
Function name Abbreviation
Value
b/c
sin 
sine of 
a/c
cos
cosine of 
b/a
tan 
tangent of 
c/b
cosecant of 
csc 
c/a
secant of 
sec 
a /b
cotangent of 
cot 
Pathagorean Theorem
c
b
a
a2 + b2 = c2
Find the value of each of the six
trigonometric functions of the angle  .
c = Hypotenuse = 13
12
13
b = Opposite = 12
a b  c
2
Adjacent
2
a  12  13
2

2
2
2
a  169  144  25
2
a 5
a  Adjacent = 5
b  Opposite = 12
c  Hypotenuse = 13
Opposite
12 csc  Hypotenuse  13
sin 

Opposite
12
Hypotenuse 13
Hypotenuse 13
Adjacent
5
sec 

cos 

Adacent
5
Hypotenuse 13
Adjacent 5
Opposite 12
cot  

tan 

Opposite 12
Adjacent 5
b a c
2
2
2
2
2
b a
c


2
2
2
c
c
c
c
b
2
2
2
 b   a   1
 c  c 

90
a
sin   cos   1
2
2
Pythagorean Identities
The equation
sin2θ + cos2 θ
along with
tan2 θ + 1 = sec2 θ
and
1 + cot2 θ = csc2 θ
are called the Pythagorean identities.
More Identities
Reciprocal
Identities
1
csc  
sin 
1
sec  
cos 
1
cot  
tan 
Quotient
Identities
sin 
tan  
cos 
cos 
cot  
sin 
Complementary Angles Theorem
Cofunctions of complementary angles
are equal.
Two acute angles are complementary if
the sum of their measures is a right
angle…90 degrees.
Complementary Angles in Right
Triangles
β
α
The angles α and β are complementary in a
right triangle, α + β = 90 degrees.
Cofunctions
D
e
g
r
e
e
s
R
a
d
i
a
n
s
sin   cos(90   ) cos   sin( 90   )


tan   cot(90   ) cot   tan( 90   )


sec   csc(90   ) csc   sec(90   )





sin   cos    cos   sin    
2

2





tan   cot     cot   tan    
2

2





sec   csc    csc   sec   
2

2


Using the Complementary Angle Theorem
Find the exact value (no calculator) of the following
expressions.

a.
b.
cos 40

sin 50
2

2

1  cos 20  cos 70

a.
cos 40

sin 50
cos 40  sin( 90  40 )  sin 50




cos 40 sin 50


1


sin 50
sin 50


b.
1  cos 20  cos 70

2
cos 20  sin 70

2

2
2

so
1  sin 70  cos 70
2

2

 1  (sin 70  cos 70 )
2
 1  (1)  0

2
