Transcript o - Nmsu

Mathematics
Trig and Geometry
This is the unit circle…
sine
The
second
Also
note
that to
The
distance
quadrant
has
the
is the
thecosine
right
on
negative
negative
ifisthe
cos-axiscosines
the and
positive
sines.
line
is drawn
cosine
of theto
and

It axes are
Trigonometry
Geometry
the
left onIIthe
angle.
cos-axis.
We
speak of
Hereoften
are some
four
otherquadrants.
examples.
1
sin
3
2
cosine
1
2
Note that the angle
The
thirdgoes
quadrant
always
from
has
thenegative
positivecosines
cos-axis
and
sines.
counterclockwise.
1
The first
quadrant
All lines
drawn
has positive
cosines
here have
a
and sines.
length of 1
and an angle
I equal to the
angle we are
working with.
150o
o
330
1
2
3
2
o
45
30o
1
12
2
11

1
1
2
3
2
III
IV
cos
Why
is the sine
The height
negative
along thehere?
sinaxis
the sine
The is
fourth
of
the angle.
quadrant
has
positive cosines
and negative
sines.
Mathematics
Trig and Geometry
Given a right triangle, the trigonometric functions for either
non-right angle are given by the following…
Trigonometry
opposite
(o)
hypotenuse
(h)
Geometry
θ
adjacent
(a)
o
sin  
h
h
csc  
o
a
cos  
h
h
sec  
a
o
tan  
a
a
cot  
o
The value of the angle can also be determine by using any two of the
sides. For example,
o
tan    
a
1
2
Mathematics
Trig and Geometry
Here is an example of how to use it…
3
5
csc  36.87  
5
3
4
5
h=5
cos  36.87  
sec  36.87  
5
4
Note: This
is NOT
3
4
tan  36.87  
cot  36.87  
drawn to
4
3
scale!
o
θ=36.87
sin 36.87 
Trigonometry
Geometry
o=3
a=4
The value of the angle can also be determine by using any two of the
sides. For example,
3
tan    36.87
4
1
3
Mathematics
Trig and Geometry
Here are some useful angle relations…
a
a
b
a b
b a
a b
b b
b
a  b  180 
a
Trigonometry
a  b  180
Geometry
a  b  180 

b
a
a  b  c  180
a
a
4
C
c

a
b
A
c
B
A
B
C


sin a sin b sin c
a
Mathematics
Trig and Geometry
For example…
Trigonometry
Geometry
a
b
a  b  180
5
If a  30 then b  150

Mathematics
Trig and Geometry
Here are some basic geometric and trigonometric formulae which we
will use often in this and the next class…
Circumference of a Circle
Trigonometry
Geometry
Trigonometric Formulae
sin 2   cos 2   1
sin  A  B   sin A cos B  cos A sin B
cos A  B   cos A cos B  sin A sin B
Area of a Circle
A  r
Surface Area of a Sphere
A  4r 2
Volume of a Sphere
4 3
V  r
3
Quadratic Formula
A  2rL
Ax 2  Bx  C  0
V  r 2 L
 B  B 2  4 AC
x
2A
Surface Area of a Cylinder
(not including end faces)
Volume of a Cylinder
6
C  2r
2