Transcript 7.3 The Sine and Cosine Functions

8.2.1 THE
SINE AND
COSINE
FUNCTIONS..
SLOWED DOWN
UNIT CIRCLE
Remember from geometry that we used sine, cosine,
and tangent while working with right triangles. Here
we also want to work with right triangles. So when
you are provided an angle measurement in degrees
or radians, you will want to draw a picture of that
angle in the coordinate plane, and then construct a
line from a point on the circle of that angle down to
or up to the x-axis. Thus creating a right angle.
We are going to start with talking about circles of
any radius, then scale everything down to the unit
circle.
Learn this idea and then we can
look at any angle and do the
same thing (just have to pay
attention to the quadrant)
r
X
So if we remember SOH CAH TOA
We can say the following.
Y
y
sin  
r
cos  
tan  
Consider the angle ϴ with a terminal ray in the second quadrant.
The terminal ray
is shared by the
right triangle and
the angle ϴ.
Thus what we
learn about the
triangle will also
be true for the
angle ϴ.
Y
X
EXAMPLE
If the terminal ray of an angle ϴ in standard position
passes through (-3,4) find sin(ϴ) and cos(ϴ).
sin( ) 
4
cos( ) 
-3
5
If ϴ is a 4th quadrant angle and sin( )  
13
find cos(ϴ).
If we let the radius of the circle be 1. Then r=1 and
when we think about the following functions
P(x,y)
1
y
ϴ
y
sin( )   y
1
x
cos( )   x
1
x
When we deal with a
circle of radius 1, it is
referred to as the UNIT
CIRCLE.
P(x,y) = (cosϴ, sinϴ)
1
y
ϴ
x
If working in the unit
circle, your x value is
simply the same as
cosϴ and your y
value is the same as
your sinϴ.
EXAMPLES .. NOT THE UNIT CIRCLE. HOW WOULD YOU FIND
THETA IN THIS PICTURE? YOU MUST CREATE A RIGHT TRIANGLE.
Find sin  and cos 
EXAMPLES.. NOT THE UNIT CIRCLE
Find sin  and cos 
EXAMPLES.. NOT THE UNIT CIRCLE
Find sin  and cos 
STATE WHETHER EACH EXPRESSION
IS POSITIVE, NEGATIVE, OR ZERO.
sin165
0
sin 268
0
cos 210
5
sin
6
2
cos
3
0
What would be the equation of a circle with its center at the
origin and a radius of 1?
Lets take a look at the points that we found yesterday on the Unit
Circle. Did those points all satisfy the equation x2+y2=1?
Now lets remember from yesterday that on the unit circle,
x=cos(ϴ) and y=sin(ϴ). So does it make since then that we can
take the unit circle equation and substitute the x and y values
and get (cosϴ)2+(sinϴ)2=1
This equation is referred to as an IDENTITY (or an equation that
is true regardless of what values are chosen) and it is the
PYTHAGOREAN IDENTITY.
Evaluating Sine and Cosine using
Reference Angles
Using Reference Angles to Evaluate Trig
Functions at Special Angles
THIS ALL TAKES PLACE USING THE UNIT
CIRCLE
Special right triangles
While working in the unit circle we come across angles that are
multiples of 30, 60, and 45. If you remember from geometry there
was a need to work with 30-60-90 and 45-45-90 right triangles, that
need resurfaces within the unit circle.
1st quadrant special angles
1
2n
1
2
n
3
2
n 𝟑
What does this
picture look like if
our radius is 1?
What are the coordinates of this point?
Remember that this is the unit circle, and
in the unit circle for any P(x,y), the x
value is cosine and the y value is sine.
1
𝟏
So what is the sin(30)? cos(30)?
𝟐
𝟑
𝟐
Lets try the same thing with a 60 degree angle for ϴ
Can it be done with a 45 degree
• If we look at the unit circle note that it is
symmetric about both the x and y axis.
Because of this symmetry we can learn a lot about the first
quadrant and then use that information as a reference (or
guide) for the remaining three quadrants.
Take an angle that has a terminal ray in the 1st quadrant, what will the sign of
cos(ϴ) be and what will the sin(ϴ) be?
Will those signs be true for any angle that has a terminal ray in the 1st quadrant?
Now do the same procedure for the 2nd, 3rd, and 4th quadrants.
S
Sine is positive
T
Tangent is
positive
A
All are positive
C
Cosine is
positive
• A reference angle is an angle that will always be
between 0 and 90 degrees and will always be
coterminal with the angle ϴ you are interested in.
Thus what you are able to learn about the reference
angle should then be true about the interested
angle ϴ. So if the terminal ray of a reference angle
passes through (x,y) then the terminal ray of your
initial angle ϴ should pass through (x,y)…Reference
angles are only needed when the initial angle ϴ is
bigger than 90 degrees. When the initial angle ϴ is
less between 0-90 degrees, then the reference angle
and the initial angle ϴ are the same exact angle.
To create a reference angle.
• Draw out the angle ϴ.
• Take a perpendicular line to the x axis from the terminal
ray
• Note what quadrant the terminal ray is in.
• Determine the angle of the new triangle.
• Reproduce the same triangle in the first quadrant, label
that angle α (alpha)
0
sin(120 )
ϴ
α
Now if we can figure out what
sin(60) is we can determine
what sin(120) is. Making sure
that our sign matches up to that
of a terminal angle in the 2nd
quadrant.
Practice Creating Reference angles
sin(135o )
cos(330o )
tan(210o )
 5 
sin 

 4 
 5 
cos 

 6 
 2 
tan 

 3 
Use reference angles to find the
following values
 2 
cos 

 3 
Use reference angles to find the
following values
 19 
sin 

 4 
Use reference angles to find the
following values
 
tan   
 3
Because we know that sin(45⁰) is the same value as sin(45⁰+360⁰)
it is evident that the sine and cosine functions repeat their values
every 360⁰ degrees or every 2π radians. Thus these functions are
referred to as being periodic and have a fundamental period of 360⁰ or 2π
radians. This is useful in analyzing repetitive phenomena such as
tides, sound waves, orbital paths, etc.