Transcript 4 2 Unit Circle

Objectives:
1. To construct all
aspects of the unit
circle (angles and
points)
2. To use the unit circle
to find the values of
trig functions
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Assignment:
P. 299: 1-4 S
P. 299: 13-22 S
P. 299: 23-28 S
P. 299: 29-36 S
P. 300: 50, 60
Memory quiz next
class (30 Q in 30 min)
You will be
able to
construct all
aspects of
the unit
circle
(angles and
points)
Graph the equation
x2 + y2 = 1. Identify
the center and the
radius.
• Center: (0, 0)
• Radius: 1 unit
This tiny circle is called
the unit circle since
its radius is 1 unit.
This circle may be
tiny, but it will give us
a way to remember
102 exact trig values.
That’s pretty useful.
We already know how
to find various angle
measures on the unit
circle in both degrees
and radians. Now we
will find the exact
coordinates of
various points on the
circle. But first…
The hypotenuse of a
45-45-90 right
triangle has length 1
unit. Find the length
of the other two
sides of the triangle.
Add this to your
worksheet.
The hypotenuse of a
30-60-90 right
triangle has length 1
unit. Find the length
of the other two
sides of the triangle.
Add this to your
worksheet.
Now let’s build our unit
circle one triangle at a
time. Except the first
set of points do not
really make a triangle.
Oh, well.
I think at some point we
said the radius of this
circle is 1. Which four
points does this
nugget of information
help us find?
Next, let’s find the
coordinates of the
points that create
angles whose
measures are
multiples of 45°.
Well, 45°, 135°,
225°, and 315°
anyway.
To get to the point in
quadrant 1 rotated
45°, we could do
one of two things.
We could rotate 45°
and then travel 1
unit. That’s called
polar coordinates.
The second thing we
could do is travel to
the right a bit and
then up a bit. This
forms a 45-45-90
right triangle.
1. What are the leg
lengths?
2. What are the
coordinates of the
point?
The next point could
be found by rotating
135° and then
traveling 1 unit.
Alternatively, we could
use rectangular
coordinates by going
left a bit and up a
bit. Another 45-4590 right triangle.
One thing to bear in
mind for these
coordinates is that
since we had to go
left, our xcoordinate is
negative.
Next, let’s rotate
225° and travel 1
unit.
Or maybe we want
to go left a bit and
then down a bit.
Watch your signs.
Finally, we’ll rotate
315° and travel 1
unit.
Which is the same as
going right a bit
and then down a
bit.
Watch your signs
again.
Use a special right triangle to find the exact and
approximate value of each of the following.
1. sin 45°
1. sin π/4
2. cos 45°
2. cos π/4
3. tan 45°
3. tan π/4
Use a calculator to find the approximate value of
each of the following.
1.
2.
3.
4.
5.
6.
sin 135°
sin 225°
sin 315°
cos 135°
cos 225°
cos 315°
1.
2.
3.
4.
5.
6.
sin 3π/4
sin 5π/4
sin 7π/4
cos 3π/4
cos 5π/4
cos 7π/4
Now that we’ve
finished with the
45-45-90
triangles, let’s find
the points at 30°,
150°, 210°, and
330°
For that first point,
we could rotate 30°
and the travel 1
unit, using polar
coordinates, or we
could use good oldfashioned
rectangular
coordinates by
moving right some
and then up a bit
less.
Using rectangular
coordinates, of
course, creates
another right
triangle, a 30-60-90.
1. What are the
lengths of the legs?
2. What are
coordinates of the
point?
Now we’ll rotate 150°
and then travel 1
unit.
Which is really the
same as moving
left some and up a
bit less.
What do you know,
another 30-60-90
right triangle.
Next we’ll rotate
210° and then
travel 1 unit.
Or maybe we
should move left
some and down a
bit less.
Watch your signs.
Perhaps you’re
beginning to get
the picture. Let’s
do one more
point here,
rotated at 330°.
Keep an eye on
those signs.
Use a special right triangle to find the exact and
approximate value of each of the following.
1. sin 30°
1. sin π/6
2. cos 30°
2. cos π/6
3. tan 30°
3. tan π/6
Use a calculator to find the approximate value of
each of the following.
1.
2.
3.
4.
5.
6.
sin 150°
sin 210°
sin 330°
cos 150°
cos 210°
cos 330°
1.
2.
3.
4.
5.
6.
sin 5π/6
sin 7π/6
sin 11π/6
cos 5π/6
cos 7π/6
cos 11π/6
Finally, let’s move on
to a completely
different triangle,
the 60-30-90 right
triangle so we can
get the coordinates
of the points at
60°, 120°, 240°,
and 300°.
Use a special right triangle to find the exact and
approximate value of each of the following.
1. sin 60°
1. sin π/3
2. cos 60°
2. cos π/3
3. tan 60°
3. tan π/3
Use a calculator to find the approximate value of
each of the following.
1.
2.
3.
4.
5.
6.
sin 120°
sin 240°
sin 300°
cos 120°
cos 240°
cos 300°
1.
2.
3.
4.
5.
6.
sin 2π/3
sin 4π/3
sin 5π/3
cos 2π/3
cos 4π/3
cos 5π/3
You will be able to use the unit
circle to find the values of trig
functions
Let’s recap: We know all
the angle measures on
the unit circle in both
degrees and radians,
and we know the
coordinates of the
points along the unit
circle. Maybe this
would be a bit more
useful if we put it all
together on one circle.
For a unit circle, let t be an angle in standard
position whose terminal side intersects the point
(x, y) on the circle.
1
csc t  , y  0
sin t  y
y
1
cost  x
sec t  , x  0
x
y
tan t  , x  0
x
cot t  , y  0
x
y
Recall that the domain of a function is the set of
inputs. For trig functions, this is the set of
angle values that we are allowed to evaluate.
For both sine and cosine, the domain is all real
numbers. In other words, you can evaluate
the sine or cosine of an angle, positive or
negative, even angles over 360°.
Recall that the range of a function is the set of
outputs. On the unit circle, that’s the set of all
x-coordinates for cosine and the set of all ycoordinates for sine. Since the radius of the
unit circle is 1, this means that sine and cosine
will always be between -1 and 1.
1  sin t  1
1  cos t  1
Now we can use the unit circle to find trig values
of angles from 0° to 360° or 0 to 2π radians.
There’s at least a hundred of the things. Let’s
organize it all in a handsome table.
Evaluate the six trig functions at t = −2π/3
• Cosine and Secant are even functions:
cos(t )  cos t
sec(t )  sec t
• Sine, Cosecant, Tangent, and Cotangent are
odd functions:
sin(t )   sin t
csc(t )   csc t
tan(t )   tan t
cot(t )   cot t
Evaluate the six trig functions at t = 13π/6.
Of course we can measure our angles over 2π
radians. Likewise, we can evaluate trig
functions at these angles; it’s just that they
start over after 2π.
Then they repeat again after 4π.
And again after 6π.
Functions that have this cyclical behavior are
called periodic functions.
A function f is periodic if there exists a positive
real number c such that
f (t  c)  f (t )
For all t in the domain of f. The smallest number
c for which f is periodic is called the period of
f.
For sine and cosine, the period is 2π. This
means that all of the values for sine and
cosine repeat after multiples of 2π.
sin(t  2 n)  sin t
cos(t  2 n)  cos t
Objectives:
1. To construct all
aspects of the unit
circle (angles and
points)
2. To use the unit circle
to find the values of
trig functions
•
•
•
•
•
•
Assignment:
P. 299: 1-4 S
P. 299: 13-22 S
P. 299: 23-28 S
P. 299: 29-36 S
P. 300: 50, 60
Memory quiz next
class (30 Q in 30 min)