Transcript x - Miami Beach Senior High School

1
3
 ,

2

2


Definitions of Trigonometric
functions
• Let t be a real number and let (x,y) be the
point on the unit circle corresponding to t
• Sin t = y
• Cos t = x
• Tan t = y/x
csc t = 1/y
sec t = 1/x
cot t = x/y
A circle with center at (0, 0) and radius 1 is called a unit circle.
The equation of this circle would be
x  y 1
2
2

2
(0,1)

(-1,0)
(1,0)
(0,-1)
3
2
So points on this circle must satisfy this equation.
Let's pick a point on the circle. We'll choose a point where the
x is 1/2. If the x is 1/2, what is the y value?
You can see there
2
2
x  y 1
are two y values.
2
They can be found
x
=
1/2
1
 
2
by putting 1/2 into

y
1
 
the equation for x
(0,1)  1 3 
2
,


and solving for y.
3
2
2
2


y 
4
3
y
2
(-1,0)
(1,0)
1
3
 ,

2

2

(0,-1) 
We'll look at a
larger version
of this and
make a right
triangle.
We know all of the sides of this triangle. The bottom leg is just
the x value of the point, the other leg is just the y value and
the hypotenuse is always 1 because it is a radius of the circle.
(0,1)
1
(-1,0)
1 3
 ,

2 2 


3
2

sin  
(1,0)
1
2
tan  
(0,-1)
cos
3
2  3
1
2
1
21
 1 2
3
2  3
1
2
Notice the sine is just the y value of the unit circle point and
the cosine is just the x value.
You
should
memorize
this. This
is a great
reference
because
you can
figure out
the trig
functions
of all
these
angles
quickly.
1
3
 ,

2
2 

Let’s think about the function f() = sin 
What is the domain? (remember domain means the “legal”
things you can put in for  ).
You can put in anything you want
so the domain is all real
numbers.
What is the range? (remember range means what you get out
of the function).
The range is: -1  sin   1
(0, 1)
Let’s look at the unit circle to
answer that. What is the
lowest and highest value
you’d ever get for sine?
(sine is the y value so what
is the lowest and highest y
value?)
(1, 0)
(-1, 0)
(0, -1)
Let’s think about the function f() = cos 
What is the domain? (remember domain means the “legal”
things you can put in for  ).
You can put in anything you want
so the domain is all real
numbers.
What is the range? (remember range means what you get out
of the function).
The range is: -1  cos   1
(0, 1)
Let’s look at the unit circle to
answer that. What is the
lowest and highest value
you’d ever get for cosine?
(cosine is the x value so
what is the lowest and
highest x value?)
(-1, 0)
(1, 0)
(0, -1)
Let’s think about the function f() = tan 
What is the domain? (remember domain means the “legal” things
you can put in for  ).
Tangent is y/x so we will have an
“illegal” if x is 0. x is 0 at 90° (or /2)
or any odd multiple of 90°
The domain then is all real numbers except odd multiples of
90° or  /2.
What is the range? (remember range means what you get out
of the function).
If we take any y/x, we could end up getting any value
so range is all real numbers.
Let’s think about the function f() = csc 
What is the domain? Since this is 1/sin , we’ll have trouble if
sin  = 0. That will happen at 0 and
multiples of  (or 180°). The domain
then is all real numbers except multiples
of .
Since the range is: -1  sin   1,
What is the range?
sine will be fractions less than
one. If you take their reciprocal
you will get things greater than 1.
The range then is all real
numbers greater than or equal to
1 or all real numbers less than or
equal to -1.
Let’s think about the function f() = sec 
What is the domain? Since this is 1/cos , we’ll have trouble if
cos  = 0. That will happen at odd
multiples of /2 (or 90°). The domain
then is all real numbers except odd
multiples of /2.
What is the range?
Since the range is: -1  cos   1,
cosine will be fractions less than
one. If you take their reciprocal
you will get things greater than 1.
The range then is all real numbers
greater than or equal to 1 or all real
numbers less than or equal to -1.
Let’s think about the function f() = cot 
What is the domain? Since this is cos /sin , we’ll have
trouble if sin  = 0. That will happen at
0 and multiples of  (or 180°). The
domain then is all real numbers except
multiples of .
What is the range?
Like the tangent, the range will be
all real numbers.
The domains and ranges of the trig functions are
summarized in your book in Table 6 on page 542. You need
to know these. If you know the unit circle, you can figure
these out.
Reciprocal functions have the same period.
PERIODIC PROPERTIES
sin( + 2) = sin 
cosec( + 2) = cosec 
cos( + 2) = cos 
sec( + 2) = sec 
tan( + ) = tan 
cot( + ) = cot 
9
tan
1
4
This would have the

same value as tan
(you can count around on unit circle
or subtract the period twice.)
4
Now let’s look at the unit circle to compare trig functions
of positive vs. negative angles.
What is cos
1
2

3
?
 
What is cos   ?
 3
1
2
Remember negative
angle means to go
clockwise
1
3
 ,

2

2


If a function is even, its reciprocal function will be
also. If a function is odd its reciprocal will be also.
EVEN-ODD PROPERTIES
sin(-  ) = - sin  (odd)
cosec(-  ) = - cosec  (odd)
cos(-  ) = cos  (even) sec(-  ) = sec  (even)
tan(-  ) = - tan  (odd)
cot(-  ) = - cot  (odd)
sin  60  what in terms of a positive angle?
 sin 60
 2
sec  
 3

  what in terms of a positive angle?

2
sec
3