A2M2L6­SB Tangent.notebook
December 09, 2015
The Tangent Function
Let
be the point where the terminal ray intersects the
unit circle after rotation by
degrees, as shown in the diagram below. a. Using triangle trigonometry, what are the values of
and
in terms of
?
b. Using triangle trigonometry, what is the value of
in terms of
c. What is the value of
and
?
in terms of
?
A2M2L6­SB Tangent.notebook
December 09, 2015
Let be any real number. In the Cartesian plane, rotate the
nonnegative ‐axis by degrees about the origin. Intersect the
.
resulting terminal ray with the unit circle to get a point
If
, then the value of
is
.
In terms of the sine and cosine functions,
for
.
Why do we specify cos( ) cannot = 0?
Which segment in the unit circle has measure equal to sin ( )? cos ( )?
When will tan ( ) be undefined? Describe all numbers of for which cos ( ) = 0.
90 + 180k, for any integer k
A2M2L6­SB Tangent.notebook
December 09, 2015
Describe the domain of the tangent function?
We know the domain of the tangent function is all real numbers with cos( ) ≠ 0
The domain of the tangent function is all real numbers
such that
, for all integers
.
Show that
.
Using a segment in the figure, make a
conjecture why mathematicians
named the function
the tangent function. A2M2L6­SB Tangent.notebook
December 09, 2015
Suppose that point is the point on the unit circle obtained by
rotating the initial ray through
)
. Find
Rotate the initial ray
about the origin the
stated number of
degrees. Draw a
sketch and label the
coordinates of point
where the terminal
ray intersects the unit
circle. What is the
slope of the line
containing this ray? b) 45
c) 60 A2M2L6­SB Tangent.notebook
Rotate the initial ray
about the origin the
stated number of
degrees. Draw a
sketch and label the
coordinates of point
where the terminal
ray intersects the unit
circle. What is the
slope of the line
containing this ray? a) 300 b) 315 c) 330 December 09, 2015
A2M2L6­SB Tangent.notebook
December 09, 2015
Summary tan 0 = 0
§ A working definition of the
tangent function is
tan 30 = , where
.
tan 60 =
tan 90 = undefined §The domain of the tangent function is all real numbers
such that
, for all integers
§ The range of the tangent function is all real numbers.