Trig Functions of Real
Numbers
Characteristics of the six trig graphs
(5.3)(2)
POD
If sin θ = 4/5, and θ is in quadrant II, find
cos θ
sin (π-θ)
sin (-θ)
What can we say about the sine of any
obtuse angles?
How about the sine of opposite angles?
Review from last time
Using the unit circle and the graphs on the
handout or calculator, compare
cos (30°)
cos (-30°)
sin (π/4)
sin (-π/4)
tan (π/6)
tan (-π/6)
What might that tell us about the nature of
these functions?
Consider a reciprocal function
What do you think the graph of
y = csc θ would look like? Let’s
build it off of the sine graph.
Consider a reciprocal function
Start with the sine
graph.
Plot reciprocal yvalues for xvalues.
Where do we not get
y-values?
Consider a reciprocal function
See how the ranges of
the reciprocal functions
are related?
If we remove the sine
graph, we have this.
Where are the vertical
asymptotes?
What are the domain and
range?
Is it even, odd, neither?
Consider a reciprocal function
y = csc (x)
an odd function
asymptotes at x = ±πn
where sin(θ) = 0
D :   n
R :  ,1  1,  
Consider another reciprocal function
How would the graph
of y = sec θ
compare with this?
Consider another reciprocal function
How would the graph of
y = sec θ compare with
this?
Where are the vertical
asymptotes?
What are the domain and
range?
Even, odd, or neither?
Consider another reciprocal function
y = sec θ
an even function
vertical asymptotes at
x = π/2±πn,
where cos(θ) = 0
D: 

 n
2
R :  ,1  1,  
Consider the third reciprocal function
y = cot θ
Where are the vertical
asymptotes? Why?
What are the domain
and range?
Consider the third reciprocal function
y = cot θ
vertical asymptotes at
x = ±πn,
where sin(θ) = 0
or where tan (θ) = 0
D :   n
R:
Summary chart– do we need to do
this?
Fill in the chart below for the characteristics of the trig
functions.
Function domain range even/odd symmetric element
Summary chart—let’s do this.
Fill in the chart below for the characteristics of three
primary trig functions.
Function
period amplitude
asymptotes
The full chart for all six trig functions is on p. 401.
Formulas for negative angles
Since sine and tangent are odd functions,
sin(-x) = -sin(x)
tan(-x) = -tan(x)
csc(-x) = -csc(x)
cot(-x) = -cot(x)
In other words, change the sign of the
angle, change the sign of the trig value.
You can see this especially clearly on the
graph.
Formulas for negative angles
Since cosine is an even function
cos(x) = cos(-x)
sec(x) = sec(-x)
In other words, change the sign of
the angle, the trig value stays the
same. You can see this on the
graph.
Practice an identity
Use the negative angle formulas to
verify the identity.
sin(  x) tan(  x)  cos(  x)  sec x
Practice an identity
Use the negative angle formulas to
verify the identity.
sin(  x) tan(  x)  cos(  x)  sec x
( sin x)(  tan x)  cos x  sec x
sin x  tan x  cos x  sec x
sin 2 x cos 2 x

 sec x
cos x
cos x
1
 sec x
cos x
sec x  sec x
Finally…
… an interesting graph. On
calculators, graph f(x) = sin(x)/x on
the interval   ,   . What does the
graph do as x  0  and x  0 ?
Finally…
Although we know
there is a hole at
x = 0, it appears
that f ( x)  1 as x
approaches 0 from
either direction.
Finally…
An interesting result from this interesting
graph is that, if x is in radians and close to
0, then
sin x
1
x
which means that sin x  x
for very small angles. Test if for
x = .03, .02, .01.