LESSON 5
Section 6.3
Trig Functions of Real Numbers
UNIT CIRCLE
Remember, the
sine of a real
number t (a number
that corresponds to
radians) is the y
value of a point on
a unit circle and
the cosine of that
real number is the
x value of the point
on a unit circle.
APPENDIX IV of your
textbook shows a
good unit circle.
Make a table of x and y values for the equation y = sin x.
x
0
π/6
π/4
π/3
π/2
2π/3
3π/4
5π/6
π
y
0
0.5
0.707
0.866
1
0.866
0.707
0.5
0
x
7π/6
5π/4
4π/3
-3π/2
5π/3
7π/4
11π/6
2π
13π/6
y
-0.5
-0.707
-0.866
-1
-0.866
-0.707
-0.5
0
0.5
This is a second revolution around the unit circle.
This is another ‘period’ of the curve.
x
2π
13π/6
9π/4
7π/3
5π/2
8π/3
11π/4
17π/6
3π
y
0
0.5
0.707
0.866
1
0.866
0.707
0.5
0
x
19π/6
13π/4
10π/3
7π/2
11π/3
15π/4
23π/6
4π
25π/6
y
-0.5
-0.707
-0.866
-1
-0.866
-0.707
-0.5
0
0.5
y = sin x
• This is a periodic function. The period is
2π.
• The domain of the function is all real
numbers.
• The range of the function is [-1, 1].
• It is a continuous function. The graph is
shown on the next slide.
Graphing the sine curve for -2π ≤ x ≤ 2π.
y  sin( x)
Domain :  ,  
Range :  1,1
Period : 2
(π/2, 1)
1
(2π, 0)
(π, 0)
0
-2π
-3π
2
-π
-π
2
(0, 0)
π
2
π
3π
2
-1
(3π/2, - 1)
2π
UNIT CIRCLE
Remember, the
sine of a real
number t (a number
that corresponds to
radians) is the y
value of a point on
a unit circle and
the cosine of that
real number is the
x value of the point
on a unit circle.
Make a table of x and y values for y = cos x
Remember, the y value in this table is actually the x value on the unit circle.
x
0
π/6
π/4
π/3
π/2
2π/3
3π/4
5π/6
π
y
1
0.866
0.707
0.5
0
-0.5
-0.707
-0.866
-1
x
7π/6
5π/4
4π/3
-3π/2
5π/3
7π/4
11π/6
2π
13π/6
y
-0.866
-0.707
-0.5
0
0.5
0.707
0.866
1
0.866
y = cos x
• This is a periodic function. The period is
2π.
• The domain of the function is all real
numbers.
• The range of the function is [-1, 1].
• It is a continuous function. The graph is
shown on the next slide.
Graphing the cosine curve for -2π ≤ x ≤ 2π.
y  co s( x)
Domain :  ,  
Range :  1,1
(0, 1)
Period : 2
(2π, 1)
1
(π/2, 0)
0
-2π
-3π
2
-π
-π
2
π
2
(3π/2, 0)
π
(π, - 1)
-1
3π
2
2π
How do the graphs of the sine
function and the cosine function
compare?
•
•
•
•
They are basically the same ‘shape’.
They have the same domain and range.
They have the same period.
If you begin at –π/2 on the cosine
curve, you have the sine curve.

y  sin x  cos( x  )
2

 
x    , sin x  ___
6

The notation above is interpreted as: ‘as x approaches the number π/6
from the right (from values of x larger than π/6), what function value is
sin x approaching?’ Since the sine curve is continuous (no breaks or
jumps), the answer will be equal to exactly the sin (π/6) or ½ .
The notation below is interpreted as: ‘as x approaches the number π/6
from the left (from values of x smaller than π/6), what function value is
sin x approaching?’ Again, since the sine curve is continuous, the answer
will be equal to exactly the sin (π/6) or ½ .

 
x    , sin x  _____
6
Answer the following.
As x   , cos x  _____


 
As x    , sin x  _____
4
Find all the values x in the interval
[0, 2) that satisfy the equation.
Use the graph to verify these values.
1
cos x 
2
1
0
-2π
-3π
2
-π
-π
2
-1
π
2
π
3π
2
2π
Find all the values x in the interval
[0, 2) that satisfy the equation.
1
cos x 
2
1
QI
0
-2π
-3π
2
-π
-π
2
-1
π
2
Q IV
π
3π
2
2π
Find all the values x in the interval
[0, 2) that satisfy the equation.
1
2
 5
x ,
3 3
cos x 
1
0
-2π
-3π
2
-π
-π
2
-1
π
2
π
3π
2
2π
Find all the values x in the interval
[0, 2) that satisfy the equation.
Use the graph to verify these values.
sin x  
1
2
1
0
-2π
-3π
2
-π
-π
2
-1
π
2
π
3π
2
2π
Find all the values x in the interval
[0, 2) that satisfy the equation.
sin x  
1
2
1
0
-2π
-3π
2
-π
-π
2
-1
π
2
π
Q III
3π
2
2π
Q IV
Find all the values x in the interval
[0, 2) that satisfy the equation.
sin x  
x
1
2
5 7
,
4 4
1
0
-2π
-3π
2
-π
-π
2
-1
π
2
π
3π
2
2π
Make a table of x and y values for y = tan x
Remember, tan x is (sinx / cosx).
x
y
x
y
-π/2 undefined 0.49π 31.821
-0.49π -31.821
π/2 undefined
-π/3
-1.732 0.51π -31.821
-π/4
-1
2π/3
-1.732
-π/6
-0.577
3π/4
-1
0
0
5π/6
-0.577
π/6
0.577
π
0
π/4
1
7π/6
0.577
π/3
1.732
5π/4
1
y = tan x
• This is a periodic function. The period
is π.
• The domain of the function is all real numbers,
except those of the form
π/2 +nπ.
• The range of the function is all real numbers.
• It is not a continuous function. The function is
undefined at -3π/2, -π/2, π/2, 3π/2, etc. There
are vertical asymptotes at these values. The
graph is shown on the next slide.
Graphing the tangent curve for -2π ≤ x ≤ 2π.
y  tan  x 
Domain :{x | x   / 2  n where n is an integer}
Range :  ,  
Period : 
-2π
-3π
2
-π
10
8
6
4
2
(-π/4, -1)0
-π
-2
2
-4
-6
-8
-10
(π/4, 1)
π
2
π
3π
2
2π

 
As x     , tan x  _____
 2

 
As x     , tan x  _____
 2
For all x values where the tangent curve is continuous, approaching
from the left or the right will equal the value of the tangent at x.
However, the two cases above are different; because there is a
vertical asymptote when x = -π/2. If approaching from the left (the
smaller side), the answer is infinity. If approaching from the right (the
larger side), the answer is negative infinity.
Find the answers.
As x   , tan x  _____

 
As x    , tan x  _____
2

 
As x    , tan x  _____
2
Find all the values x in the interval
[0, 2) that satisfy the equation.
tan x = 1
10
8
-2π
-3π
2
-π
-π
2
6
4
2
0
-2
-4
-6
-8
-10
QI
Q III
π
2
π
3π
2
2π
Find all the values x in the interval
[0, 2) that satisfy the equation.
tan x  1
x
-2π
-3π
2
 5
4
,
-π
10
8
4
-π
2
6
4
2
0
-2
-4
-6
-8
-10
π
2
π
3π
2
2π
Find all the values x in the interval
  3 
  2 , 2 
tan x 
-2π
-3π
2
that satisfy the equation.
1
3
-π
10
8
-π
2
6
4
2
0
-2
-4
-6
-8
-10
π
2
π
3π
2
2π
Find all the values x in the interval
  3 
  2 , 2 
tan x 
-2π
-3π
2
that satisfy the equation.
1
3
-π
10
8
-π
2
6
4
2
0
-2
-4
-6
-8
-10
QI
Q III
π
2
π
3π
2
2π
Find all the values x in the interval
  3 
  2 , 2 
that satisfy the equation.
1
3
 7
x ,
6 6
tan x 
-2π
-3π
2
-π
10
8
-π
2
6
4
2
0
-2
-4
-6
-8
-10
π
2
π
3π
2
2π
Find all the values x in the interval
  3 
  2 , 2 
that satisfy the equation.
tan x  1
10
8
-2π
-3π
2
-π
-π
2
6
4
2
0
-2
-4
-6
-8
-10
π
2
π
3π
2
2π
Find all the values x in the interval
  3 
  2 , 2 
that satisfy the equation.
tan x  1
10
8
Q
-2π
-3π
2
-π
-π
2
6
4
2
IV
0
-2
-4
-6
-8
-10
Q II
π
2
π
3π
2
2π
Find all the values x in the interval
  3 
  2 , 2 
that satisfy the equation.
tan x  1
x
-2π
-3π
2
 3
4
-π
,
10
8
4
-π
2
6
4
2
0
-2
-4
-6
-8
-10
π
2
π
3π
2
2π
Sketch the graph of y = sin x + 1
This will be a graph of the basic sine function,
but shifted one unit up.
The domain will be all real numbers. What
would be the range?
Since the range of a basic sine function is [-1, 1], the
domain of the function above would be [0, 2].
Sketch the graph of y = sin x + 1
3
1
-2π
-3π
2
-π
π
2
-π
2
-1
π
3π
2
2π
Sketch the graph of y = cos x - 2
This would be the graph of a basic cosine
function shifted 2 units down.
The domain is still all real numbers. What is
the range?
The basic cosine function has a range of [-1, 1]. The range of the
function above would be [-3, -1].
Sketch the graph of y = cos x - 2
1
-2π
-3π
2
-π
0
-π
2
-1
-2
-3
-4
π
2
π
3π
2
2π
Find the intervals from –2π to 2π where the graph of
y = tan x is:
a) Increasing
b) Decreasing
Remember: No brackets should be used on values of
x where the function is not defined.
a) Increasing: [-2π, -3π/2)
(-3π/2, -π/2)
(-π/2, π/2)
(π/2, 3π/2)
(3π/2, 2π]
b) The function never decreases.