MA 15400
Lesson 4
Trigonometric Values of Real Numbers
Section 6.3
I want to talk about the basis of trigonometry, the unit circle. Draw a coordinate plane
and only mark one unit in each direction.. Draw a circle that is centered at the origin and
has a radius of one. This is the unit circle. Mark you x-axis and y-axis and also write
cosine next to the x and sine next the y.
y-axis
sine θ
1
x-axis
cosine
θ
Draw a right triangle with a central angle  = 60,
mark point P on the circle. The x-value of the
coordinate is equal to the cos(60) and the
y-value of the coordinate is equal to sin(60)
y-axis
sine θ
P  1 ,
3
 2 2 


60°
1
x-axis
cosine
θ
3
1
cos(60) = and the sin(60) =
2
2
If we draw the corresponding right triangle for a 45 central angle, the corresponding
 1 1 
1
1
point is P , , therefore cos(45) =
and sin(45) =
 2 2 
2
2 y-axis
sine θ
1
P

2
,
1
2

45°
1
x-axis
cosine
Draw the corresponding right triangle for a 30 central angle and the corresponding point
 3 1 
3
1
y-axis
is P
 , 
, therefore cos(30) = 2 and the sin(30) =
2
 2 2 
sine θ
3
sin 60
3 2
sin
Since tangent =
then; tan(60) =
 2 
  3,
cos60 1
2 1
cos
2
1
1
sin( 45)
sin 30
1 2
1
3
2

1
tan(30)=
and, tan(45)=
 2  


1
cos(45)
cos30
3 2 3
3
3
2
2
P
 
3 1
,
2 2
30°
1
x-axis
cosine
MA 15400
Lesson 4
Trigonometric Values of Real Numbers
Section 6.3
From the unit circles shown with angles of 30°, 45°, and 60° in standard position, you can
see that angles less than 45° have an x-coordinate for the point on the unit circle greater
than the y-coordinate. (The cosine θ is larger than the sine θ.) Angles greater than 45°
have an x-coordinate for the point on the unit circle less than the y-coordinate. (The
cosine θ is smaller than the sine θ.) For the 45° angle, the sine and cosine values are
equal. If you keep these pictures in mind, it will help you ‘memorize’ and distinguish
between the sine and cosine values for the 30 and 60 degrees.
cos   sin  for angles between 0 and 45
cos   sin  for angles between 45 and 90
1
cos 45  sin 45 
2
What do the pictures say about the tangent values? Since tan  
sin 
, the following
cos 
0  tan   1 for angles between 0 and 45
must be true. tan   1 for angles between 45 and 90
tan   1 for 45
Find the exact value of x and y.
y
5
8
y
60
30
x
y
x
x
45
6
MA 15400
Lesson 4
Trigonometric Values of Real Numbers
Section 6.3
y
(0, 1)
We can pick points on the unit circle that are not in QI
and find the sine and cosine of the angle.
Find the exact value of the six trigonometric functions:
3
 = 90
=
 = 0
2
(1, 0)
(-1, 0)
(0, -1)
Review:
cos  =
1
, tan  < 0
2
csc  =
Use your calculator to find the following:
cos(33) and cos(– 33)
sin(18) and sin(– 18)
FORMULAS FOR NEGATIVES
Therefore:
cos(–) = cos()

sin(–) = – sin()
5
, sec  < 0
2
tan(63) and tan(–63)
tan(-) = – tan()
x
MA 15400
Lesson 4
Trigonometric Values of Real Numbers
Section 6.3
A point P(x, y) is on the unit circle. Find the value of the six trig functions.
 4 3
P , 
 5 5 
 8 15 
P , 
 17 17 
Where would the following
points be found?
P (t   ) 
P (t   ) 
y-axis
sine θ
t
P(t) = (-x,- y)
x-axis
cosine
P (t ) 
P (t   ) 
Let P(t) be a point on the unit circle. Find:
P(t + ), P(t – ), P(–t), P(–t – )
 8 15 
 , 
 17 17 
Compare P(t) with P(-t). Notice the cosines are the same and the sines are the opposites.
This reinforces what the formulas for negatives stated.
Let P be a point on the unit circle. Find the exact values of the six trig functions.

3
(This would be a 45° angle in QII.)
4
MA 15400

Lesson 4
Trigonometric Values of Real Numbers
3
2
Use the formula for negative to find the exact value.
  
cos135 (Think: 45°)
sin 
 6 
  
tan 
 2 
Verify the identity:
csc(–x)cos(–x)=–cot(x)
Section 6.3