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MA 154
Section 6.3
Lesson 4
Delworth
Trigonometric Functions of Real Numbers
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.
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)
Therefore the cos(60) =
3
1
and the sin(60) =
2
2
If we draw the corresponding right triangle for a 45 central angle, the corresponding point
 1 1 
1
1
is P , , therefore cos(45) =
and sin(45) =
 2 2 
2
2
Draw the corresponding right triangle for a 30 central angle and the corresponding point is
 3 1 
3
1
P
 , 
, therefore cos(30) = 2 and the sin(30) =
2
 2 2 
3
sin 60
3 2
sin
Since tangent =
then; tan(60) =
 2 
  3,
1
cos60
2 1
cos
2
1
1
sin( 45)
sin 30
1 2
1
3
tan(30)=
and, tan(45)=
 2 1
 2  


1
cos(45)
cos30
3 2 3
3
3
2
2
1
MA 154
Lesson 4
Section 6.3
Delworth
Trigonometric Functions of Real Numbers
Find the exact value of x and y.
8
y
y
y
5
x
60
30
x
45
x
6
y
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
cos  =
1
, tan  < 0
2
csc  =
x
5
, sec  < 0
2
Find the exact value of:
cos(45) and cos(– 45)
sin(45) and sin(– 45)
tan(45) and tan(–45)
Therefore:
cos(–) = cos()
sin(–) = – sin()
tan() = – tan()
2
MA 154
Section 6.3
Lesson 4
Delworth
Trigonometric Functions of Real Numbers
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 
Let P(t) be a point on the unit circle. Find:
P(t + ), P(t – ), P(–t), P(–t – )
 8 15 
 , 
17 17 
Let P be a point on the unit circle. Find the exact values of the six trig functions.

3
4

Use the formula for negative to find the exact value.
  
cos135
sin 
 6 
Verify the identity:
csc(–x)cos(–x)=–cot(x)
3
3
2
  
tan 
 2 
MA 154
Section 6.3
Lesson 4
Delworth
Trigonometric Functions of Real Numbers
y
sine
x
cosine
4