Do Now:
• Graph the equation:
X2 + y2 = 1
• Draw and label the special right triangles
• What happens when the hypotenuse of each
triangle equals 1?
ALGEBRA II HONORS
PRE-CALC: 4.2: TRIG FUNCTIONS:
THE UNIT CIRCLE
Trigonometry
• The word trigonometry means measurement of triangles.
• Initially trigonometry dealt with the relationships among the
sides and angles of triangles.
y
sin( ) 
r
r
y

x
cos( ) 
x
r
y
tan( ) 
x
Trigonometry
r

y
sin( ) 
r
r
csc( ) 
y
x
cos( ) 
r
r
sec( ) 
x
y
tan( ) 
x
x
cot( ) 
y
y
x
SIX TRIGONOMETRIC FUNCTIONS
• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• Cosecant (csc)
• Secant (sec)
• Cotangent (cot)
UNIT CIRCLE: X2 + Y2 = 1
(0,1)
(-1,0)
(1,0)
(0,-1)
UNIT CIRCLE: X2 + Y2 = 1
(0,1)
r
(-1,0)

x
(0,-1)
(x,y)
y
(1,0)
UNIT CIRCLE: X2 + Y2 = 1
(0,1)
1
(-1,0)

x
(0,-1)
(x,y)
y
(1,0)
Unit Circle Trig
1

x
y
sin( ) 
1
1
csc( ) 
y
x
cos( ) 
1
1
sec( ) 
x
y
tan( ) 
x
x
cot( ) 
y
y
UNIT CIRCLE: X2 + Y2 = 1
(0,1)
1
(-1,0)

cos 
(0,-1)
(cos , sin  )
sin 
(1,0)
UNIT CIRCLE
• Let the radius = 1.
• Graph x2 + y2 = 1
• Find the (x, y)
coordinates using
special right
triangle ratios for
45-45-90.
(0,1)
(-1,0)
(1,0)
(0,-1)
UNIT CIRCLE
• Find all the (x, y)
coordinates using
special right
triangle ratios for
45-45-90.
(0,1)
(-1,0)
(1,0)
(0,-1)
UNIT CIRCLE
• Let the radius = 1.
• Find the (x, y)
coordinates using
special right
triangle ratios for
30-60-90.
(0,1)
(-1,0)
(1,0)
(0,-1)
UNIT CIRCLE
• Let the radius = 1.
• Find the (x, y)
coordinates using
special right
triangle ratios for
30-60-90.
(0,1)
(-1,0)
(1,0)
(0,-1)
UNIT CIRCLE
• Let the radius = 1.
• Find the (x, y)
coordinates using
special right
triangle ratios for
30-60-90.
(0,1)
(-1,0)
(1,0)
(0,-1)
UNIT CIRCLE
• Let the radius = 1.
• Find the (x, y)
coordinates using
special right
triangle ratios for
30-60-90.
(0,1)
(-1,0)
(1,0)
(0,-1)
UNIT CIRCLE: FOR EACH POINT ON THE UNIT
CIRCLE LABEL THE ORDERED PAIR (COS, SIN)
AND THE ANGLE IN DEGREES AND RADIANS.
Closing:
• What special triangles did we use to help us learn
the unit circle?
Homework:
• worksheet
DO NOW: FOR EACH POINT ON THE UNIT
CIRCLE LABEL THE ORDERED PAIR (COS, SIN)
AND THE ANGLE IN DEGREES AND RADIANS.
UNIT CIRCLE WORKSHEET
1.
2.
3.
4.
5.
6.
7.
8.
radians
0

6

4

3

2
2
3
3
4
5
6
degrees
cos 
sin 
tan 
sec 
csc 
cot 
UNIT CIRCLE WORKSHEET
radians
9.

10.
7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
11.
12.
13.
14.
15.
16.
17.
degrees
cos 
sin 
tan 
sec 
csc 
cot 
Practice
Practice
Practice
Trigonometry
1

x
y
sin( ) 
1
1
csc( ) 
y
x
cos( ) 
1
1
sec( ) 
x
y
tan( ) 
x
x
cot( ) 
y
y
Trigonometry:
• Given that cos = x and sin = y
• Find a new way to write tan, cot,
sec, and csc.
TRIG FUNCTIONS: UNIT CIRCLE
sin 
tan  
cos 
cos 
cot  
sin 
1
sec  
cos 
(reciprocal of cosine)
1
csc  
sin 
(reciprocal of sine)
Practice
Homework:
• Packet
• 1-28