Download 4.2 - The Unit Circle

CH. 4 – TRIGONOMETRIC
FUNCTIONS
4.2 – The Unit Circle
FUNDAMENTAL TRIG IDENTITIES

Reciprocal Identities:
1
sec  
cos 

1
csc  
sin 
Quotient Identities:
sin 
tan  
cos 

Pythagorean Identities:
sin 2   cos 2   1
tan 2   1  sec 2 
1  cot 2   csc 2 
cos 
cot  
sin 
1
cot  
tan 
THE UNIT CIRCLE

The unit circle follows the equation x2 + y2 = 1
Radius = 1, center at the origin
 Angles always have the initial side on the positive xaxis


Consider the functions of an angle in the first
quadrant that intersects the circle at (x, y):
cos(θ) = x/1 = x
 sin(θ) = y/1 = y
 tan(θ) = y/x
(x, y)


From these 3 functions, we get…



sec(θ) = 1/x
csc(θ) = 1/y
cot(θ) = x/y
1

y
x
SPECIAL RIGHT TRIANGLES

Recall the dimensions of a 30-60-90 and a 45-45-90
right triangle:
2x
x 2
x
x
30º
x 3
x

60º
These relationships give us trigonometric values for
common angles!

 3 1
• At  
, (x, y) =  , 
6
 2 2
(x, y)

 2
2

• At  
, (x, y) =  ,

2 
4
1
 2
y
• At  

x

3
, (x, y) =  1 , 3 


2
2 




 1 3
 ,


 2
2   2 2 


,
 2
2 

3 1
2π/3
, 
3π/4
2 2
5π/6
(-1, 0)
135º
π
π/2
π/3
π/4
120º
150º
1 3
 ,

2 2  
2

  2

 2 , 2 


(0, 1)
60º 45º
90º
30º
180º
π/6
 3 1


 2 , 2


(1, 0)
0º
0
210º
7π/6
330º
270º
300º 315º
225º 240º

3 1
5π/4


,

 2
2 

4π/3

2
2


 2 , 2 

 1
3
 , 

 2

2


11π/6
7π/4
5π/3
3π/2
(0, -1)
 3 1


,

 2
2 

 2
2


,

1
2 
3   2
 ,

2

2


PROPERTIES OF TRIG FUNCTIONS

Cosine and secant are even functions
cos(-θ) = cos(θ)
 sec(-θ) = sec(θ)


Sine, cosecant, tangent, and cotangent are odd
functions
sin(-θ) = -sin(θ)
 csc(-θ) = -csc(θ)
 tan(-θ) = -tan(θ)
 cot(-θ) = -cot(θ)


Trig functions are periodic
sin(θ + 2πn) = sin(θ)
 cos(θ + 2πn) = cos(θ)
 tan(θ + πn) = tan(θ)

To memorize the unit circle, know your reference
angles!!!
 Ex: Find the 6 trigonometric functions at θ = 2π/3.

Reference angle = π/3   1 , 3 
2 2 


 Since 2π/3 is in quadrant II, the x is negative 

 1 3
 ,

 2 2 


Using this coordinate, we can find the 6 trig functions:
1
2
1
2
3
2
1

cos

sin

sec

1  2
3
2

3
2
3 cos
2
3
3
2 sin
2
1
 3
 3
2
1

tan




cot

1
1
3 cos
3

3
tan
 3
2
1
2

2
1
2 3

csc

3

3
3 sin
3
2

7
EVALUATE WITHOUT A CALCULATOR: sin
4
2.
3.
4.
5.
1

3
2
2
2
1
2
2

2
0%
1
1.
0%
0%
0%
0%
EVALUATE WITHOUT A CALCULATOR: tan
4
1
2.
3
3
3.
3
4.
2
5.
2
2
0%
1
1.

0%
0%
0%
0%
EVALUATE WITHOUT A CALCULATOR: tan
2
1
2.
0
3.
3
3
4.
3
5.
undefined
0%
1
1.

0%
0%
0%
0%
5
EVALUATE WITHOUT A CALCULATOR: sec
6
1.
2.
3.
4.
5.
3

2
2
2 3

3
3
2
 2
0%
0%
0%
0%
0%
15
EVALUATE WITHOUT A CALCULATOR: cos
4
1.
3
2
2
2
2.
3.
1

2
4.

5.
2

2
3
2
0%
0%
0%
0%
0%