4.2, 4.4 – The Unit Circle, Trig Functions
The unit circle is defined by the equation x2 + y2 = 1.
It has its center at the origin and radius 1.
(0 , 1)
(1 , 0)
(1 , 0)
1
(0 , 1)
4.2, 4.4 – The Unit Circle, Trig Functions
If the point (x , y) lies on the terminal side of θ, the six trig
functions of θ can be defined as follows:
A reference triangle is made
by “dropping” a perpendicular
line segment to the x-axis.
(x , y)
(− , +)
r
θ
(− , −)
y
x
(+ , −)
r 2 = x2 + y 2
y
r
tan θ 
y
x
sec θ 
r
x
x
cos θ 
r
cot θ 
x
y
csc θ 
r
y
sin θ 
4.2, 4.4 – The Unit Circle, Trig Functions
Evaluate the six trig functions of an angle θ whose
terminal side contains the point (−5 , 2).
2
2 29

29
29
5
5 29
cos θ  

29
29
2
tan θ  
5
sin θ 
(−5 , 2)
29
2
−5
cot θ  
sec θ  
csc θ 
5
2
29
5
29
2
4.2, 4.4 – The Unit Circle, Trig Functions
For a unit circle (radius 1)
sin = y
(x , y)
1
cos = x

(1 , 0)
1
y
tan =
x
4.2, 4.4 – The Unit Circle, Trig Functions
1 3
 ,

2 2 


1
π
(1 , 0)
3
1
4.2, 4.4 – The Unit Circle, Trig Functions
4.2, 4.4 – The Unit Circle, Trig Functions
Find the six trig functions of 0º
(1 , 0)
r=1
y
sin θ 
r
x
cos θ 
r
y
tan θ 
x
x
cot θ 
y
r
sec θ 
x
r
csc θ 
y
0
 0
1
1
 1
1
0
 0
1
1
  undef.
0
1
 1
1
1
  undef.
0
4.2, 4.4 – The Unit Circle, Trig Functions
Summary
Deg.
Rad.
Sin
Cos
Tan
0º
0
π
6
π
4
π
3
π
2

3π
2
2
0
1
2
1
0
3
3
2
2
30º
45º
60º
90º
180º
270º
360º
3
2
2
2
1
3
1
3
1
2
0
undef.
0
−1
0
−1
0
undef.
0
1
0
2
4.2, 4.4 – The Unit Circle, Trig Functions
Basic Trig Identities
Reciprocal
Quotient
Pythagorean
1
csc θ 
sin θ
1
sec θ 
cos θ
sin θ
tan θ 
cos θ
sin2θ + cos2θ = 1
cos θ
cot θ 
sin θ
tan2θ + 1 = sec2θ
Even
Odd
Cofunction
cos(θ) = cosθ
sin(θ) = sinθ
sinθ = cos(90  θ)
sec(θ) = secθ
tan(θ) = tanθ
cot θ 
1
tan θ
cot2θ + 1 = csc2θ
tanθ = cot(90  θ)
cot(θ) = cotθ
secθ = csc(90  θ)
csc(θ) = cscθ
4.2, 4.4 – The Unit Circle, Trig Functions
Use trig identities to evaluate the six trig functions of an
angle θ if cos θ = 4 5 and θ is a 4th quadrant angle.
sin2θ = 1 − cos2θ
sin θ  1  cos 2θ
3
sin θ  
5
 5
  1 4
  1 16
25
25  16

25
9

25
3
5
2
4
cos θ 
5
3
5 3
4
4
5
4
cot θ  
3
tan θ 
sec θ 
5
4
csc θ  
4
5
5
3
−3
4.2, 4.4 – The Unit Circle, Trig Functions
For any angle θ, the reference angle for θ, generally
written θ', is always positive, always acute, and always
made with the x-axis.
θ'
θ
4.2, 4.4 – The Unit Circle, Trig Functions
For any angle θ, the reference angle for θ, generally
written θ', is always positive, always acute, and always
made with the x-axis.
θ'
θ
4.2, 4.4 – The Unit Circle, Trig Functions
For any angle θ, the reference angle for θ, generally
written θ', is always positive, always acute, and always
made with the x-axis.
θ'
θ
4.2, 4.4 – The Unit Circle, Trig Functions
Find the reference angles for α and β below.
α = 217º
37º
β = 301º
59º
α' = 217º − 180º = 37º
β' = 360º − 301º = 59º
4.2, 4.4 – The Unit Circle, Trig Functions
The trig functions for any angle θ may differ from the
trig functions of the reference angle θ' only in sign.
θ'
θ
θ = 135º
θ' = 180º − 135º = 45º
sin 135º =  sin 45º
= 22
= 22
cos 135º = − 2 2
tan 135º = −1
4.2, 4.4 – The Unit Circle, Trig Functions
A function is periodic if
f(x + np) = f(x)
for every x in the domain of f,
every integer n,
and some positive number p (called the period).
0 , 2π
sine & cosine
period = 2π
secant & cosecant period = 2π
tangent & cotangent period = π
4.2, 4.4 – The Unit Circle, Trig Functions
π
2
tan π 3 = 3
sin π  2π = 3
3
2

tan π 3  π = 3
sin π 3  4π  =
tan π 3  2π = 3
sin 3 =

3
3
2




Find the exact value of each.
sin300
7
cot
4
cos( 240)
  
csc 

 4 