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)
(1 , 0)
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:
(x , y)
y θ x
x
yθtan
y
xθcot
x
rθsec
y
rθ csc
r
yθsin
r
xθ cos
A reference triangle is made by “dropping” a perpendicular
line segment to the x-axis.
r2 = x2 + y2
r(− , +)
(− , −) (+ , −)
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).
(−5 , 2)
2
−5 5
2θtan
2
5θcot
5
29θsec
2
29θ csc
29
292
29
2θsin
29
295
29
5θ cos
29
4.2, 4.4 – The Unit Circle, Trig Functions
For a unit circle (radius 1)
1 (1 , 0)
1
(x , y)
sin = y
cos = x
tan = x
y
4.2, 4.4 – The Unit Circle, Trig Functions
1
(1 , 0) 1
3π
2
3 ,
2
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)x
yθtan
y
xθcot
x
rθsec
y
rθ csc
r
yθsin
r
xθ cos
r = 1
undef.0
1
01
0
11
1
undef.0
1
01
0
11
1
4.2, 4.4 – The Unit Circle, Trig Functions
Deg. Rad. Sin Cos Tan
0º 0 0 1 0
30º
45º 1
60º
90º 1 0 undef.
180º 0 −1 0
270º −1 0 undef.
360º 2 0 1 0
Summary
21
22
23
21
22
23
33
33π
6π
4π
2π
23π
4.2, 4.4 – The Unit Circle, Trig Functions
Basic Trig Identities
θtan
1θcot
θ cos
1θsec
θsin
1θ csc
Reciprocal Quotient Pythagoreansin2θ + cos2θ = 1
tan2θ + 1 = sec2θ
cot2θ + 1 = csc2θθsin
θ cosθcot
θ cos
θsin θtan
Cofunctionsinθ = cos(90
θ)
tanθ = cot(90 θ)
secθ = csc(90 θ)
Evencos(θ) = cos θ
sec(θ) = sec θ
Oddsin(θ) = sin θ
tan(θ) = tan θ
cot(θ) = cot θ
csc(θ) = csc θ
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 = and θ is a 4th quadrant angle.
sin2θ = 1 − cos2θ5
4
θcos1θsin 2
2541
25161
25
1625
25
9
53
4
3
54
53
θtan
3
4θcot
4
5θsec
3
5θ csc
5
3θsin
5
4θ cos
4
5 −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 β = 301º
α' = 217º − 180º = 37º
β' = 360º − 301º = 59º
37º 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 = 135º
θ' = 180º − 135º = 45º
sin 135º = sin 45º
=
=
cos 135º = −
tan 135º = −1
22
22
22
θ θ'
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
sin =
sin =
sin =
3π
23
3π 3tan =
tan =
tan =
π23π 2
3
π43π 2
3
π3π
π23π
3
3
Find the exact value of each.
7
sin300 cot4
cos( 240 ) csc4