Graphing Trigonometric Functions
The sine function
Imagine a particle on the unit circle, starting at (1,0) and rotating
counterclockwise around the origin. Every position of the particle
corresponds with an angle, θ, where y = sin θ. As the particle moves
through the four quadrants, we get four pieces of the sin graph:
sin θ
y
90°
135°
45°
II
I
I
0°
180°
II I
IV
225°
315°
270°
II
x
0
90°
180°
360° θ
270°
III
IV
θ
sin θ
0
0
π/2
1
π
0
3π/2
−1
2π
0
Sine is 2π Periodic
sin θ
−3π
−2π
−π
0
π
One period
2π
sin θ: Domain: all real numbers, (−∞, ∞)
Range: −1 to 1, inclusive [−1, 1]
sin θ is an odd function; it is symmetric about the origin.
sin(−θ) = −sin(θ)
2π
3π θ
The cosine function
Imagine a particle on the unit circle, starting at (1,0) and rotating
counterclockwise around the origin. Every position of the particle
corresponds with an angle, θ, where x = cos θ. As the particle moves
through the four quadrants, we get four pieces of the cos graph:
y
cos θ
90°
135°
45°
II
I
I
0°
180°
II I
IV
225°
315°
270°
IV
x
0
90°
270°
180°
II
III
θ
360°
θ
cos θ
0
1
π/2
0
π
−1
3π/2
0
2π
1
Cosine is a 2π Periodic
cos θ
θ
−3π
−2π
−π
0
π
One period
2π
cos θ: Domain: all real numbers, (−∞, ∞)
Range: −1 to 1, inclusive [−1, 1]
cos θ is an even function; it is symmetric about the y-axis.
cos(−θ) = cos(θ)
2π
3π
The Tangent Function
tan 
sin 
cos 
When cos θ = 0, tan θ is undefined.
This occurs every odd multiple of π/2: { … −π/2, π/2, 3π/2, 5π/2, … }
Table from θ = −π/2 to θ = π/2 .
Tanθ is π periodic.
θ
sin θ
cos θ
tan θ
θ
tan θ
−π/2
−1
0
−∞
−π/2
−∞
2
2
2
2
−1
−π/4
−1
0
0
1
0
0
0
π/4
2
2
2
2
1
π/4
1
π/2
1
0
∞
π/2
∞
−π/4

Graph of Tangent Function: Periodic
tan θ
Vertical asymptotes
where cos θ = 0
tan 
θ
tan θ
−π/2
−∞
−π/4
−1
0
0
π/4
1
π/2
∞
−3π/2
−π/2
0
π/2
One period: π
tan θ: Domain: θ ≠ π/2 + πn; i.e., odd multiple of π/2 .
Range: all real numbers (−∞, ∞)
tan θ is an odd function; it is symmetric about the origin.
tan(−θ) = −tan(θ)
3π/2
θ
sin 
cos 
The Cotangent Function
cot  
cos 
sin 
When sin θ = 0, cot θ is undefined.
This occurs every π intervals, starting at 0: { … −π, 0, π, 2π, … }
Table from θ = 0 to θ = π.
cotθ is π periodic.
cos θ
cot θ
θ
cot θ
0
1
∞
0
∞
π/4
2
2
2
2
1
π/4
1
π/2
1
0
0
π/2
0
3π/4
2
2
2
2
−1
3π/4
−1
–1
−∞
π
−∞
θ
0
π
sin θ
0

Graph of Cotangent Function: Periodic
Vertical asymptotes
where sin θ = 0
cos
cot 
sin
cot θ
θ
tan θ
0
∞
π/4
1
π/2
0
3π/4
−1
π
−∞
−3π/2
-π
−π/2
π/2
cot θ: Domain: θ ≠ πn
Range: all real numbers (−∞, ∞)
cot θ is an odd function; it is symmetric about the origin.
tan(−θ) = −tan(θ)
π
3π/2
Cosecant is the reciprocal of sine
Vertical asymptotes
where sin θ = 0
csc θ
−3π
θ
0
−2π
−π
π
2π
3π
sin θ
One period: 2π
sin θ: Domain: (−∞, ∞)
Range: [−1, 1]
sin θ and csc θ
csc θ: Domain: θ ≠ πn
are odd
(where sin θ = 0)
(symmetric about the origin)
Range: |csc θ| ≥ 1
or (−∞, −1] U [1, ∞]
Secant is the reciprocal of cosine
Vertical asymptotes
where cos θ = 0
sec θ
θ
−3π
−2π
−π
0
π
2π
3π
cos θ
One period: 2π
cos θ and sec θ
cos θ: Domain: (−∞, ∞) sec θ: Domain: θ ≠ π/2 + πn
(where cos θ = 0)
are even
Range: [−1, 1]
Range: |sec θ | ≥ 1
(symmetric about the y-axis)
or (−∞, −1] U [1, ∞]
Summary of Graph Characteristics
Function
Definition
Period
Domain
Range
Even/Odd
−1 ≤ x ≤ 1 or
[−1, 1]
odd
∆
о
sin θ
opp
hyp
y
r
2π
(−∞, ∞)
csc θ
1
.sinθ
r
.y
2π
θ ≠ πn
cos θ
adj
hyp
x
r
2π
(−∞, ∞)
sec θ
1 .
sinθ
r
y
2π
θ ≠ π2 +πn
tan θ
sinθ
cosθ
y
x
π
θ ≠ π2 +πn
All Reals or
(−∞, ∞)
odd
cot θ
cosθ
.sinθ
x
y
π
θ ≠ πn
All Reals or
(−∞, ∞)
odd
|csc θ| ≥ 1 or
(−∞, −1] U [1, ∞)
All Reals or
(−∞, ∞)
|sec θ| ≥ 1 or
(−∞, −1] U [1, ∞)
odd
even
even