Graphing Trigonometric Functions

The sine function

y

x

I I

I I I IV

I

45°

90°

135°

180°

270°

225°

0°

315°

90° 180° 270°0 360°

I II

III IV

sin θ

θ

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:

θ sin θ

0 0

π/2 1

π 0

3π/2 −1

2π 0

Sine is 2π Periodic

One period

2π

0 3π2ππ−2π −π−3π

sin θ

θ

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)

The cosine function

y

x

I I

I I I IV

I

45°

90°

135°

180°

270°

225°

0°

315°

90° 180° 270°0 360°

IV

cos θ

θ

III

I

II

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:

θ cos θ

0 1

π/2 0

π −1

3π/2 0

2π 1

Cosine is a 2π Periodic

One period2π

π 3π−2π 2π−π−3π 0θ

cos θ

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)

The Tangent Function

θ sin θ cos θ tan θ

−π/2

−π/4

0

π/4

π/2

θ tan θ

−π/2 −∞

−π/4 −1

0 0

π/4 1

π/2 ∞

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.

cossin

tan

22

22

22

22

1

1

1

−1

−1

0

0 0

∞

−∞

0

θ tan θ

−π/2 −∞

−π/4 −1

0 0

π/4 1

π/2 ∞

tan θ is an odd function; it is symmetric about the origin. tan(−θ) = −tan(θ)

0 θ

tan θ

−π/2 π/2

One period: π

tan θ: Domain: θ ≠ π/2 + πn; i.e., odd multiple of π/2 . Range: all real numbers (−∞, ∞)

3π/2−3π/2

Vertical asymptotes where cos θ = 0

Graph of Tangent Function: Periodic

cossin

tan

The Cotangent Function

θ sin θ cos θ cot θ

0

π/4

π/2

3π/4

π

θ cot θ

0 ∞

π/4 1

π/2 0

3π/4 −1

π −∞

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.

sin

coscot

22

2

2

22

2

2

0

−1

0

0

1

1

1 0

−∞

∞

–1

θ tan θ

0 ∞

π/4 1

π/2 0

3π/4 −1

π −∞

cot θ is an odd function; it is symmetric about the origin. tan(−θ) = −tan(θ)

cot θ: Domain: θ ≠ πn Range: all real numbers (−∞, ∞)

3π/2−3π/2

Vertical asymptotes where sin θ = 0

Graph of Cotangent Function: Periodic

sin

coscot

π-π −π/2 π/2

cot θ

Cosecant is the reciprocal of sine

sin θ: Domain: (−∞, ∞) Range: [−1, 1]

csc θ: Domain: θ ≠ πn (where sin θ = 0) Range: |csc θ| ≥ 1 or (−∞, −1] U [1, ∞]

sin θ and csc θare odd

(symmetric about the origin)

One period: 2π

π 2π 3π0

−π−2π−3π

Vertical asymptotes where sin θ = 0

θ

csc θ

sin θ

Secant is the reciprocal of cosine

cos θ: Domain: (−∞, ∞) Range: [−1, 1]

One period: 2π

π 3π−2π 2π−π−3π 0θ

sec θ

cos θ

Vertical asymptotes where cos θ = 0

sec θ: Domain: θ ≠ π/2 + πn (where cos θ = 0) Range: |sec θ | ≥ 1 or (−∞, −1] U [1, ∞]

cos θ and sec θare even

(symmetric about the y-axis)

Summary of Graph Characteristics

FunctionDefinition

∆ о Period Domain Range Even/Odd

sin θ opp hyp

y r 2π (−∞, ∞)

−1 ≤ x ≤ 1 or [−1, 1]

odd

csc θ 1 .sinθ

r .y 2π θ ≠ πn

|csc θ| ≥ 1 or(−∞, −1] U [1, ∞)

odd

cos θadj hyp

x r 2π (−∞, ∞)

All Reals or (−∞, ∞)

even

sec θ 1 . sinθ

r y 2π θ ≠ π2 +πn

|sec θ| ≥ 1 or(−∞, −1] U [1, ∞)

even

tan θsinθ cosθ

y x π θ ≠ π2 +πn

All Reals or (−∞, ∞)

odd

cot θ

cosθ .sinθ

x y π θ ≠ πn

All Reals or (−∞, ∞)

odd