Graphing Trigonometric Functions. The sine function 45° 90° 135° 180° 270° 225° 0° 315° 90°...
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Transcript of Graphing Trigonometric Functions. The sine function 45° 90° 135° 180° 270° 225° 0° 315° 90°...
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