1. Graphing Trigonometric Functions

2. The sine function 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: θsin θ 00 π/21 π0 3π/2−1 2π2π0

3. Sine is 2π Periodic One period 2π2π 0 3π3π 2π2π π −2π−2π−π −3π−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)

4. The cosine function 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: θcos θ 01 π/20 π−1 3π/20 2π2π1

5. Cosine is a 2π Periodic One period 2π2π π 3π3π −2π2π2π −π −3π 0 θ 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)

6. The Tangent Function θsin θcos θtan θ −π/2 −π/4 0 π/4 π/2 θtan θ −π/2 −∞−∞ −π/4−1 00 π/41 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. 1 1 1 −1 0 00 ∞ −∞ 0

7. θtan θ −π/2 −∞−∞ −π/4−1 00 π/41 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

8. The Cotangent Function θsin θcos θcot θ 0 π/4 π/2 3π/4 π θcot θ 0 ∞ π/41 π/20 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. 0 −1−1 0 0 1 1 10 −∞−∞ ∞ –1

9. θtan θ 0 ∞ π/41 π/20 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 π -π-π −π/2 π/2 cot θ

10. 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 θ are odd (symmetric about the origin) One period: 2π π 2π2π 3π3π 0 −π −2π −3π Vertical asymptotes where sin θ = 0 θ csc θ sin θ

11. Secant is the reciprocal of cosine cos θ: Domain: (−∞, ∞) Range: [−1, 1] One period: 2π π 3π3π −2π 2π2π −π −3π 0 θ 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 θ are even (symmetric about the y-axis)

12. Summary of Graph Characteristics Function Definition ∆ о PeriodDomainRangeEven/Odd sin θ opp hyp y r 2π2π(−∞, ∞) −1 ≤ x ≤ 1 or [−1, 1] odd csc θ 1.sinθ r.y 2π2πθ ≠ πn |csc θ| ≥ 1 or (−∞, −1] U [1, ∞) odd cos θ adj hyp xrxr 2π2π(−∞, ∞) All Reals or (−∞, ∞) even sec θ 1. sinθ r y 2π2πθ ≠ π2 +πn |sec θ| ≥ 1 or (−∞, −1] U [1, ∞) even tan θ sinθ cosθ yxyx πθ ≠ π2 +πn All Reals or (−∞, ∞) odd cot θ cosθ.sinθ x y πθ ≠ πn All Reals or (−∞, ∞) odd