Chapter 7: Trigonometric Functions

Chapter 7: Trigonometric Functions L7.4 & 5: Graphing the Trigonometric Functions (Part 2)

90° 135° 45° 0° 180° 180° 360° 0 90° 270° 225° 315° sin θ 270° I II θ III IV 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: I. From 0° to 90° the y-coordinate increases from 0 to 1 II. From 90° to 180° the y-coordinate decreases from 1 to 0 III. From 180° to 270° the y-coordinate decreases from 0 to −1 IV. From 270° to 360° the y-coordinate increases from −1 to 0 Interactive Sine Unwrap

sin θ 3π −2π −π θ π 0 −3π 2π One period 2π Sine is a periodic function: p = 2π sin θ: Domain (angle measures): all real numbers, (−∞, ∞) Range (ratio of sides): −1 to 1, inclusive [−1, 1] sin θis an odd function; it is symmetric wrt the origin. sin(−θ) = −sin(θ)

90° 135° 45° 0° 180° 225° 315° 270° cos θ I IV θ 0 90° 270° 360° 180° II III 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: I. From 0° to 90° the x-coordinate decreases from 1 to 0 II. From 90° to 180° the x-coordinate decreases from 0 to −1 III. From 180° to 270° the x-coordinate increases from −1 to 0 IV. From 270° to 360° the x-coordinate increases from 0 to 1

cos θ θ π 0 −2π 2π −3π −π 3π One period 2π Cosine is a periodic function: p = 2π cos θ: Domain (angle measures): all real numbers, (−∞, ∞) Range (ratio of sides): −1 to 1, inclusive [−1, 1] cos θis an even function; it is symmetric wrt the y-axis. cos(−θ) = cos(θ)

Tangent Function Recall that . Since cos θ is in the denominator, when cos θ = 0, tan θ is undefined. This occurs @ π intervals, offset by π/2: { … −π/2, π/2, 3π/2, 5π/2, … } Let’s create an x/y table from θ = −π/2 to θ = π/2 (one π interval), with 5 input angle values. −∞ 0 −1 −1 1 0 0 1 0 1 ∞

tan θ θ −π/2 π/2 0 One period: π Graph of Tangent Function: Periodic Vertical asymptotes where cos θ = 0 −3π/2 3π/2 tan θ: Domain (angle measures): θ≠π/2 + πn Range (ratio of sides): all real numbers (−∞, ∞) tan θis an odd function; it is symmetric wrt the origin. tan(−θ) = −tan(θ)

Cotangent Function Recall that . Since sin θ is in the denominator, when sin θ = 0, cot θ is undefined. This occurs @ π intervals, starting at 0: { … −π, 0, π, 2π, … } Let’s create an x/y table from θ = 0 to θ = π (one π interval), with 5 input angle values. ∞ 1 0 1 0 1 0 −1 −∞ –1 0

Graph of Cotangent Function: Periodic Vertical asymptotes where sin θ = 0 cot θ −3π/2 -π −π/2 π/2 π 3π/2 cot θ: Domain (angle measures): θ≠πn Range (ratio of sides): all real numbers (−∞, ∞) cot θis an odd function; it is symmetric wrt the origin. tan(−θ) = −tan(θ)

Cosecant is the reciprocal of sine Vertical asymptotes where sin θ = 0 csc θ θ 0 −3π −π π −2π 2π 3π sin θ One period: 2π sin θ and csc θ are odd(symm wrt origin) csc θ: Domain: θ ≠ πn (where sin θ = 0) Range: |csc θ| ≥ 1 or (−∞, −1] U [1, ∞] sin θ: Domain: (−∞, ∞) Range: [−1, 1]

Vertical asymptotes where cos θ = 0 sec θ θ 0 2π 3π −3π π −2π −π cos θ One period: 2π Secant is the reciprocal of cosine 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(symm wrt y-axis) cos θ: Domain: (−∞, ∞) Range: [−1, 1]

Summary of Graph Characteristics

Graphing Trig Functions on the TI89 • Mode critical – radian vs. degree • Graphing: ZoomTrig sets x-coordinates as multiple of π/2 • Graph the following in radian mode: sin(x), cos x [use trace to observe x/y values] • Switch to degree mode and re-graph the above • What do you think would happen if you graphed –cos(x), or 3cos(x) + 2? [We’ll study these transformations in the next chapter]

Chapter 7: Trigonometric Functions