Graphing the Trigonometric Functions : Graphing the Trigonometric Functions All 6 functions
The sine function : 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: 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 θ sin θ 0 0 π /2 1 π 0 3 π /2 −1 2 π 0
Sine is a periodic function: p = 2π : Sine is a periodic function: p = 2 π One period 2 π 0 3 π 2 π π − 2 π −π − 3 π sin θ θ sin θ : Domain (angle measures): all real numbers, (−∞, ∞) Range (ratio of sides): −1 to 1, inclusive [−1, 1] sin θ is an odd function; it is symmetric wrt the origin. sin(− θ ) = −sin( θ )
The cosine function : 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: 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 θ cos θ 0 1 π /2 0 π −1 3 π /2 0 2 π 1
Cosine is a periodic function: p = 2π : Cosine is a periodic function: p = 2 π One period 2 π π 3 π −2 π 2 π −π −3 π 0 θ cos θ cos θ : Domain (angle measures): all real numbers, (−∞, ∞) Range (ratio of sides): −1 to 1, inclusive [−1, 1] cos θ is an even function; it is symmetric wrt the y-axis. cos(− θ ) = cos( θ )
Tangent Function : 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 ∞ 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, 3 π /2, 5 π /2, … } Let’s create an x/y table from θ = − π /2 to θ = π /2 (one π interval), with 5 input angle values. 1 1 1 −1 −1 0 0 0 ∞ −∞ 0
Graph of Tangent Function: Periodic : θ tan θ − π /2 − ∞ − π /4 −1 0 0 π /4 1 π /2 ∞ tan θ is an odd function; it is symmetric wrt the origin. tan(− θ ) = −tan( θ ) 0 θ tan θ −π /2 π /2 One period: π tan θ : Domain (angle measures): θ ≠ π /2 + π n Range (ratio of sides): all real numbers (−∞, ∞) 3 π /2 − 3 π /2 Vertical asymptotes where cos θ = 0 Graph of Tangent Function: Periodic
Cotangent Function : Cotangent Function θ sin θ cos θ cot θ 0 π /4 π /2 3 π /4 π θ cot θ 0 ∞ π /4 1 π /2 0 3 π /4 −1 π − ∞ 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 , 2 π , … } Let’s create an x/y table from θ = 0 to θ = π (one π interval), with 5 input angle values. 0 − 1 0 0 1 1 1 0 − ∞ ∞ –1
Graph of Cotangent Function: Periodic : θ tan θ 0 ∞ π /4 1 π /2 0 3 π /4 −1 π −∞ cot θ is an odd function; it is symmetric wrt the origin. tan(− θ ) = −tan( θ ) cot θ : Domain (angle measures): θ ≠ π n Range (ratio of sides): all real numbers (−∞, ∞) 3 π /2 − 3 π /2 Vertical asymptotes where sin θ = 0 Graph of Cotangent Function: Periodic π - π −π /2 π /2 cot θ
Cosecant is the reciprocal of sine : Cosecant is the reciprocal of sine sin θ : Domain: (−∞, ∞) Range: [−1, 1] csc θ : Domain: θ ≠ π n (where sin θ = 0) Range: |csc θ | ≥ 1 or (−∞, −1] U [1, ∞] sin θ and csc θ are odd (symm wrt origin) One period: 2 π π 2 π 3 π 0 −π −2 π −3 π Vertical asymptotes where sin θ = 0 θ csc θ sin θ
Secant is the reciprocal of cosine : Secant is the reciprocal of cosine cos θ : Domain: (−∞, ∞) Range: [−1, 1] One period: 2 π π 3 π −2 π 2 π −π −3 π 0 θ sec θ cos θ Vertical asymptotes where cos θ = 0 sec θ : Domain: θ ≠ π /2 + π n (where cos θ = 0) Range: | sec θ | ≥ 1 or (−∞, −1] U [1, ∞] cos θ and sec θ are even (symm wrt y-axis)
Summary of Graph Characteristics : Summary of Graph Characteristics Def’n ∆ о Period Domain Range Even/Odd sin θ csc θ cos θ sec θ tan θ cot θ
Summary of Graph Characteristics : Summary of Graph Characteristics Def’n ∆ о Period Domain Range Even/Odd sin θ opp hyp y r 2 π (−∞, ∞) −1 ≤ x ≤ 1 or [−1, 1] odd csc θ 1 . sin θ r . y 2 π θ ≠ π n | csc θ | ≥ 1 or (−∞, −1] U [1, ∞) odd cos θ adj hyp x r 2 π (−∞, ∞) All Reals or (−∞, ∞) even sec θ 1 . sin θ r y 2 π θ ≠ π 2 + π n | sec θ | ≥ 1 or (−∞, −1] U [1, ∞) even tan θ sin θ cos θ y x π θ ≠ π 2 + π n All Reals or (−∞, ∞) odd cot θ cos θ . sin θ x y π θ ≠ π n All Reals or (−∞, ∞) odd
Graphing Trig Functions on the TI89 : 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] Enter the function. Use ZoomTrig. This is the graph.
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