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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Aim: What do the graphs of Trig functions look like? Do Now: You and a friend are the last people seated on Ferris wheel. Once the ride begins, the wheel moves at a constant speed. It takes 36 seconds to complete one revolution. 5’ 40’ When the ride starts, how high above the ground are you? At what height are you after 9 s.? after 18 s.? 27 s.? At what height are you after 126 s.? How many revolutions have you made? Predict where you will be after 3 minutes.

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Periodic Functions What would a graph showing the relationship between your height above the ground and the time since the ride began. Use 0 ≤ t ≤ 144 for the domain, where t = 0 is the time when the ride began. 5 10 15 20 25 30 35 40 0 24 48 72 96 120 144

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Periodic Functions 5 10 15 20 25 30 35 40 A periodic function repeats a pattern of y-values (outputs) at regular intervals. A period of a function is the horizontal length of one cycle. 0 24 48 72 96 120 144 One complete pattern is called a cycle,which may begin anywhere on the graph. 1 cycle 36 s.

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Periodic Functions 5 10 15 20 25 30 35 40 0 24 48 72 96 120 144 The amplitude of a periodic function is half the difference between the minimum and maximum values of the function. max 45’ min 5’ amplitude y = 20

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. y = sin x unit circle y = sin x 1 π2π 3π/2 π/2 For what value of x does the graph y = sinx reach the maximum amplitude? sine curve or wave What is the cycle? What is the period? π/2 360º or 2π radians

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Model Problem y = sin x 1 π2π 3π/2 π/2 Determine whether sin x increases or decrease in each quadrant. sine curve or wave radians QI – increasing from 0 to 1 QII – decreasing from 1 to 0 QIII – decreasing from 0 to -1 QIV – increasing from -1 to 0 QIQIIQIIIQIV

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. y = cos x unit circle y = cos x 1 π2π 3π/2 π/2 For what value of x does the graph y = cos x reach the maximum amplitude? cosine curve What is the cycle? What is the period? 0 & 2π 360º or 2π radians

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Model Problem y = cos x 1 π2π 3π/2 π/2 Determine whether cos x increases or decrease in each quadrant. cosine curve radians QI – decreasing from 1 to 0 QII – decreasing from 0 to -1 QIII – increasing from -1 to 0 QIV – increasing from 0 to 1 QIQIIQIIIQIV

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Comparing sine and cosine curves y = sin x 1 π2π 3π/2 π/2 radians y = cos x 1 π2π 3π/2 π/2 radians Both curves have amplitudes of 1 and maximums of 1 and minimums of -1.

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Comparing sine and cosine curves y = sin x radians y = cos x 1 π2π 3π/2 π/2 radians 1 π2π 3π/2 π/2 π2π 3π/2 π/2 π 3π/2 π/2 Both curves are cyclical and have periods of 2π. period - 2π cos x = cos(x + 2πk) for any integer k sin x = sin(x + 2πk) for any integer k

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Comparing sine and cosine curves y = sin x 1 π2π 3π/2 π/2 radians y = cos x 1 π2π 3π/2 π/2 radians The cosine curve is a translation of the sine curve sin = cos(90º – )cos = sin(90º – ) Co-

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Sine Curve & Trig Values y = sin x 1 π2π 3π/2 π/2 radians x0 0 306090 120150 0 210240270300330360 y0100

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Cosine Curve & Trig Values x0 0 306090 120150 0 210240270300330360 y1011 y = cos x 1 π2π 3π/2 π/2 radians

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig.

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Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig.

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EXAMPLE 1 Graph a vertical translation Graph y = 2 sin 4x + 3. SOLUTION STEP 1 Identify the amplitude, period, horizontal shift, and vertical shift. Amplitude:

EXAMPLE 1 Graph a vertical translation Graph y = 2 sin 4x + 3. SOLUTION STEP 1 Identify the amplitude, period, horizontal shift, and vertical shift. Amplitude:

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