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How do you graph sine and cosine by unwrapping the unit circle? CHAPTER 4 – LESSON 1 Warm-Up/Activator Fill in the table (separate sheet) with the radian measure of the angles and then both the exact and approximate values for sine, cosine, and tangent of these angles. 30-60-90 Angle Chart for Unit Circle 0 30 45 60 90 120 135 150 180 210 225 0 30 45 60 90 60 45 30 0 30 45 0 1 2 2 2 3 2 1 3 2 2 2 1 2 0 1 2 2 2 0 .5 .707 .866 1 .866 .707 .5 0 -.5 -.707 -.866 1 3 2 2 2 1 2 0 1 2 2 2 1 .866 .707 .5 0 -.5 -.707 -.866 1 3 0 0 .577 --1 1 1 1 3 1.7 2 3 1.7 1 1 2 -- 2 1 3 .577 2 1.15 1.414 -- 3 2 1.414 2 2 2 3 1.15 --0 3 -1.7 1 3 -1 -1 1 0 -.577 -1 -- 2 -- -2 1 1 2 3 2 3 2 -1 3 2 0 1 3 -.577 0 .577 3 -- 3 -1.7 -- 1.7 2 3 -1 2 3 2 2 2 -1 --- 2 2 -.866 -.707 1 3 -1.414 -1.15 1.15 1.414 -1 1 1 1 1 240 270 300 315 330 360 60 90 60 45 30 0 3 2 -1 3 2 2 2 -2 -.866 -.707 0 3 2 1 1 2 -.5 0 .5 .707 .866 3 -1 1 3 -- 3 1.7 -- -1.7 2 2 -1 0 -.5 0 1 0 -.577 0 3 -- -1.7 -- 1 3 0 .577 0 2 -- 2 2 2 3 1 -- 2 1.414 1.15 1 2 2 -- 2 2 -1 1 2 1 2 -1.15 -1.414 -2 2 45-45-90 2 3 -1.414 -1.15 -1 -1 1 3 1 -.577 -1 2 3 -1.15 -1.414 -2 -- Graphs of Functions Sine y 1.5 1 0.5 x 30 -0.5 -1 -1.5 60 90 120 150 180 210 240 270 300 330 360 Graphs of Functions Cosine y 1.5 1 0.5 x 30 -0.5 -1 -1.5 60 90 120 150 180 210 240 270 300 330 360 Graphs of Functions Tangent y 1.5 1 0.5 x 30 -0.5 -1 -1.5 60 90 120 150 180 210 240 270 300 330 360 Graphs of Functions Sine Cosine Graphs of Functions Tan Cotangent Graphs of Functions Secant Cosecant Chapter 4 - Lesson 2 Transforming Trig Functions Essential Question: How can we use the amplitude, period, phase shift and vertical shift to transform the sine and cosine curves? Key Question: How do the values of A, B, H, and K impact the shape of the trigonometric functions? Warm-Up/Activator Complete the Exploring Sine Graphs Activity and Report findings to the class. Alternate Activator Graph each equation without a calculator Y = 2(x -3)2 + 1 y = - (x + 2)2 - 3 Transformations: Vertical Shift: the vertical movement of the graph (“new” x-axis) Phase Shift: the horizontal movement of the graph (“new” y-axis) Period: the number of degrees or radians required to draw one complete cycle of the curve Amplitude: the distance the curve is from the “new” x-axis Transformation Equation Period Vertical Movement y A sin( B H ) K Amplitude and Inversion Combine to give Horizontal Movement Transformations Example 1 y y = 2 cos (3x) 4 3 2 1 x -π π 2π 3π -1 -2 -3 -4 amp = phase = period = vertical = 4π 5π 6π 7π Example 2 y 4 y = cos (1/3x) 3 2 1 x -π π 2π 3π -1 -2 -3 -4 amp = phase = period = vertical = 4π 5π 6π 7π Example 3 y 4 y = cos(4x) + 2 3 2 1 x -π π 2π 3π -1 -2 -3 -4 amp = phase = period = vertical = 4π 5π 6π 7π Example 4 y 4 y = -sin(4x) – 2 3 2 1 x -π π 2π 3π -1 -2 -3 -4 amp = phase = period = vertical = 4π 5π 6π 7π Example 5 y 4 y = cos(x+Π) + 1 3 2 1 x -π π 2π 3π -1 -2 -3 -4 amp = phase = period = vertical = 4π 5π 6π 7π Example 6 y 4 y = 3 sin(2x – Π) + 1 3 2 1 x -π π 2π -1 -2 -3 -4 amp = phase = period = vertical = 3π 4π 5π 6π 7π Example 7 y 4 y = ½ cos(2x) + 2 3 2 1 x -π π 2π 3π -1 -2 -3 -4 amp = phase = period = vertical = 4π 5π 6π 7π Example 8 y 4 y =2 cos(1/2x+Π) – 1 3 2 1 x -π π 2π 3π -1 -2 -3 -4 amp = phase = period = vertical = 4π 5π 6π 7π Example 9 : Degrees y 5 y =1.5 cos(1/2x+90) +2 4 3 2 1 x -540 -450 -360 -270 -180 -90 90 -1 -2 -3 -4 -5 amp = phase = period = vertical = 180 270 360 450 540 Chapter 4 - Lesson 3 Sinusoidal Regressions Essential Question: How can sinusoidal regressions be used to model periodic data? Key Question: How do you use the calculator to find sinusoidal regressions? Your Turn