November 29, 2011 At the end of today you will be able to understand where the sine and cosine curve derive from. DO NOW: Discuss Final Review Questions. 14. C18. A22. E 15. E19. D23. B 16. D20. C24. A 17. A21. A25. C HW: Finish back side of activity sheet. Start Final Review Part 2 #1-7

Unit Circle and Function Graph Worksheet 1 should be done in 10 minutes Worksheet 2 completed by the end of the period. You will be graded by accuracy and completion of the function graph and worksheet 2. Each group will present 2 questions to the class.

December 5, 2011 At the end of today, you will understand the period and amplitude of a sin and cos curve. Warm-up: Correct Final Review #1-7 1.(x – 1) 2 + (y – 3) 2 = 16 2.y = -4/5x + 17/5 3a. -1/8 3b. -1/283c ≤ x ≤ 10 5.Zeros: (-.75, 0), (0, 0), (.75, 0), min: (-.61, -.30), (.61,.30) 6.Zero: (6, 0), min (6, 0) 7.Zero: (0, 0), max (2, 8) HW 4.5: Pg #1-4, Final Rev 2 due Thurs.

The Sine Graph Looking at the unit circle, make a table of values of the Sine values at each angle: sin 0 π/2 Π 3π/2 2π2π

The Sine Graph Find the values of sin and plot those points on the coordinate axis below: sin 0 π/2 Π 3π/2 2π2π 1 π/2π3π/22π2π

The Cosine Graph Looking at the unit circle, make a table of values of the cosine values at each angle: cos 0 π/2 Π 3π/2 2π2π

The Cosine Graph Using the values from the table, plot those points on the coordinate axis below: cos 0 π/2 Π 3π/2 2π2π 1 π/2π3π/22π2π

Sketch the sine and cosine curve from -2π to 2π Always use the intercepts, maximum, and minimum points to sketch sine or cosine curve. y = sin x y = cos x θ θ

Sine curve

Cosine curve

Range of Sine and Cosine Graphs The range for the sine and cosine curve is The amplitude of the graphs is how high (or low) your graph goes from the x-axis. BUT it is not always 1 and given as: y = asin ory = acos |a| is the magnitude or the amplitude of the graph. [-1, 1]

The amplitude What is the amplitude of y = 2sinx? 1 π/2π3π/22π2π st : graph the zeroes 2 nd : graph max and min

Sketch a graph of each, then identify the intercepts, max, and min. π/2π3π/22π2ππ/2π3π/22π2π Vertical Shrink Vertical Stretch

Domain of Sine and Cosine Graphs What is the domain of sine and cosine? What is the period of the sine and cosine curves? (the angle at which the curve repeats its pattern) (-∞,∞) 2π

The Period of Sine and Cosine The period changes if you multiply the angle by some factor, b: y = sin(b)y = cos(b) Period of sine or cosine = What is the period of y = cos(2x)? Since b = 2, This means that the curve will repeat after π.

Graph y = cos(2x), include 2 full periods 1)Find the period and amplitude. 2) Divide the period into four parts (these will be our zeros, max and mins since sine and cosine always follow the same pattern!) 3) Use your answer from 2 to scale your x-axis amp = 1, per = π Horizontal Shrink 1

Graph, include 2 full periods 1 Horizontal Stretch amp = 1, per = 4π

Start HW 4.5 #1 and #37, 39 1.Find the period and amplitude: y = 3 sin 2x, then sketch 37.Sketch the graph 39. Sketch the graph

Challenge Problem A boat on a lake bobs up and down with the waves. The difference between the lowest and highest points of the boat is 10 inches. The boat is at equilibrium when it is halfway between the lowest and highest points. Each cycle of the period motion lasts 4 seconds. a)Write an equation for the motion of the boat. Let h represent the height in inches and let t represent the time in seconds. Assume that the boat is at equilibrium at t = 0 seconds. b) Draw a graph