1. Warm up Use the Pythagorean identity to determine if the point (.623,.377) is on the circumference of the unit circle Using Pythagorean identity, solve for cosine of 45° if sine is

2. Graphing Sine and Cosine Objectives: 1)Identify and describe sine and cosine graphs and the terms associated with them 2)Determine the amplitude, period, and shifts of a sine/cosine graph

3. Terms Amplitude: the height of a graph Period: the length of one full cycle of a graph Vertical shift: moving a graph up or down Phase shift: moving a graph horizontally (left and right) Frequency: the number of cycles completed on a certain interval

4. Sine and Cosine graphs Both are called "wave" graphs because of their rolling wave-like appearance Both graphs have an amplitude of 1 Both graphs have a period length of

5. Sine Function: y = sin(x) Sin(x) graph starts at the origin

6. Cosine Function: y = cos(x) Cos(x) graph starts at 1 (above the origin)

7. Changes to the graphs We can change the graphs of sin(x) and cos(x) in several ways The functions can be written as Asin(B x +or- C) +or- D The A causes a change in amplitude The B causes a change in the period length The C causes a horizontal or phase shift The D causes a vertical shift

8. Change in Amplitude y = Asin(x) Does not matter if A is positive or negative

10. Examples Identify the amplitude in each graph – Y = 9cos(x) – Y = sin(x) – Y = 3 + 4sin(x) – Y = -4sin(x) + 6

11. Change in Period y = sin(Bx) results in a change in the period length To find the length of the period we ALWAYS do the following operation: /IBI For example the graph of y = sin(5x) has a period length of /5

12. Examples Find the period length of each of the following functions – Y = 4sin(2x) – Y = cos(-4x) – Y = sin(x)

13. Phase shift y = sin(x + C) results in a phase shift The phase shift is always C/B – If you subtract a number it shifts to the right – If you add a number it shifts to the left

14. Examples Identify the phase shift in each function – Y = sin(x + 4) – Y = cos(2x - 7) – Y = 4sin(x) + 3 – Y = sin(x - /3)

15. Vertical shifts of sine and cosine y = sin(x) + or – D – Adding a constant shifts the graph up – Subtracting a constant shifts the graph down

16. Examples Identify the vertical shift in each graph – Y = sin(x) – Y = sin(x) + 8 – Y = 3 + sin(x) – Y = -12 + cos(x)

17. Combining A and D Combining A and D gives us the functions: y = A sin(x) + D or y = A sin(x) - D The maximum value can be found by adding A and D The minimum value can be found by adding -A and D – Max = – Min =

18. PUT IT ALL TOGETHER Identify the Vertical Shift, Amplitude, Maximum and Minimum, Period, and Phase Shift of each function Y = 4cos(x - 2) Y = 2sin(3x + 7) + 3 Y = 2 + cos(5x - 2)

19. Cumulative Practice Identify the Vertical Shift, Amplitude, Maximum and Minimum, Period, and Phase Shift of each function Y = 4cos(x - 3) Y = 2 + sin(x + 4) Y = 1 + 7cos(4x) Y = 4sin( x - 2) - 2 Y = 4cos(x - )