1. 6.3 – GRAPHING SINE AND COSINE FUNCTIONS

2. Periodic Function and Period  A function is periodic if, for some real number α, f(x + α) = f(x) for each x in the domain of f.  The least positive value of α for which f(x) = f(x + α) is the period of the function.

3. Which function is periodic?

4. Determine the period of the function

5. Graph of Sine Look at the value for sin(x) for the domain value between -2 π and 2 π in multiples of π /4 xsin(x) -2 π 0 -7 π /4 √2/2 -3 π /2 1 -5 π /4 √2/2 -π-π 0 -3 π /4 -√2/2 - π /2 - π /4 -√2/2 xsin(x) 00 π /4 √2/2 π /2 1 3 π /4 √2/2 π 0 5 π /4 -√2/2 3 π /2 7 π /4 -√2/2 2π2π 0

6. Graph of Sine

7. Properties of the Graph of Sine  What is the domain?  What is the range?  What is the period?  Where are all the x-intercepts?  Where is the y-intercept?  Where are the maximums?  Where are the minimums? All real numbers Set of real numbers between -1 and 1 At y = 1 and occur when x = π /2 + 2 π n at (0, 0) at πn, where n is any integer 2π2π At y = -1 and occur when x = 3 π /2 + 2 π n

8. Find each value by referring to the graph of sine 1. sin(5 π /2) 2. sin(3 π )

9. Find the values of θ for which sin θ = -1 is true.

10. Graph y = sinx for the interval of 2 π to 4 π

11. Graph of Cosine Look at the value for cos(x) for the domain value between -2 π and 2 π in multiples of π /4 xcos(x) -2 π 1 -7 π /4 √2/2 -3 π /2 0 -5 π /4 -√2/2 -π-π -3 π /4 -√2/2 - π /2 0 - π /4 √2/2 xcos(x) 01 π /4 √2/2 π /2 0 3 π /4 -√2/2 π 5 π /4 -√2/2 3 π /2 0 7 π /4 √2/2 2π2π 1

12. Graph of Cosine

13. Properties of the Graph of Cosine  What is the domain?  What is the range?  What is the period?  Where are all the x-intercepts?  Where is the y-intercept?  Where are the maximums?  Where are the minimums? All real numbers Set of real numbers between -1 and 1 At y = 1 and occur when x = π n, n is an even integer at (0, 1) at π/2 + πn, where n is any integer 2π2π At y = -1 and occur when x = π n, n is an odd integer

14. Graphing in your calculator  Make sure you are in radians!  Type equation into the y= ‘s menu  Go to Zoom Trig before you graph  Then your graph will appear accurately