Warm-Up

1. Warm-Up Draw the Unit Circle on a sheet of paper and label ALL the parts. DO NOT LOOK AT YOUR NOTES 

2. So far, in our study of trigonometric functions we have: defined all of them learned how to evaluate them used them on the unit circleSo logically, the next step would be to study the graphs of the functions.

3. Unit 2: Graphs and Inverses of Trigonometric Functions LG 2-1 Graphing Trig Functions (Quiz 9/6 LG 2-2 Circular Functions (Quiz 9/10) LG 2-3 Evaluating Inverse Trig Functions (Quiz 9/12) LG 2-4 Graphing Inverse Trig Functions (Quiz 9/14) TEST 9/17

4. What’s Your Temperature? Scientists are continually monitoring the average temperatures across the globe to determine if Earth is experiencing Climate Change (Global Warming!). One statistic scientists use to describe the climate of an area is average temperature. The average temperature of a region is the mean of its average high and low temperatures.

5. What’s Your Temperature? A function that repeats itself in regular intervals, or periods, is called periodic. a. If you were to continue the temperature graph, what would you consider its interval, or period, to be? b. Choose either the high or low average temperatures and sketch the graph for three intervals, or periods.

7. Periodic Functions • Aperiodic functionis a function whose values repeat at regular intervals. • Sine and Cosine are examples of periodic functions • The part of the graph from any point to the point where the graph starts repeating itself is called a cycle. • The period is the difference between the horizontal coordinates corresponding to one cycle. • Sine and Cosine functions complete a cycle every 360°. So the period of these functions is 360°.

8. Periodicity is Common in Nature • Day/night cycle (rotation of earth) • Ocean Tides • Pendulums and other swinging movements • Ocean waves • Birth/marriage/death cycle • Menstrual cycle • Eating and sleeping cycle • Musical rhythm • Linguistic rhythm • Dribbling, juggling • Calendars • Fashion cycles, for example, skirt lengths or necktie widths • Economic and political cycles, for example, boom and bust economic periods, right-wing and left-wing political tendencies

9. LG 2-1 Graphing Trigonometric Functions Whenever you have to draw a graph of an unfamiliar function, you must do it by point-wise plotting, or calculate and plot enough points to detect a pattern. Then you connect the points with a smooth curve or line. Objective: Discover by point-wise plotting what the graphs of the six trig functions look like. Homework: Finish the characteristics table and do the discovery task

10. Graphing Calculator Introduction • Put your calculator in degree mode. • Graph y = sin Θ. Trace along the graph. What do you observe? • Repeat for y = cosΘ but this time put your calculator on radian mode.

11. Graphing Sine and Cosine Functions • The graph of the sine and cosine functions are made by evaluating each function at the special angles on the unit circle. • The input of the function is the angle (in degrees OR radians) measure on the unit circle. • The output is the value of the function for that angle. • We can “unwrap” these values from the unit circle and put them on the coordinate plane.

12. Exploration: Parent Sinusoids Sinusoid – a graph of a sine or cosine function “sinus” coming from the same origin as “sine,” and “– oid” being a suffix meaning “like.”

13. The Graph of Sine

14. The Graph of Cosine

15. Important Words • Sinusoidal axis - the horizontal line halfway between the local maximum and local minimum: y = 0 for parent function • Convex – bulging side of the wave • Concave – hollowed out side of the wave • Concave up • Concave down • Point of inflection - point on a curve at which the sign of the curvature (the concavity) changes.

16. The Graph of Tangent

17. Co-Trig Functions (Reciprocal Functions) • Each of the co-functions relate to the original graph. • Plot the “important points” for the sine function on the cosecant graph and then sketch the sine curve LIGHTLY in pencil (borrow one if you need to!)

18. The Graph of Cosecant

19. The Graph of Secant

20. The Graph of Cotangent

21. Discontinuous Functions • The graphs of tan, cot, sec, and csc functions are discontinuous where the function value would involve division by zero. What happens to the graph when a function is discontinuous?

22. Trig Function Characteristics • As always, we need to talk about domain, range, max and min, etc. • You will fill out the table for each of the 6 trig functions (and finish for HW).

23. Period • The period of a trig function is how long it takes to complete one cycle. • What is the period of sine and cosine? • What about cosecant and secant? • Tangent and cotangent? • The period of the functions tangent and cotangent is only 180° instead of 360°, like the four trigonometric functions.

24. Domain and Range • When we think about the domain and range, we have to make sure we are considering the entire function and not just the part on the unit circle.

25. Maximum and Minimum • Local max & min • Absolute max & min

26. Intercepts • x-intercepts • y-intercepts

27. Points of Inflection • point on a curve at which the sign of the curvature (the concavity) changes. • Will all graphs have points of inflection?

28. Intervals of Increase and Decrease • Positive slope? • Negative slope?

29. Classwork… • Fill out the characteristics table for each of the 6 trig functions

31. What’s Your Temperature? Sine and Cosine functions can be used to model average temperatures for cities. Based on what you learned about these graphs, why do you think these functions are more appropriate than a cubic function? Or an exponential function?