Algebra II- Trigonometric Functions

1. Integrating Technology Into Instruction Algebra II- Trigonometric Functions

2. Principles and Standards • In the NCTM’s call for change, the Standards recognized the impact that technology tools have played on the way mathematics is taught, by freeing students from time-consuming, mundane tasks and giving them the means to see and explore interesting relationships.

3. TEAM-Math Curriculum • G5. Verify simple trigonometric identities using Pythagorean and/or reciprocal identities. • G6. Solve general triangles, mathematical problems, and real-world applications using the Law of Sins and the Law of Cosines. • G7. Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions.

4. Continued… • A1. a. Identify and graphically represent: y=sin x,y=cos x, y=tan x Constructing graphs by analyzing their functions as sums, differences, or products • b. Translate, rotate, dilate, and reflect trigonometric functions. • d. Determine period and amplitude of sine, cosine, and tangent functions from graphs or basic equations. • e. Solve equations and inequalities including: Trigonometric

5. AlabamaCourseofStudy • 7. Solve equations, inequalities, and applied problems involving absolute values, radicals, and quadratics over the complex numbers, as well as simple trigonometric, exponential, and logarithmic functions. • 9. Graph trigonometric functions of the form y=a sin(bx), y=a cos(bx), and y=a tan(bx). • a. Determining period and amplitude of sine, cosine, and tangent functions from graphs or basic equations • b. Determining specific unit circle coordinates associated with special angles

6. Continued… • 10. Solve general triangles, mathematical problems, and real-world applications using the Law of Sines and the Law of Cosines. a. Deriving formulas for Law of Sines and Law of Cosines b. Determining area of oblique triangles • 11. Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions. • 12. Verify simple trigonometric identities using Pythagorean and/or reciprocal identities.

7. Technology • “Students should learn to recognize how the values of parameters shape the graphs of functions in a class” (NCTM).

8. Types of Technology • Graphing Calculator • TI-Nspire: Interactive Classroom • Geometer’s Sketchpad • Smart Tools: *Smart Board *Smart Slate • www.wolframalpha.com *Computational knowledge engine

9. Amplitude • Definition:The graph of y= a sin(x) or y= a cos(x), with a ≠ 0, will have the same shape as the graph of y= sin(x) or y= cos(x), respectively, except with range [-|a|, |a|]. The amplitude is |a|. • The amplitude changes the “height” of the graph.

10. Period • Definition:For b>0, the graph of y=sin(bx) will resemble that y=sin(x), but with period 2π/b. Also, the graph of y= cos(bx) will resemble that of y=cos(x), but with period 2π/b. • The period is the “length” of the graph. When b changes it will affect the period.

11. Graphing Calculator • We are going to use the graphing calculator to graph sin(x), cos(x), and tan(x) functions. We will notice how changing certain parameters affects the period and amplitude of the graphs. The graphing calculator will make it easy for the students to notice how these changes in functions affect the graphs. If we had to do these calculations by hand it would be very tedious for the student.

12. Graphing • “Students should learn to recognize how the values of parameters shape the graphs of functions in a class” (NCTM). • CAS- Computer Algebra Systems -Graphs functions and relations -Can compute values of functions or solutions to equations -Carries out manipulations of symbolic expressions

13. Graphing Functions • Y=sin(x) • Y=cos(x) • Y=tan(x) • Y=2sin(x) • Y=2cos(x) • Y=1+sin(x) • Y=1+cos(x) • Y= - cos(x) • Y= - sin(x) • Y=sin(2x) • Y=cos(2x) For each of the following: 1. Hit Y= 2. Type in the function and close parenthesis 3. Hit graph

14. Graphing Calculator: Cosine • Y1=cos(x) • Y1=cos(3x) • Y2=1+cos(x) • Y1=2cos(x)

15. Geometer’s Sketchpad • Y=sin(x) • Y=2sin(x) • Y=sin(2x) • Y=1+sin(x) • Y=cos(x) • Y=2cos(x) • Y=cos(2x) • Y=1+cos(x) • Y= - cos(x) • Y=tan(x) 1. After opening GSP, go to the toolbar and click on graph. 2. Go to grid form and click on rectangular 3. Next go to graph again and click on plot new function 4. Type in the function 5. It will ask if you want to change your graph into radians. Click yes so you can see the graph continued.

16. Geometer’s Sketchpad 1. After opening GSP, go to the toolbar and click on graph. 2. Go to grid form and click on rectangular 3. Next go to graph again and click on plot new function 4. Type in the function 5. It will ask if you want to change your graph into radians. Click yes so you can see the graph continued. • Let’s graph sin(x) first.

17. Changing the Coefficient • Now let’s change the coefficient to 2. • F(x)= 2sin(x) • If you graph the picture on the same as before it gives you the chance to see the change. • By changing the coefficient notice that the graph goes higher.

18. Adding a Constant • Let’s now graph f(x)=1+sin(x) • Adding or subtracting a number to sin(x) is going to shift the graph up or down.

19. Cosine • Now let’s look at cosine. • Graph f(x)=cos(x) • Next graph f(x)= - cos(x) on top of f(x)=cos(x).

20. Wolframalpha.com: Tangent • Go to www.wolframalpha.com • Type in tan(x) • Change the function to make it negative - tan(x)

21. Illuminations • Illuminations has over 100 online activities. • http://illuminations.nctm.org/ActivityDetail.aspx?ID=174 • This activity is on graphing trig functions. It is easy to use.

22. Solving Triangles • “In grades 9-12 all students should use trigonometric relationships to determine lengths and angle measures” (NCTM). • “Some Old Horse Caught Another Horse Taking Oats Away.” • Sinx=opposite / hypotenuse • Cosx=adjacent / hypotenuse • Tanx=opposite / adjacent

23. Geometer’s Sketchpad: Solving Triangles • Let’s use sin x to find side AB. • * sin(x)=opposite/hypotenuse • Sin(46.63)=AB/9.21, so AB=6.69

24. Webmath.com • Solving Right Triangles • Interactive • Visual • Find sides or angles • http://www.webmath.com/rtri.html

25. Youtube.com • Solving Triangles http://www.youtube.com/watch?v=XFh_JC7OSrg&feature=fvw • Solving Identities http://www.youtube.com/watch?v=9uoKutwuCio

26. In summary… • Technology is beneficial • Makes graphing easier • Can create triangles themselves • Visual • Hands on • Fun!!!

27. Technology • “When students can see math in different ways, they are able to broaden their critical thinking skills and discover meaningful real-world connections” (Ouellette).

28. References • National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Nation Council of Teachers of Mathematics. • Ouellette, Steve. Guide to TI-Nspire Technology. Cliff Notes. Wiley Publishing. December 3, 2009. • TEAM-Math. Retrieved December 1, 2009, from TEAM-Math Curriculum Guide Web site: http://team-math.net/curriculum/index.htm • Illumination Activities • Webmath. Retrieved December 1, 2009, from webmath website: http://www.webmath.com/rtri.html • Youtube Videos