ALGEBRA II- TRIGONOMETRIC

FUNCTIONS

Integrating Technology Into Instruction

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.

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.

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

Alabama Course of Study 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

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.

Technology

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

Types of Technology

Graphing Calculator TI-Nspire: Interactive Classroom Geometer’s Sketchpad Smart Tools:

*Smart Board

*Smart Slate www.wolframalpha.com

*Computational knowledge engine

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.

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.

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.

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

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

Graphing Calculator: Cosine Y1=cos(x)

Y1=cos(3x)

Y2=1+cos(x)

Y1=2cos(x)

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.

Geometer’s Sketchpad1. 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.

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.

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.

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).

Wolframalpha.com: Tangent Go to

www.wolframalpha.com Type in tan(x)

Change the function to make it negative

- tan(x)

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.

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

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

Webmath.com

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

Youtube.com

Solving Triangles

http://www.youtube.com/watch?v=XFh_JC7OSrg&feature=fvw

Solving Identities

http://www.youtube.com/watch?v=9uoKutwuCio

In summary…

Technology is beneficial Makes graphing easier Can create triangles themselves Visual Hands on Fun!!!

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).

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