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Graphing of Trigonometric Functions Lesson Plan By: Douglas A. Ruby Class: Pre-Calculus II Date: 10/10/2002 Grades: 11/12 INSTRUCTIONAL OBJECTIVES: At the end of this lesson, the student will be able to: 1. Evaluate trigonometric functions with the TI-83+ using either degree or radian measure. 2. Use a TI-83+ Graphing Calculator to graph all six basic trigonometric functions in the form of y = sin(x) , y = cos(x), y=yan(x), y=sec(x), y=csc(x), and y=cot(x). 3. Given a trigonometric function in the form of y=sin(x), y=cos(x), etc., be able to sketch the curve. 4. Use the ZOOM, WINDOW, and TRACE functions to evaluate the trigonometric functions for various angles. Relevant Massachusetts Curriculum Framework PC.P.3 - Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). Relate the functions to their geometric definitions. MENTAL MATH – (5 Minutes) Question 1: Recall the discussion on trigonometric functions of a right triangle discussed yesterday. What does the acronym SOHCAHTOA stand for? Question 2: What are the values for the six trigonometric functions for the following angles: = /3 Sin = Cos = Tan = = 11 /4 Csc = Sec = Cot = Sin = Cos = Tan = Page 1 Csc = Sec = Cot = Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby CLASS ACTIVITIES – (Note: 45 Minute Lesson Plan) 1. Basics of using the TI-83+ to graph trigonometric functions. Ok, we are going to start this class by using the TI-83+ to graph the basic trigonometric functions of sin, cos, tan, csc, sec, and cot. To start with, let’s look at the basic TI-83+ keyboard. Your keyboard should look like this: Arrow Keys Start by hitting the ON key at the lower left of the keypad. Next, let’s set ourselves up to graph trigonometric functions using degree measurement. To do this, let’s hit the MODE key next to the 2nd key. You should see the following window: Page 2 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby Use the Down Arrow (the purple one in the upper right quadrant) to position the blinking cursor to “radians”. Next, use the Right Arrow to move the cursor to “degrees” and then hit Enter followed by Clear. We now have set the TI-83+ up so that the “x-axis” and any angular data we enter are assumed to be in units of degrees. We now want to set up our display window so that the vertical y-axis is 5 units (+ or -) and the horizontal axis is 360 degrees plus and minus. To do this, hit the WINDOW key next to the Y= key. You should see: Let’s change the display windows size so that Xmin=-360, Xmax=+360, Ymin=-5, and Ymax=5. Do this by using the Arrow Keys to position the cursor and overtyping the previous numbers. Use DEL to clear unwanted characters. When you are done, this window should look like this: Ok, now we are ready to start graphing. Hit the Y= key to get this window: Page 3 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby Now we will enter our first trigonometric function. Please enter sin(x) by hitting the SIN key, followed by the X,T, ,n key, followed by a closed parentheses “)”. Your screen should look like: Now, let’s graph the y = sin x function by hitting the GRAPH key. Your screen should look like: Use the Arrow Keys to move the arrow around the screen. See if you can find the value of x when y=1? What do you get? Try following the curve and see if you can find some values for y=sin x? Notice that it is difficult to get exactly correct values? Hit the CLEAR button twice to get back to the basic entry screen. Let’s calculate the value of y=sin(67.5o). You should see: Now go back and try to find y=sin(67.5o) by tracing the graph from before. Hit the GRAPH button again. Now hit the TRACE button. Notice that the trace cursor is at x=0 and y=0. Move the trace cursor to the right until x=67.5. When I try to do this, I can only get as accurate as x=68.93617 and y=.96774194. However, we can ZOOM in to see this part of the curve more closely. Let’s hit the ZOOM button and ZOOM IN (followed by ENTER). You now should see: Page 4 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby Let’s continue to TRACE and see if we can find y=sin(67.5o). I still have a hard time getting exactly x=67.5o. Let’s now try a different form of ZOOM. Hit ZOOM and then hit ZTRIG followed by ENTER. You will see that graph looks pretty good now. When you TRACE, you will see that the cursor moves in units of 7.5o. This is nice, because we can look at the y value of y=sin(x) for x=7.5o, 15o, 22.5o, 30o, 37.5o, 45o, 52.5o, 60o, etc. By ZOOMING IN or ZOOMING OUT we can also move around in the curve and find other (x,y) values for y=sin(x). Of course, we don’t need our calculators for the 30o, 45o,60o, 90o angles, since we already have those committed to memory, right! Right now, I’d like you to try graphing y=cos(x), use the ZTRIG, ZOOM IN, and ZOOM OUT to focus on different parts of the curve. Using the trace function, please find the following: a) y = cos (67.5o) .382 e) y = tan (67.5o) 2.41 b) y =cos (367.5o) .991 f) y = tan (367.5o) .131 c) y =cos (-52.5o) .609 g) y = tan (-52.5o) -1.30 d) y = cos (-65.625o) .413 f) y = tan (-65.625o) -2.207 2. Graphing all six trigonometric functions Ok, now we will graph all six trig functions. Notice from the keypad, that we only have SIN, COS, and TAN keys. To graph the secant, cosecant, and cotangent functions, what will have to do? Now try graphing y=sec(x). First, set up your ZOOM to ZTRIG and hit ENTER. Now, hit the Y= key and clear any previous entries by positioning the cursor to the entry using the ARROW KEYS and then hitting CLEAR. Now enter y=sec(x). How do we do this? Well, do you remember that sec(x) = 1/cos(x)? Given that we know the definitions of all six trigonometric functions, we can use the COS(X) key to create the SEC(X) function by setting Y1=1 COS(x). You should see the following: Page 5 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby Now, we can graph the y=sec(x) function and we should see: Try tracing this function to values of x=90o. Notice that at X=90o, Y=<blank>. Why is this so? This is because cos(90o) = 0, so 1 cos(90o) = sec(90o) is undefined. Now go ahead and graph the cosecant and cotangent functions. To do this, please again use the zooming and trace functions to find: a) y = csc (67.5o) 1.082 e) y = cot (67.5o) .414 b) y = sec (367.5o) 1.009 f) y = csc (367.5o) 7.661 c) y = 1/tan (-52.5o) -.767 g) y = cos (-52.5o) .609 d) y = csc (-65.625o) -1.098 f) y = sec (-65.625o) 2.423 You now know how to graph the basic six trigonometric functions. Before we finish today’s lesson, let’s also look at two additional topics. Page 6 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby 3. Graphing using Radian instead of Degree measure In order to use radians instead of degrees, let’s change our calculator MODE. First, go back to Y= and make sure Y1=sin(x). Then hit the MODE button and you should see: Use the arrow keys to highlight Radian instead of Degree then hit ENTER. If you now return to the GRAPH function, you should see: Let’s reset the ZOOM to ZTRIG. Now TRACE the graph. You should notice that at x=1.5707963, y=1. This value x=1.5707963) is, in fact, an approximation for x= /2 radians. If you have done this correctly, your screen should look like: To approximate when tracing radians, use =3.14. Therefore, a decimal approximations of /6 is 3.14 6 = .523. To approximate and (n x /6), you would multiply .512 by n. The same can be done for multiples of /2 = 1.57, /4 = .785, and /3 = 1.046. Now, use the trace functions and radian measure to find the following: a) y = csc ( /2) 1 e) y = cot (3.2) 17.102 b) y = sec (3 /2) c) y = cot (- /6) (undefined) -2.95*10-8 -1.732 f) y = tan ( /3) g) y = cos (13 /2) 1.732 (0) 1.3*10-9 Page 7 d) y = cos (11 /6) .866 f) y = sec (-2 ) 1 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby Note: Some value should be undefined or zero but may not be due to numerical error. Summary: We have covered using a TI-83+ calculator to evaluate and graph trigonometric functions. Tonight’s homework asks you to continue this process. You will evaluate functions using the normal mode as well as by tracing the graph of the function. You will be asked to sketch graphs of all six functions. You will also be asked to look at the graph of a “constant” times a trigonometric function in the form of y = ½ * cos(x) and compare it with y = cos(x). When you compare these two graphs, I’d like you to see what you can learn about the impact of the constant on the shape of the graph. Does the constant “½ “change the period (cyclical nature) or amplitude (height of rise and fall) of the curve? Page 8 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby HOMEWORK: 1. Use the regular calculator functions to evaluate y=tan(x) at: a. 120o b. /2 radians c. 13o 2. Graph the y=tan(x) functions, use trace and zoom to evaluate the function at: a. 120o b. /2 radians c. 13o 3. Using the graphing calculator to guide you, sketch graphs of the six trigonometric functions over the domain x=-2 to x=2 radians (-360o to 360o). Label each graph with the name of the function. Label the x-axis with degrees or radians and label the y-axis with the correct y values. Sine Cosecant Cosine Secant Page 9 Tangent Cotangent Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby 4. Sketch a graph of y=cos(x) for x= -2 to 2 radians. Now, sketch a graph of y= ½*cos(x) on top of the first graph. Use the graph box below. 5. Now use your calculator to graph the two functions in #4 above. Graph both functions on the same screen. What is the period of y=cos(x)? What is the amplitude of y=cos(x? What is the period of y= ½*cos(x)? What is the amplitude of y= ½*cos(x)? Page 10 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby Answer Keys: MENTAL MATH – (5 Minutes) Question 1: Recall the discussion on trigonometric functions of a right triangle discussed yesterday. What does the acronym SOHCAHTOA stand for? For any angle of a right triangle, the: Sine of = Opposite side over Hypotenuse side Cosine of = Adjacent side over Hypotenuse side Tangent of = Opposite side over Adjacent side Question 2: What are the values for the six trigonometric functions for the following angles: = /3 (60o) Csc =2 =12 Sec =2 = Cot =1 Sin = Cos Tan 3 2 = 11 /4 (2 + 3 /4 or 360o + 135o) 3 3 3 Sin =1 Cos = 1 Tan = -1 Page 11 2 2 Csc = Sec = Cot = -1 2 2 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby HOMEWORK (Answer Key): 1. Use the regular calculator functions to evaluate y=tan(x) at: a. 120o = -1.732050808 (or - 3 ) b. /2 radians = get ERR:DOMAIN since tan( /2) is undefined c. 13o = .2308681911 2. Graph the y=tan(x) functions, use trace and zoom to evaluate the function at: d. 120o = -1.732051 e. /2 radians = -4.8757x10-9 f. 13o = .23086819 Page 12 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby 3. Using the graphing calculator to guide you, sketch graphs of the six trigonometric functions over the domain x=-2 to x=2 radians (-360o to 360o). Label each graph with the name of the function. Label the x-axis with degrees or radians and label the y-axis with the correct y values. (Note: x range is [-2 , 2 ] in all six graphs, y range is specified in title.) Sine (y=[-2,2]) Cosecant (y=[-6,6]) Cosine (y=[-2,2]) Secant (y=[-6,6]) Tangent (y=[-4,4]) Cotangent (y=[-4,4]) 4. Sketch a graph of y=cos(x) for x= -2 to 2 radians. Now, sketch a graph of y= ½*cos(x) on top of the first graph. Use the graph box below. 5. Now use your calculator to graph the two functions in #4 above. Graph both functions on the same screen. What is the period of y=cos(x)? What is the amplitude of y=cos(x? What is the period of y= ½*cos(x)? What is the amplitude of y= ½*cos(x)? See screen shot above. Note that period of y=cos(x) is still 360o. Amplitude of wave form is now ½ instead of 1. Page 13 Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby Powerpoint Slides to go with Lesson: Objectives At the end of this lesson, the student will be able to: Evaluate trigonometric functions with the TI83+ using either degree or radian measure. Use a TI-83+ Graphing Calculator to graph all six basic trigonometric functions in the form of y = sin(x) , y = cos(x), y=yan(x), y=sec(x), y=csc(x), and y=cot(x). Given a trigonometric function in the form of y=sin(x), y=cos(x), etc., be able to sketch the curve. Use the ZOOM , WINDO W , and TRACE functions to evaluate the trigonometric functions for various angles. Mental Math Question 1: Recall the discussion on trigonometric functions of a right triangle discussed yesterday. What does the acronym SOHCAHTOA stand for? S________ = O_______/H_____________ C________= A________/H____________ T________= O________/A____________ Question 2: What are the values for the six trigonometric functions for the two angles: = /3 (60 o) Sin = Cos = Tan = Csc Sec Cot = 11 /4 (495 o) = = = Sin = Cos = Tan = Page 14 Csc Sec Cot = = = This document was created with Win2PDF available at http://www.daneprairie.com. The unregistered version of Win2PDF is for evaluation or non-commercial use only.