Trigonometry Module
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Trigonometry/C.Akkoc/1996

MODULE No.2

Trigonometric Functions: PERIOD, AMPLITUDE, PHASE SHIFT

Construct the graphs for the following trigonometric functions on TWIDDLE . Make a hard copy of each graph (4 graphs altogether) on the dot matrix printer. Label where appropriate to identify each graph on the hard copy. Calculate and measure the AMPLITUDE, PERIOD, and PHASE SHIFT when asked and write them by hand on the graph. Type in TWIDDLElish:

f(x)=a*sin(b*x+c)

to construct the graphs for the following functions:

    AMPLITUDE MODULATION (AM): On the same screen; X[-0.5,6.4], Y[-5.1,5.1]; Calculate/measure the amplitude for each case.
  1. f(x) = sin(x)
  2. f(x) = 2 sin(x)
  3. f(x) = 5 sin(x)
  4. f(x) = -3 sin(x)

    FREQUENCY MODULATION (FM): On the same screen; X[-0.1,25.2], Y[-1,1]; Calculate/measure the period for each case.
  5. f(x) = sin(2x)
  6. f(x) = sin(x/2)
  7. f(x) = sin(x/4)

    MIXED: On the same screen; X[0,30], Y[-5,5]; Calculate/measure the amplitude/period/phase shift in graph 11 compared with graph 8.
  8. f(x) = sin(x)
  9. f(x) = sin(x/4)
  10. f(x) = sin (x/4-ã/8)
  11. f(x) = 5 sin (x/4-ã/8)

    COMPOSITE WAVE (Type separately into TWIDDLE )
  12. f(x) = sin(2*x)+cos(x)+2*sin(x/2)+3*cos(x) ; X[-17,17],Y[-7,5]
    Estimate the amplitude/period by direct measurements on the hard copy. Don't know how to calculate these parameters algebraically as of now.
 Trigonometry Module
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Trigonometry/C.Akkoc/1996

MODULE No. 3

GRAPHICAL SOLUTIONS of TRIGONOMETRIC EQUATIONS

Construct graphs for the following trigonometric functions on TWIDDLE or PLOT . Make a hard copy of each graph on the dot matrix printer. Label where appropriate to identify each graph on the hard copy. Determine by direct measurements on the hard copy the AMPLITUDE, and PERIOD, when asked, of each and write this information by hand on the graph. Estimate the SOLUTIONS to the associated trigonometric equation by eyeballing with the aid of a millimetric ruler (interpolation). Mark each solution on the hard copy and write its value on the graph. Solve the equation (part 1 only) analytically to compare with your graphical solution.
  1. Trigonometric Equation: 4 sin^2(x) tan(x) - tan(x) = 0; interval of Interest: [0,2Pi]; associated function: f(x)=4*((sin(x))^2)*tan(x)-tan(x) in Twiddelish; Suggested Window: X[-0.2,6.5]; Y[-0.5,0.5].

    Solutions to the given equation will consist of all x-axis crossings within the interval of interest. Solve above equation analytically using algebraic operations and compare your graphical findings with the analytical solutions. They should agree completely.

  2. Trigonometric Equation: sin(4x) - 4 sin(x) = 2 ; interval of interest: [-4,6]; associated function(s): f(x)=a*2+b*(sin(4*x)- 4*sin(x)) in Twiddelish; suggested window: X[-6,7]; Y[-5,5].

    On TWIDDLE use parameters "a" and "b" as shut-off switches to activate only the expression (function) desired. Solutions to the given equation will consist of x-coordinates of all points where the graph of y=sin(4x)-4sin(x) crosses the ÿgraph of the constant function y=2 within the interval of interest. Find these points (coordinates) graphically. Also determine the amplitude and period of the function y(x)=sin(4x)-4sin(x) by direct measurements on the hard copy.

 Trigonometry Module
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Trigonometry/C.Akkoc/1996

MODULE No. 4

SOLVING AN EQUATION INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS

Given the trigonometric equation


arcsin(x/sqrt(3)) + arccos(-x) = Pi
  1. Solve the given equation by analytic/algebraic procedures for a closed-form solution. You need to pay attention to the domain and range of the functions involved. You also need to confirm the "solutions" you find by substituting them back into the original equation.

  2. Graph the two sides of the equation on PLOT in the form of two separate functions; f1=arcsin(x/sqrt(3))+arccos(-x) and f2=Pi on the same screen. The solution set will consist of all points of intersection on the said graph. This graphical solution should agree with your analytical solution(s) in Part1 above.

  3. Make a hard copy of the graph on a dot matrix printer. Mark the point(s) of intersection and the associated coordinates on the hard copy by hand.

  4. On a blank sheet of paper show key steps in your calculations in Part1 together with your results.