Section outline

  • A Lissajous knotA Lissajous figure is a pattern produced by the intersection of two sinusoidal curves at right angles to each other. They are the curves you often see on oscilloscope screens depicting compound vibration. They were first studied by the American mathematician Nathaniel Bowditch in 1815, and later by the French mathematician Jules-Antoine Lissajous, and today have applications in physics and astronomy, medicine, music, and many other fields.

    In 1855 Lissajous invented a pair of tuning forks designed to visualize sound vibrations. Each tuning fork had a small mirror mounted at the end of one prong, and a light beam reflected from one mirror to the other was projected onto a screen, producing a Lissajous figure. Stable patterns appear only when the two forks vibrate at frequencies in simple ratios, such as 2:1 or 3:2, which correspond to the musical intervals of the octave and perfect fifth. 

    So, by observing the Lissajous figures, people were able to make tuning adjustments more accurately than they could do by ear. In 1942 the Dadaist artist Max Ernst punched a small hole in a can of paint, attached it to a coupled pendulum, and set it swinging to create Lissajous figures. He then used the designs in some of his paintings.

    In 2001, NASA launched the Wilkinson Microwave Anisotropy Probe (WMAP) to make fundamental measurements of our universe as a whole. The probe was positioned near a gravitational balance point between Earth and the Sun and moved in a controlled Lissajous pattern around the point. This orbit isolated the spacecraft from radio emissions from Earth. The goal of WMAP was to map the relative cosmic microwave background (CMB) temperature over the full sky. CMB radiation is the radiant heat left over from the Big Bang. Tiny fluctuations in the CMB are the result of fluctuations in the density of matter in the early universe, so they carry information about the initial conditions for the formation of cosmic structures such as galaxies, clusters, and voids. 

    From the WMAP data, scientists were able to:
      • estimate the age of the universe at 13.77 billion years old.
      • calculate the curvature of space to within 0.4% of "flat" Euclidean.
      • determine that ordinary atoms (also called baryons) make up only 4.6% of the universe.
      • find that dark matter (matter not made of atoms) is 24.0% of the universe.
    Image Caption: A Lissajous knot is a knot defined by parametric equations that take on specific forms of trigonometric functions with phase shifts (credit: https://en.wikipedia.org/wiki/Lissajous_knot)
    (Content & Image Source: Chapter 4 Introduction, Trigonometry, Katherine Yoshiwara, GNU Free Documentation License)
    Upon completion of this module, you will be able to:

    4.1 Angles and Rotation
    • Use angles to represent rotations
    • Sketch angles in standard position
    • Find coterminal angles
    • Find and use reference angles
    • Find trigonometric ratios for the special angles
    4.2 Graphs of Trigonometric Functions
    • Find coordinates
    • Use bearings to determine position
    • Sketch graphs of the sine and cosine functions
    • Find the coordinates of points on a sine or cosine graph
    • Use function notation
    • Find reference angles
    • Solve equations graphically
    • Graph the tangent function
    • Find and use the angle of inclination of a line
    4.3 Using Trigonometric Functions
    • Solve trigonometric equations, graphically and algebraically
    • Find coordinates of points on circles
    • Use bearings to determine position
    • Find and use the angle of inclination of a line
    • Identify periodic functions and give their periods
    • Sketch periodic functions
    • Sketch graphs to model sinusoidal functions
    • Analyze periodic graphs
    To achieve these objectives:
    1. Read the Module 4 Introduction (see above).
    2. Read Sections 4.1-4.3 of Chapter 4: Trigonometric Functions in Trigonometry (links to each Section provided below)
      • Note: The Algebra Refresher at the top of each Section might be beneficial before you begin
      • At the end of each Section there is a list of Vocabulary, Concepts, Study Questions, and a Self-Check H5P activity
    3. Complete the MyOpenMath Homework Assignments for each Section (links provided below) - These are graded!
    4. View the Chapter 4 Summary and Review (link provided below)
    5. Practice the problems on the Exercises Sections, checking the solutions provided (links to each Section provided below)
    6. View the Exercises: Chapter 4 Review Problems (link provided below)
    7. Complete the MyOpenMath Quiz for Chapter 4 (link provided below) - This is graded!
    8. Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
    9. Post in the Chapter 4 Q&A Discussion Forum - link provided below.

    Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.


    Module Pressbooks Resources and Activities

    You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.

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