3332.jpg (15800 bytes) What is trigonometry? How is it used? Why is it important? Answers these questions as specific and practical uses of trigonometry are shown: land surveys, marine navigation, astronomy, amusement park rides, architecture, and others. Does not teach trigonometry concepts and assumes familiarity with trigonometric terms.

ACADEMIC STANDARDS

Subject Area: Mathematics

INSTRUCTIONAL GOALS

  1. To define trigonometry.
  2. To list the six trigonometric ratios.
  3. To list some trigonometric identities.
  4. To explain how trigonometry applies to surveying, navigation, and astronomy.
  5. To introduce the concept of trigonometric parallax.
  6. To show how degrees are used when measuring circles.
  7. To define radian.
  8. To show graphs of trigonometric functions.

VOCABULARY

  1. acute angle
  2. functionhypotenuse
  3. reciprocal
  4. cofunction
  5. Pythagorean
  6. sine
  7. cosine
  8. tangent
  9. vector
  10. trigonometric parallax
  11. quadrant
  12. radian

BEFORE SHOWING  

  1. Display a flow chart showing the various branches of mathematics.
    1. Point out trigonometry and its position relevant to algebra, geometry, and calculus.
    2. Discuss the order in which these classes should be studied.
    3. Discuss reasons students take trigonometry in high school.
  1. Cut a grapefruit in half and insert two large toothpicks about three inches apart in one of the halves.
  1. Push one of the toothpicks down into the grapefruit until its top is level with the grapefruit pulp. Label this point A.
  2. Adjust the other toothpick so that it stands as tall as possible. Label this toothpick B.
  3. Use a ruler to find the length of toothpick B, the distance between point A and the base of B, and the distance between point A and the top of B.
  4. Draw a triangle to represent the measurements.
  5. Discuss the ease of direct measurement.
  1. Display a picture showing a tall object such as a tree, flagpole, or building.
  1. Discuss ways to measure the height of the object.
  2. Draw a triangle similar to the one in the grapefruit activity.
  3. Discuss the difficulty of using direct measurement in this situation.
  4. Introduce the term indirect measurement.
  1. Explain that trigonometry is a method for finding the missing parts of right triangles. Discuss possible uses of trigonometry in practical situations.
  2. Explain that there are different kinds of trigonometry: triangle, circular, hyperbolic, and spherical.

DURING SHOWING

Discussion Items and Questions

  1. View the video more than once, with one showing uninterrupted.
  2. Pause at the end of the sentence about uncooked pork. Hold up a flashcard with the word trichomoniasis printed on it. Compare the spellings.
  3. Point out the two Greek words that convey the meaning of trigonometry.
  4. Pause at the section listing the three basic trigonometric ratios.
  1. Point out that a is used to represent an angle. Explain that other symbols such as 2 (theta) are also used to denote angles.
  2. Rewrite the ratios using words instead of the letters a, b, and c. (Example: tangent = opposite/adjacent.)
  3. Suggest mnemonics to help memorize the three ratios. (SOH-CAH-TOA or Silly Old Horse Cracks All Her Teeth On Apples)
  4. Explain that cosecant, secant, and cotangent are the reciprocals of the other three ratios.
  1. Pause at the section listing the four identities of trigonometry and briefly explain each one.
  2. Pause at the part showing the calculator. Explain that sin-1, cos-1, and tan–1 represent secant, cosecant, and cotangent respectively.
  3. Pause at the section that has the caption "I like A. Definitely B." Determine what other meaning of surveying is implied here.
  4. Pause at the caption containing " . . . 3-4-5 triangle." Draw it and explain what it is.
  5. Pause after the segment on navigation. Discuss why navigation problems are usually more difficult to solve than surveying problems.
  6. Pause at the section depicting the circumference of the circle as 2B r.
  1. What would be the distance of the part of the circle formed by a 45° angle? A 90° angle? A 180° angle? A 270° angle?
  2. How did these concepts aid ancient astronomers?
  1. Pause at the segment on trigonometric parallax and do the extended arm and thumb demonstration.
  2. Pause at the part describing the directions of a pilot flying on course. Explain how degrees, minutes, and seconds are used as a means of indicating direction and location.
  3. Pause at the table showing degrees and radians. Discuss how the values for the radians are obtained.
  4. Pause at the section containing the captions "Ad infinitum. Ad nauseum." Explain the meanings.
  5. Pause at the segment about AM and FM frequencies. Explain the difference.
  6. Pause at the part showing the word metronome. Explain what it is.
  7. Pause at the quote by Pierre Bayle at the end of the video. Discuss its meaning.

AFTER SHOWING

Discussion Items and Questions

  1. What is the definition of trigonometry?
  2. Why were the early surveyors regarded as heroes?
  3. What are some situations in which the surveyor must find the height of a tall object?
  4. What are some situations where the surveyor must find the length of a horizontal line that is difficult to measure?
  5. In what ways do navigators use trigonometry?
  6. What are vectors and how are they used in navigation?
  7. What are some applications of trigonometry in astronomy?
  8. What is trigonometric parallax and how is it useful?
  9. Which direction of rotation is used to define angles on a circle?
  10. How is it possible to get angle measurements greater than 360° ? How is it possible to get negative angles?
  11. What are quadrants and how are they useful in angle measurement?
  12. What are the advantages of using a circle to represent angles?
  13. What is a radian and how is it used?
  14. What do the graphs of sine and cosine functions look like?

Applications and Activities

  1. Solve the problems presented in the video.
  2. Make cardboard models of 30° -60° -90° and 45°
    -45° -90° triangles.
  1. Measure each side with a ruler.
  2. Use the trigonometric ratios and check the answers.
  1. Devise a worksheet with drawings of right triangles with some measurements of sides and angles missing. Determine which trigonometric ratio should be used to find the answer.
  2. Draw a large circle on posterboard.
  1. Using a protractor, mark off the following angles and draw their sides: 30° , 45° , 60° , 90° , 120° , 150° , 180° , 210° , 240° , 270° , 300° , 330° , and 360° .
  2. Write the corresponding radians at the edge of the mark-off points.
  1. Report on the following:
  1. Pythagorean identity
  2. cofunction identity
  3. ratio identity
  4. reciprocal identity
  5. law of sines
  6. law of cosines
  7. Buffon’s Needle
  1. Design a time line depicting the history of trigonometry.
  2. Obtain a table of trigonometric values for tangent, sine, and cosine. Compare these values with the values on a calculator. Note the number of decimal places.
  3. Using a calculator and an x and y value table, graph the following on graph paper: y = sin x, y = cos x, and y = tan x.
  1. Use radian values on the x axis.
  2. Periodic functions have amplitude and frequency. Research these terms.
  3. Note which angles yield positive values and which angles yield negative values.
  1. Use a graphing calculator to graph the following functions and look for a pattern:
  1. y = cos (2x) and y = 2 cos x
  2. y = sin (3x) and y = 3 sin x
  3. y = cos x and y = cos (1/2 x)
  4. y = sin x and y = sin x + 5
  1. If a physical model has motion that can be described as a sine or cosine curve, the motion is called harmonic. List some objects that have harmonic motion, such as a Slinky or a yo-yo.
  2. The video does not mention the graph of a tangent function. Plot the graph of y = tan on graph paper. How does the graph of a tangent function differ from that of a sine or cosine function?

RELATED RESOURCES

World Wide Web

The following Web sites complement the contents of this guide; they were selected by professionals who have experience in teaching deaf and hard of hearing students. Every effort was made to select accurate, educationally relevant, and "kid-safe" sites. However, teachers should preview them before use. The U.S. Department of Education, the National Association of the Deaf, and the Captioned Media Program do not endorse the sites and are not responsible for their content.

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