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Chapter 5: Triangle Trigonometry


 Date

Entry Task

Activity

Assignment/Homework*

9/17
Compare objects and processes used in last night's homework at your 4-top.  A spokesperson for your group will compile a list of commonalities and explain to the class when it is your group's turn to speak.
Begin to make a Trigulator.

JR:  Use simple language to explain the meaning of "the unit circle" (a trigonometric term).
Finish the Trigulator II.
9/19
List the three ratios from Trigulator II along with their largest and smallest values. Use your results from Trigulator II to set up and solve the following problems
  • For right triangle ABC (angle C is the right angle), angle A has a measure of 70 degrees and AB = 12 cm; solve for all other unknowns.
  • For right triangle DEF (angle F is the right angle), angle D has a measure of 40 degrees and EF = 12 cm; solve for all other unknowns.
  • For right triangle GHJ (angle J is the right angle), GJ = 5 cm and HJ = 12 cm; solve for all other unknowns.
JR:  Explain how to use a table of trigonometric values to solve for unknown sides or angles in a triangle.
Finish the problems begun in class.
9/22
Explain how to solve for the measure of one of the acute angles of a right triangle if you know the lengths of two legs AND are using the table from Trigulator II.
Debrief the homework.

Begin Trigulator III.

JR:  Explain how you would improve Trigulator I and Trigulator II so you would get "more accurate" results when computing the measures of unknown parts of triangles.
Finish Trigulator III.
9/23
Get ready for the quiz!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people. Quiz: use ratios from YOUR Trigulator II to set up equations that will solve for an unknown side or angle of a triangle.  No calculator or notes.  No accommodations will be made for those who do not have their Trigulator II completed or with them.

JR:  Explain how to determine the height of a structure indirectly (i.e. by using trigonometry).
  1. Add a column to Trigulator II and label it AB ÷ AC.  Compute the ratios from 0 degrees through 90 degrees.  What is the aviation connection for the ratios in the column?
  2. What is the height above ground level (AGL) of the red beacon at the top of the West side of RAHS?  Note: either "measure" it and provide your (detailed) method or reference your source of information.
Remember to obtain and bring ruler and protractor!
9/24
YOUR ruler and protractor are required for this!

Create a scale diagram (use ruler and protractor) that satisfies the following conditions
  • The apparent "angle of elevation" to the sun is 15 degrees
  • An object casts a shadow of 55 meters.
Use your diagram from the ET to determine the height of the object that casts the 55 m shadow.

Construction of a "clinometer" from a protractor.  How to make one is available from the Exploratorium or you can use THIS file from NCTM.

JR:  Explain in simple language how to modify and use a protractor to determine angle of elevation and angle of depression.
Use your clinometer (made from a protractor) to measure the
  • angle of elevation of two structures
  • angle of depression of two different structures
Provide the location of the structures and your distance (paces, if necessary) from the structure.  Include a scale diagram of each situation.
9/26
List several applications of trigonometry.
Try Angles activity.

JR:  If the lengths of the sides of a triangle are a and b, write an expression for the maximum and minimum lengths of the third side.
Finish Try Angles.
9/29
Which trigonometric functions are associated with the last two columns of the Try Angles activity?  Explain the significance of the ratios in relation to the measure of angle B.
Debrief Try Angles.

Issue textbooks. 

Begin Textbook Scavenger Hunt.  Write answers into your homework notebook.

JR:  List some ways to protect and preserve your textbook.
Finish Textbook Scavenger Hunt.  Write answers in your homework notebook.


9/30
Sketch the situation from p. 313 #6.
Do Lesson 5.1 (Right Triangles) pp. 303-310.

See THIS graphic for the relationship between degrees and radians along with quadrant positivity for each trig function.

JR:  List what must be included on every assignment problem to receive credit.
Pg. 311 #1, 2, 3.


10/1
Construct a right triangle having legs 6.0 cm and 8.0 cm.  Measure the angles and hypotenuse.  Explain how "inverse trig functions" will solve for the angle measures. Inverses.  Lesson  5.2 (pp. 318-324).

Write a summary in your homework notebook: explain the relationship between a trig function and its resultant ratio--discuss from the standpoint of the function and "movement" on your "Trigulator III" paper.

JR:  Explain, in detail, the purpose of an "inverse function."  Give several examples (both trigonometric and non-trigonometric).
Pgs. 325-326 #3, 5.

Click the links to see a 35 mm camera and 35 mm film (from which negatives are made).
10/3
Explain how similarity of triangles is related to trigonometry.
The Sims activity.

JR:  Explain how to use similarity of triangles to solve for unknown parts of a triangle.
Use known ratios from the triangles you created in The Sims activity to solve for the unknown parts of the given mystery triangles.  DO NOT solve by scale drawing or using trigonometry.  Write ratios from the known triangles and solve the proportion using algebra.  Use only the arithmetic functions from your calculator (+, —, x , ÷).  Begin each problem with a sketch of the mystery triangle and include a sketch of the corresponding known triangle (from the first page of this activity).
  1. The triangle has an angle of 70.0 degrees and the side adjacent the angle is 5280 cm long.
  2. The legs of the triangle are 42 SM and 35 SM.
  3. The triangle has an angle of 30.0 degrees and the hypotenuse is 145 cm long.
  4. The triangle has an angle of 35.0 degrees and the side opposite the angle is 42.0 cm long (turn up your brain cells for this one!).
10/6
p. 325 #1.
Quiz 5.1 (trigonometry).  You may use YOUR calculator and a 3 x 5 note card** (writing on both sides is permitted).  Measurement devices (e.g. YOUR ruler and protractor) are also allowed.

**Note cards may not be mechanically reproduced (no photo copies, word processing, etc.).

JR:  Explain in simple language how to solve for unknown sides and angles of a right triangle given: a) two sides are known; and, b) one angle and the side opposite the angle are known.
Pg. 327 #7, 8.
10/7
The height of a mountain (relative to its surroundings) can be calculated by finding the angle of elevation to the peak, moving away from the mountain a measured distance, and finding the new angle of elevation.  Draw the situation, “invent” some measures, and solve for the height of the mountain.
Oblique triangles: Lesson  5.3 (Pgs. 329-335).  Additional notes on the laws of Sines and Cosines is HERE.

JR:  Explain when to use the Law of Sines rather than the Law of Cosines.
Pgs. 336-338 #1, 3.
10/8
Draw an arbitrary triangle ABC and measure the sides and angles.  Confirm the Law of Sines for all three angle/side ratios.  Check to see if the ratio of cosines are also equal (i.e. cosine (A)/a = cosine (B)/b).  Write a conclusion of your observation.
Work with table partners on
The Law of Sines
and
The Law of Cosines.
JR:  A baseball "diamond" is comprised of a home plate along with first, second, and third bases--all ninety feet away from the previous base.  The pitcher must throw from a "rubber" that is 60 feet and 6 inches (that is, 60.5 feet) from home plate.  How far is the pitcher's rubber from first base?  Include a diagram!
Pgs. 340-342 #9, 13. 
10/13
Make a sketch and solve for the unknown side for triangle ABC
  • AB = 12
  • AC = 15
  • The measure of angle A is 42 degrees.
Pgs. 338-339 #4, 5.

JR:  Draw equilateral triangle ABC.  Does a2 + b2 = c2?  Why/why not?  Speculate how this problem relates to other triangles.
Pg. 339 #6; p. 341 #11.
10/14
The shadow of one of Blair's Cuspids was approximately 110 meters when the Sun was at an angle of elevation of 10.9 degrees.  Compute the height of the cuspid if the ground were level, if the ground sloped downward at 5.0 degrees and if the ground sloped upward at 5.0 degrees. Begin working on the Chapter 5 Review (Pgs. 343-344 #1-10).

JR:  Under what conditions would it be impossible to solve for the unknowns of a triangle even though you know three facts (at least one of which is a side)?  Create a scale drawing.
Finish the Chapter 5 Review.

For #7, assume the towers are on the same line of latitude. 
10/17
List the domain and range for each of the following functions
  • y = sin-1(x).
  • y = cos-1(x).
  • y = tan-1(x).
Peak My Interest activity.

JR:  Why must the distances between the two mountains in today's activity be computed using Law of Cosines rather than using proportions from the distances between villages B and C?
Finish today's activity.

Recommended problems:
Pgs. 327-328 #9, 12, 13.

Note: on "concavity:" While concavity is defined more formally, loosely, we can think of concave up as a part of a function with a smile or curves upward. Concave down is a function which frowns or curves downward (courtesy Mr. G).  See also
10/20
Weather balloons provide soundings that give a profile of the atmospheric conditions.  Before GPS wind speed and direction were calculated by measuring altitude and downrange movement.  Explain how the math for this works. In-class review.  Bring questions & concerns!

JR:  Under what circumstances will the Law of Cosines give the wrong answer?  Explain why.
 Prepare for the test.  Consider beginning the Chapter 5 Take-home.  Note: the downloadable version is in color with a larger diagram on page 2.  :-)
10/21
Get ready for the test!  Move to a seat where you have ample room, obtain all the materials you need before class starts, put yourself in a positive mental state.  Test 5.1.  You may use YOUR calculator and a 3 x 5 note card** (writing on both sides is permitted).  Measurement devices (e.g. ruler and protractor) are also allowed, however they can only be the ones you brought to class.  Expect questions on
  • Solving for unknown parts of right triangles.
  • Solving for unknown parts of non-right triangles.
**Note cards may not be mechanically reproduced (no photo copies, word processing, etc.).

JR:  What were the hardest parts of Chapter 5 for you?  What were the easiest parts?  
 Prepare for the test.  Consider continuing the Chapter 5 Take-home.
10/23
Get ready for the test!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.  Test 5.2.  You may use a 3 x 5 note card* (writing on both sides is permitted).  Measurement devices (e.g. ruler and protractor) are required, however, they may only be the ones you brought to class.  Calculator NOT allowed.  Expect a problem on determining a trigonometric value using a scale diagram.

**Note cards may not be mechanically reproduced (no photo copies, word processing, etc.).

JR:   Suggest some ways Dr. Edge's teaching of this unit could be improved.
Finish the Chapter 5 Take-home.
10/24
Get ready for the test!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.

Have your homework notebook ready and computer activated.

Note: today is the last day any late papers will be scored this quarter!
Trigonometry post-test.  Calculator allowed.

Note: today is the last day any late papers will be scored this quarter!

JR:  Did you improve versus the pretest on this unit?  If so, why?  If not, why not?
Explain each of the following
  • Polar graphing.
  • Azimuth.
  • Compass heading.
  • "Altitude" (as with sighting a planet or star).
Note: today is the last day any late papers will be scored this quarter!
*Unless otherwise noted, homework is due the next class day.