GT Unit 10: Trigonometric Laws

Using the Law of Sines and Law of Cosines

TRIGONOMETRIC LAWS OVERVIEW

Go ahead and jump down to Part 1 and start taking some notes – I’ll get the rest of the info plugged in right now.

PART 1: LAW OF SINES

(50 pts)  About 3 days

The first part of our unit is all about similarity between figures.  We’ll look at the transformation the dilation and what happens to figures when they undergo dilation.  These ideas will evolve into our next group of geometric proofs: proofs for similarity!

PART 2: LAW OF COSINES

(60 pts)  About 3 days

The second part of our unit covers what similarity tells us about triangles.  We’ll look at what happens when we dissect a triangle by subdividing segments.  Here we’ll see proportions play a huge role in our understanding of similar triangles.

PART 3: BUILD A PUZZLE!

(140 pts)  About 3 days

Of course our unit will end with a review and test, but in addition you’ll get to build a model of similar figures!  You’ll be asked to create a series of laser-cut similar triangles.  You’ll ethen use those triangles to compare their perimeters, areas, and volumes (weights) to better understand proportions in similar figures!

PART 1: Law of Sines

Using the Law of Sines to solve problems with triangles

Dilations & Similarity

The first part of our unit is all about similarity between figures and what happens when a shape undergoes the dilation transformation.  We’ll start by reviwing similarity and dilations.  Last semester we looked carefully at different translations (rotations, reflections, etc); now we get to look at mathematical operations that scale figures to different sizes.  You’ll start by taking a full page of notes on the four videos provided below.  These videos cover the essential components of similarity, and will give us a good foundation to begin our serious work in trigonometry.  In addition, you’ll work through three homework assignments on these topics before completing the part 1 Quiz 17: Similarity!

GRADING & PROCESS

 Use your engineering notebook to take a full page of notes on similarity between transformed shapes, dilations, and proving similarity

 Complete HW37: Dilations and Proof

 Add to your notes about similarity as you watch the Corresponding Parts and AA Similarity presentations

 Complete HW38: Corresponding Parts

 Complete HW39: AA Similarity

 Take Quiz 17: Similarity!

The Law of Sines

The Law of Sines

Example Law of Sines Problem

What’s Due In Similarity Part 1: Similarity

  • DIlations/Similarity Notes 1
  • HW37
  • Corresponding parts/AA Notes
  • HW38
  • HW39
  • Quiz 17: Similarity

Here’s what’s due in Part 1: Similarity

  1.  Take notes on dilations, transformations, and shape similarity
  2.  Complete HW37: Dilations & Proof
  3.  Next, add to your notes on similarity with information about corresponding parts and AA Similarity
  4.  Complete HW38: Corresponding Parts 
  5.  Complete HW39: AA Similarity
  6.  Take Quiz 17: Similarity!

PART 2: Law of Cosines

Using the Law of Cosines to solve triangles

Similar Triangles

The second part of our unit is all about what happens when we divide triangles into smaller parts by subdividing opposing sides.  In short: if we draw a line through a triangle that is parallel to one of the sides, it creates a smaller triangle that is similar to the original triangle.  We’ll develop this idea more formally throughout the next few topics and assignments so that we can eventually turn these ideas into the six trigonometric ratios!

GRADING & PROCESS

 Use your Engineering Notebook to take a full page of notes on the Triangle Proportionality and Subdividing Triangles presentations

 Complete HW40: Triangle Proportionality

 Take one more complete page of notes on the Proportions in Triangles and Similarity in Right Triangles presentations.  Add examples from your homework to your notes on these topics!

 Complete HW41: Subdividing & Proportions

 Complete HW42: Right Triangle Similarity

 Take Quiz 18: Similar Triangles!

Law of Cosines

Law of Cosines 2

Law of Cosines 3

Law of Sines vs Cosines

What’s Due In Similarity Part 2: Similar Triangles

  • Triangle Proportionality Notes
  • HW40
  • Proportions & Right Triangle Notes
  • HW41
  • HW42
  • Quiz 18: Similar Triangles

Here’s what’s due in Part 2: Similar Triangles

  1.  Take notes on triangle proportionality and subdividing triangles!
  2.  Complete HW40: Triangle Proportionality
  3.  Take one more page of notes on proportions in triangles and similarity in right triangles
  4.  Complete HW41: Subdividing & Proportions
  5.  Complete HW42: Right Triangle Similarity
  6.  Take Quiz 18: Similar Triangles

PART 3: Model It!

Create a laser-cut series of models

The last part of our unit gives us the chance to build a series of laser-cut similar triangles and investigate their properties to confirm both the functions of Autodesk Inventor as well as the mathematical relationships between similar shapes!  First you’ll need to plan out your 4-triangle series in your engineering notebook, and then build the proper triangles of various sizes within Autodesk Inventor.  You’ll use the iProperties tab to record perimeter, area, and volume of each before exporting a .DXF file for use with the laser.  After you’ve laser-cut your 4 triangles, you’ll measure their perimeters, areas, and weights to see how these numbers compare to the values calculated by Autodesk.  You’ll also get to compare these values to see what they tell use mathematically about similar figures.

Of course our unit will also end with a unit review, a chance to get questions answered, and our final unit test to wrap things up!

GRADING & PROCESS

 Use your Engineering Notebook to make a plan for creating your custom-cut laser models.

 Use Autodesk Inventor to create your triangles each of the proper size.  Use the iProperties menu to record the perimeter, area, and volume of each of your four triangle models.

 Export .DXF files of your triangles and cut them on the laser out of acrylic.  Measure and record the perimeter, area, and weight of each of your four triangle models.

 Compare your measured values to those calculated by Autodesk, and to investigate the relationships between similar figures

 Complete the Unit 7 Review

 Schedule time to talk with Benshoof and get questions answered

 Take Unit Test 7: Similarity!

What’s Due In Similarity Part 3: Similar Laser Triangles!

  • Triangle Plan
  • Autodesk & iProperties
  • Laser Cut
  • Compare Triangles
  • Unit 7 Review
  • Question Time
  • Unit 7 Test

Here’s what’s due in Part 3: Build It!

  1.  Make a plan for creating your custom-cut laser models.
  2.  Use Autodesk Inventor & iProperties to create your triangles each of the proper size
  3.  Laser cut them out of acrylic & measure properties
  4.  Compare your measured values to Autodesk, and investigate the relationships
  5.  Complete the Unit 7 Review
  6.  Schedule time to talk with Benshoof and get questions answered
  7.  Take Unit Test 7: Similarity!