GT Unit 8: Right Triangle Trigonometry

Using sin, cos, tan, csc, sec, cot to solve triangles


Our eighth unit here in STEM Geometry Honors is possibly one of the most important in the entire year!  Ratios that naturally exist in right triangles can be used to help identify similar triangles, and even be used to solve for missing sides and angles of triangles of different kinds!  Throughout this unit we’ll build our knowledge of the six (6) most important trig functions:  sin, cos, tan, csc, sec, cot.   

Unit 8 is all about these main trig functions and how they get used in solving for the properties and measurements of right triangles. During this unit we’ll go through three major topics:

  1. Ratios within all right triangles and how those ratios are defined
  2. Measuring angles using degrees and using radians
  3. The functions sin, cos, tan and how to use them to solve triangles
  4. The functions csc, sec, cot and what they can do for us
  5. Inverse trig functions and how we use those to find missing angles

When you’re done, you’ll have covered the basics of right triangle trigonometry!


(50 pts)  About 3 days

The first part of our unit is all about the basics of right triangle trigonometry.  We’ll start with an overview of the three main ratios that exist within all right triangles, and then we’ll give official names to each of them. We’ll also look at how angles are measured in triangles and we’ll make sure that we understand the degree and radian angle systems.


(60 pts)  About 3 days

IN the second par of our unit, you’ll look into the details surrounding a special group of right triangles.  Here you’ll investigate some of the key ratios that exist in these triangles, and use the ratios and proportions to solve more triangles.  Then, we’ll look at some of the various applications of these special right triangles.


(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: Trig Ratios & Angles

Definitions of the basic trigonometric ratios

Trig Ratio Reference

The first part of this unit looks at definitions for the three main trig functions:  sin, cos, tan.  Each of these functions is simply a formal organization of the ratio between two carefully chosen sides of a triangle.  These work well for us because if two right triangles are similar, then the ratio between corresponding sides will be equal.  We can then take advantage of these common ratios to help identify similar triangles in different contexts.  Using that, we’ll even be able to start with half-a-triangle and use the ratios to solve for the missing half of the triangle.


 Use your engineering notebook and take a full page of notes on the three trig ratios.  Make sure that your notes include pictures and details about how each ratio is defined!

 Complete HW43: Tangent Ratio

 Add to your notes about trig ratios as you keep working.  Make sure to add at least one (1) example of all three trig ratios being used to solve triangles!

 Complete HW44: Sine Ratio

 Complete HW45: Cosine Ratio

 Take Quiz 19: Trig Ratios

Trig Ratio Overview




What’s Due In Right Triangle Trigonometry Part1: Trig Ratios & Angles

  • Notes on Tan, Sin, Cos
  • HW43
  • Trig Usage Notes
  • HW44
  • HW45
  • Quiz 19: Trig Ratios

Here’s what’s due in Part 1: Trig Ratios & Angles

  1.  Take notes on all three trig ratios:  tan sin, cos
  2.  Complete HW43: Tangent Ratio
  3.  Next, add to your notes on trig ratios by including at least 1 example of how tan, sin, and cos are used!
  4.  Complete HW44: Sine Ratio
  5.  Complete HW45: Cosine Ratio
  6.  Take Quiz 19: Trig Ratios

PART 2: Special Right Triangles

Using common ratios to define a set of special right triangles

Special Right Triangle Reference

The second part of the unit looks at two very important “Special Right Triangles”. The Isosceles Right Triangle and the 30-60-90 triangle are both super useful in math and engineering because they are relatively easy to identify, simple to work with, and can make solving triangles a trivial job.  Your responsibility is to learn how to identify these two types of triangles, learn the ratios that describe their side lengths, and then get practice using these relationships in solving problems!


 Use your Engineering Notebook to take a full page of notes on the three presentations below about Special Right Triangles.  Make sure that your notes include a detailed picture of the Right Isosceles and 30-60-90 triangles.

 Complete HW46: Special Right Triangles

 Complete HW47: Special Right Triangle Applications

 Complete HW48: TrigStar Practice 1

 Take Quiz 20: Special Right Triangles!

Intro to Special Right Triangles

Right Triangle Details

More Right Triangles

What’s Due In Similarity Part 2: Special Right Triangles

  • Special Triangles Notes
  • HW46
  • HW47
  • HW48
  • Quiz 20: Special Right Triangles

Here’s what’s due in Part 2: Special Right Triangles

  1.  Take notes on special right triangles, their ratios, and their definitions

  2.  Complete HW46: Special Right Triangles

  3.  Complete HW47: Special Right Triangle Applications

  4. Complete HW48:  TrigStar Practice 1
  5. Take Quiz 18: Special Right 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!


 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!