Math 20-1

Chapter 2 Trigonometry

Lesson 2.3  The Sine Law

Support Documents

Answer pages

You should be checking your work for EVERY question you do!

To make that easy, you can download the solutions files which have the answers for all the practice problems in the text.

You can print it, though it would be a lot of pages.

I think the best choice would be to keep the files handy, and have them open and ready every time you are working on math.

This saves you the time of flipping to the back of the book every time you need to check an answer.

FOLDER LINK

Formula Page

Math 20-1 is an academic course and you will be expected to memorize a lot of what you are covering in class.

Your teacher will have expectations for the things you will see on a formula page.

The link below is to formula page that you can download and print to use with your course.

You may or may not be allowed to use this on tests, and you may or may not be allowed to ADD your own notes to these pages.

Use of this page depends on your teacher.

What goes on this page is also subjective, so it is subject to change!

Formula Page

Add ons

The Ambiguous Case  (Sine Law)
THIS PAGE summarizes all the possible scenarios that result when you are given SSA for a triangle

Focus on...

Lesson Objectives

•	using the primary trigonometric ratios to solve problems involving triangles that are not right triangles

•	recognizing when to use the sine law to solve a given problem

•	sketching a diagram to represent a problem involving the sine law

•	explaining a proof of the sine law

•	solving problems using the sine law

•	solving problems involving the ambiguous case of the sine law

Focus on...  Activity

The FOCUS section introduces the main concept that will be covered in the lesson.

TEXT Page 100

Read through the  Focus  section at the top of page 100 in  your textbook.

Demonstration

Before you go through the investigation, you might want to check out the demonstration that follows.
If provides the motivation for the ideas of this lesson.

Video Tutorial (DEMO)

Interactive web site

Here is another demonstration of “the sine law”,  giving you some information to hopefully prepare you for the ideas of this lesson

Video Tutorial  Geogebra A

LINK

Investigate  the Sine Law

The investigation provides the background for the ideas presented in the lesson.

TEXT Pages 100 - 101

Read/Work through the  Investigate  section on Pages 100 - 101 in your textbook.

A list of materials  you need for the Investigation is given in the side bar.

Your work will be a lot more efficient if you PLAN for and GATHER these items BEFORE you start the Investigation!

You may not have other students to work with.  
No worries!  
If you are working on your own,  answer the questions, AS BEST YOU CAN, before you view the Tutorial that follows.
Once you have done that,  view the video for help with this section.

Investigate the Sine Law

Video Tutorial Question 1   INV 1A

Video Tutorial Question 1   INV 1B

Video Tutorial Questions  2 - 5   INV B 1

Video Tutorial Questions  2 - 5   INV B 2

Reflect and Respond

Question 6  Hopefully you can answer this based on the previous questions

Video Tutorial Question 7   INV C

Question 8  You should do this in your own words !

Video Tutorial Question 9   INV D

Link the Ideas

Link the Ideas summarizes the math in the lesson.
A presentation of ideas connected to the lesson focus.

TEXT Pages 102 - 103

Read/Work through the  Link the Ideas  section on Pages 102 - 103 in your textbook.

The proof presented here was covered in the Investigation.

TEXT Pages 104 - 105  (you may want to do Examples 1 - 2 first)

Read/Work through the  Link the Ideas  section on Pages 104 - 105 in your textbook.

I’ve created a demonstration and a video (see below) to cover this part of the lesson.

The Ambiguous case

Introduction  AMB 1

Angle A is acute, side length a is less than the height , no solution  AMB 2

Angle A is acute, side length a is equal to the height , one solution  AMB 3

Angle A is acute, side length a is greater than the height ,  but less than side length b,  one solution  AMB 4

Angle A is acute, side length a is greater than the height , and greater than or equal to side length b,  one solution  AMB 5

Angle A is obtuse, side length a is less than side b , no solution
Angle A is obtuse, side length a is equal to  side b , no solution
Angle A is obtuse, side length a is greater than  side b , one solution      AMB 6

Interactive

You can use the interactive demonstration below,  or the links to Geogebra Applets online, to explore the ambiguous case of the sine law.

Ambiguous case explorer

The Sine Law is a powerful tool for solving oblique triangles.
There is a situation where the information provided may not be enough to uniquely determine a triangle.
This is called the SSA case,  where you are given two side lengths, and one angle (and the angle is NOT between the given sides)
If you draw this,  it might look something like what you see in the demonstration below.

Your textbook gives ALL the possible scenarios,  and I’ve listed them below.
You can visually explore ALL of these scenarios using the demonstration.

I will put this summary on a single page that you can download and print to have in your notes.
(see above, under formula page add ons)

If the given angle (A) is acute:
Find the height h, from the third side to the end of side b.   h = b sin A
I)	If  the length of side a is less than h,  no possible triangle can be drawn.
	(side a is too short to reach the third side)  
II)	If a = h,  side a just reaches the third side and is perpendicular to it, one possible triangle can be drawn,  a right triangle. 
III)	If a > b > h,  one possible triangle can be drawn.  
IV)	If h < a < b,  two possible triangles can be drawn, one with angle B acute and one with angle B obtuse

If the given angle is obtuse:
V)	If the length of side a is less than the length of side b,  there is no possible solution 
VI)	If the length of side a is equal to the length of side b,  there is no possible solution 
VII)	If the length of side a is greater than the length of side b,  there is one possible solution.

Geogebra

The Ambiguous Case  LINK

The Ambiguous Case  LINK  (use this one for angles greater than 90°)

Examples

Examples will give you TOOLS for solving problems, and also show you that there may be different ways to approach a question.

Example 1  Determine an Unknown Side Length

TEXT Pages 102 - 103

Work through Example 1 on Pages 102 - 103 in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Example 2  Determine an Unknown Angle Measure

TEXT Page 104

Work through Example 2 on Page 104  in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Example 3  Use the Sine Law in an Ambiguous Case

TEXT Pages 106 - 107

Work through Example 3 on Pages 106 - 107 in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Key Ideas

Key Ideas summarizes the main new concepts of the lesson.

•	Carefully read through each of the bullet points in this section.

•	If you are unclear about any of these ideas,  ASK YOUR TEACHER FOR HELP!

Check Your Understanding    (Practice Problems)

Doing Practice Problems

Practice problems reinforce the math presented in this lesson.

•	Work through the practice problems for this lesson.
	These problems are for practice, you don’t need to hand them in!
	(your teacher may assign specific questions for you to complete)

•	Use the answers in the back of the book to CHECK YOUR WORK!

•	If you make a mistake, TRY TO FIX IT!

•	If you need help or want more information, view the video solutions provided.

•	If you still need more help,  CONTACT YOUR TEACHER!

NOTE!
There are MANY practice questions.
In the interests of time,  I will be selectively skipping some of them.
You have access to the solutions, and you can use those for help as needed.
If the presentation seems straight-forward, I won’t create a video  (or just do one part of a multi-part question).
I’m hoping that as this course continues to be developed, that if MORE HELP is needed for a specific question,  the missing videos can be added at a later date.

Types of Practice Problems

Practice 
These questions allow you to check your understanding of the concepts. 
You can often do the first few questions by checking the Link the Ideas notes or by following one of the worked Examples.

Apply 
These questions ask you to apply what you have learned to solve problems. 
You can choose your own methods of solving a variety of problem types.

Extend 
These questions may be more challenging. 
Many connect to other concepts or lessons. 
They also allow you to choose your own methods of solving a variety of problem types.

Create Connections
These questions focus your thinking on the Key Ideas and also encourage communication. 
Many of these questions also connect to other subject areas or other topics within mathematics.

Mini Lab
These questions provide hands-on activities that encourage you to further explore the concept you are learning.

Notes

Question 7,    I didn’t do a video for this one.

Question 10,    I didn’t do a video for this one.

Question 12,    I didn’t do a video for this one.

Question 15,    I didn’t do a video for this one.

Question 19,    I didn’t do a video for this one.

Question 20,    I didn’t do a video for this one.

Video Solution Links

Check your work for every problem you do!
If you make a mistake,  try to fix it!
Use the video solutions for more information.
If a problem still doesn’t make sense, ASK YOUR TEACHER FOR HELP!

({{"Question 1", "Question 2", "Question 3", "Question 4", "Question 5"}, {"Question 6",  , "Question 8", "Question 9",  }, {"Question 11",  , "Question 13", "Question 14",  }, {"Question 16", "Question 17", "Question 18",  ,  }, {"Question 21", "Question 22", "Question 23", , }})

I didn’t create a video for any of the Create Connections problems.

Question CC 26  is a great one,  with lots of connections between trig values, spirals, the golden ratio, etc.
But...  certainly an extend problem.
Below is a demonstration of the steps in the process,   I haven’t had time yet to add the spiral, but I’m sure you can see where it would go.