Lesson 2

Lesson 2: Solving Linear Trigonometric Equations

Focus

As shown with the semicircular arches, architecture makes extensive use of symmetry. Some of the world’s more impressive buildings are based on circle geometry. The Calgary Tower and the Leaning Tower of Pisa are significantly based on circle symmetry—the same symmetry you’re using to study trigonometry.

The circle symmetry in the Calgary Tower makes the Tower’s famous revolving restaurant possible. Circle symmetry also makes the structure resistant to earthquakes and strong winds. The Tower of Pisa could be seen as circle symmetry gone wrong. On closer examination, however, the symmetry is sound—an engineering error causes the tilt. The fact that the Leaning Tower of Pisa stands nearly 1000 years after construction was started speaks to the strength of the circle.

In Lesson 1 you used Primary Trigonometric Ratios: Formal Definitions to determine the two angles at which a boom repairing a semicircular arch would be 4 m above the ground. The equation to be solved was sin θ = 0.8. In Lesson 2 you will learn to use your calculator and symmetry to solve equations like these.

Outcomes

At the end of this lesson you will be able to

• explain how symmetry can be used to solve equations of the form sin θ = k, where −1 ≤ k ≤ 1

• solve equations of the form sin θ = k, where −1 ≤ k ≤ 1 (for each of sine, cosine, and tangent)

Lesson Question

In this lesson you will investigate the following question:

• How does circle symmetry help solve equations of the form sin θ = k, where −1 ≤ k ≤ 1?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 2 Assignment (Download the Lesson 2 Assignment and save it in your course folder now.)
  
  
• course folder submissions from Math Lab, Try This, and Share activities
  
  
• additions to Module 2 Glossary Terms and Formula Sheet

Materials

You will need the following supplies:

• three copies of Circle of Angles
  
• a pencil
• a ruler

Discover

In Lesson 1 you used the applet Primary Trigonometric Ratios: Formal Definitions to solve equations like cos θ = 0.7. You found that there were always two answers to this equation. In previous math courses you may have found one solution to this by writing θ = cos−1(0.7) then entering cos−1(0.7) into your calculator to get the answer 46°. How do you get the second solution (314°) without having to use the applet Primary Trigonometric Ratios: Formal Definitions?

In Try This 1 you will explore the relationship between the answer given by your calculator and the second answer given by the applet.

Try This 1

If you are a PC user, you can copy the image in Circle of Angles by clicking on the image and then pressing the print screen key on your keyboard. “PrtSc” is next to the F12 key. You can then paste the image into a blank document. To copy and paste using a Mac, press Command + Shift + 4, followed by the spacebar, and then click on the active window. This will take a screenshot of the window and save it as a file on the desktop.

Print three copies of Circle of Angles. You will use one copy in each part of Try This 1. Do you prefer to work digitally? If so, you can paste a screenshot of Circle of Angles into a drawing program on your computer.

You will be using Primary Trigonometric Ratios: Formal Definitions, which you first used in Lesson 1.

Part 1

Step 1: Use Primary Trigonometric Ratios: Formal Definitions to determine the two values of θ that make the equation cos θ = 0.309 true.

Step 2: Draw these two angles in standard position on one of the Circle of Angles worksheets you printed. Mark a large point where each terminal arm intersects the circle.

Step 3: Fold the worksheet along the thick horizontal line (the x-axis). Record what you notice, if anything, about the two points on the terminal arms.

Step 4: Fold the worksheet along the thick vertical line (the y-axis). Record what you notice, if anything, about the two points on the terminal arms.

Repeat Steps 1 to 4 for the following equations.

• cos θ = 0.9135
  
  
• cos θ = −0.6691
  
  
• cos θ = −0.9976

Part 2

Step 1: Use Primary Trigonometric Ratios: Formal Definitions to determine the two values of θ that make the equation sin θ = 0.4695 true.

Step 2: Draw these two angles in standard position on the second Circle of Angles worksheet you printed. Mark a large point where each terminal arm intersects the circle.

Step 3: Fold the sheet along the thick horizontal line (the x-axis). Record what you notice, if anything, about the two points on the terminal arms.

Step 4: Fold the sheet along the thick vertical line (the y-axis). Record what you notice, if anything, about the two points on the terminal arms.

Repeat Steps 1 to 4 for the following equations:

• sin θ = 0.9613
  
  
• sin θ = −0.9272
  
  
• sin θ = −0.4384

Part 3

Step 1: Use Primary Trigonometric Ratios: Formal Definitions to determine the two values of θ that make the equation tan θ = 1.8807 true.

Step 2: Draw these two angles in standard position on the third Circle of Angles worksheet you printed. Mark a large point where the terminal arms intersect the circle.

Step 3: Fold the sheet along the thick horizontal line (the x-axis). Record what you notice, if anything, about the two points on the terminal arms.

Step 4: Fold the sheet along the thick vertical line (the y-axis). Record what you notice, if anything, about the two points on the terminal arms.

Repeat Steps 1 to 4 for the following equations:

• tan θ = 0.364
  
  
• tan θ = −0.2126
  
  
• tan θ = −2.4751

Save your responses in your course folder.

Share 1

For each part of Try This 1, discuss the following questions with a partner or group.

• Does there appear to be a pattern to the relationship between each pair of angles you plotted?
  
  
• Can you see any relationship between the patterns you identified in Try This 1 and the primary trigonometric ratios introduced in Lesson 1?

Save a summary of the patterns and relationships you discussed in your course folder.

You will refer to your notes in Explore.

Explore

In Try This 1 you worked through three activities to identify some patterns for finding the solutions to equations in the form cos θ = k; sin θ = k, where −1 ≤ k ≤ 1; and tan θ = a, . In Try This 2 you will identify and confirm the patterns you may have noticed in Try This 1.

Try This 2

1. Work through Cosine Reflection Equation to help you identify methods for finding solutions to equations of the form cos θ = k, where −1 ≤ k ≤ 1.
   
   
    
   
   
   
   
2. Work through Sine Reflection Equation to help you identify methods for finding solutions to equations of the form sin θ = k, where −1 ≤ k ≤ 1.
   
   
    
   
   
   
   
3. Work through Tangent Reflection Equation to help you identify methods for finding solutions to equations of the form tan θ = a, .
   
   
    
   
   
   
   
4. Look back at the patterns you described in Share 1. How do they compare with the patterns you observed in the animations above?
   
   
5. Summarize your observations by creating three general rules that describe the relationship between the axis of reflection of the two solutions for cos θ = k, sin θ = k (−1 ≤ k ≤ 1), and tan θ = a, .

Save your responses in your course folder.

The Relationship Between the Solutions and the Axis of Reflection

In Try This 2 you developed three general rules to describe the relationship between the solutions for cos θ = k, sin θ = k (−1 ≤ k ≤ 1), and the axis of reflection. Compare the rules you developed to the following rules.

Here is a way to help you remember which trigonometric ratio is reflected in which axis:

• The definition for cosine involves the x-coordinate of the endpoint of the terminal arm P(x, y), . The second solution of an equation like cos θ = k is a reflection in the x-axis.
  
  
• The definition for sine involves the y-coordinate of the endpoint of the terminal arm P(x, y), . The second solution of an equation like sin θ = k is a reflection in the y-axis.
  
  
• The definition for tan involves both the x- and y-coordinates of the endpoint of the terminal arm P(x, y), . The second solution to an equation like tan θ = k is a reflection in both the x- and y-axes.
  

Remember to watch your positive and negative signs when determining x- and y-values.

Lesson 2 Summary

In this lesson you learned that, with the exception of sin θ = ±1 and cos θ = −1, all equations of the form cos θ = k, where −1 ≤ k ≤ 1, have two solutions when angles are limited to 0° ≤ θ ≤ 360°.

You learned about each type of equation.

Sine: sin θ = k, where −1 ≤ k ≤ 1

• The first solution, θ1, can be obtained by entering sin−1(k) in your calculator.
  
  
• The second solution will have a terminal arm that is a reflection in the y-axis of θ1’s terminal arm.
  
  
• Your calculator may give you a negative angle. This negative angle corresponds to a clockwise rotation and can be converted to a positive angle using 360° − θ1.

Cosine: cos θ = k, where −1 ≤ k ≤ 1

• The first solution, θ1, can be obtained by entering cos−1(k) in your calculator.
  
  
• The second solution will have a terminal arm that is a reflection in the x-axis of θ1’s terminal arm.

Tangent: tan θ = a, a ∈

• The first solution, θ1, can be obtained by entering tan−1(k) in your calculator.
  
  
• The second solution will have a terminal arm that is a reflection in the x-axis and y-axis of θ1’s terminal arm. As a result, the two terminal arms form a straight line.

In all cases, circle symmetry was used to determine the second solution.