Lesson 3

Lesson 3: Trigonometry Without Technology

 

Focus

 

Hemera/Thinkstock

Bridges have always been needed to cross rivers and ravines. Bridges require sturdy trusses. The exact origin of the truss web, made up of identical repeating triangles, is unclear. Today, many other types of trusses exist. Triangles are one element all the different types of trusses have in common. You probably don’t have to look hard to see the triangles that make up the truss bridge shown in the photo.

 

In 1848, James Warren obtained a British patent for a design of repetitive equilateral triangles that could support a roadbed on either its top or bottom dimension. Warren's name became synonymous with this form of truss design.

 

How did the mathematicians and engineers who worked on bridges solve trigonometric equations without calculators? The vast majority of them used tables of values. One of the oldest known tables of values was created by the Greek astronomer and mathematician Hipparchus (ca. 190 BC to 120 BC). How were those tables created?

 

This lesson will help you understand how to calculate the cosine, sine, or tangent of a number of special angles without a calculator.

 

Outcomes

 

At the end of this lesson you will be able to

• determine, without the use of technology, the cosine, sine, or tangent of 0°, 30°, 45°, 60°, and 90° angles, and their reflections in the other three quadrants of the Cartesian plane
  
  
• determine the cosine, sine, or tangent of angles given the coordinates of a point on the terminal arm

Lesson Question

 

You will investigate the following question:

• How can the cosine, sine, or tangent of an angle be determined without technology?

Assessment

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

• completion of the Lesson 3 Assignment (Download the Lesson 3 Assignment and save it in your course folder now.)
  
  
• course folder submissions from Try This and Share activities
  
  
• additions to Module 2 Glossary Terms and Formula Sheet

Discover

 

Consider the following circle of radius 2 units.

 

One could imagine superimposing angles in standard position on top of this circle. For instance, this is what a 180° angle would look like:

 

 

Since you know the coordinates of a point on the terminal arm and the distance from that point to the origin (r), you can determine the cosine, sine, and tangent of 180° without technology by using the definitions introduced in Lesson 1.

 

 

Try This 1

 

Use the definitions process to determine the cosine, sine, and tangent of the following angles.

 

θ	cos θ	sin θ	tan θ
0°
90°
180°	−1	0	0
270°
360°

 

Save your chart in your course folder.

 

Share 1

 

Compare your results with a partner.

• Discuss any differences and try to come to agreement on all values in the table.
  
  
• How would your values change if the circle had been drawn with a radius of 3 or with any other radius?

If required, save a copy of your discussion in your course folder.

Explore

 

In Discover you determined the cosine, sine, and tangent for 0°, 90°, 180°, 270°, and 360° angles using points on a circle with a radius of 2. In Share 1 you may have discussed the fact that the values for these trigonometric ratios are always the same and don’t depend on the radius of the circle.

 

Finding the trigonometric ratios for these angles was straightforward because the terminal arms of each angle landed on either the x- or y-axis, making the point P(x, y) easily known. But what about other angles whose terminal arms do not lie on the x- or y-axis?

Using Special Triangles to Find Angles

 

In Try This 2 you saw how cosine, sine, and tangent can be determined for a 45° angle without using technology. Watch Finding Trigonometric Ratios for 30° and 60° Angles Without Technology.

You can now create a very basic table of trigonometric values based on the ratios found using

• angles with terminal arms that lie on the x- or y-axis of a circle (0°, 90°, 180°, 270°, and 360°)
  
  
• angles found in special triangles (30°, 45°, 60°)

θ	cos θ	sin θ	tan θ
0°	1	0	0
30°
45°			1
60°
90°	0	1	undefined
180°	−1	0	0
270°	0	−1	undefined
360°	1	0	0

 

So far you have used these special triangles only for angles in the first quadrant (0° ≤ θ ≤ 90°). These triangles can be used to find angles in other quadrants too. The key is to use one of these special triangles to determine the coordinates of a point on the terminal arm of the angle in question. Watch Using a 30-60-90 Triangle for a 150° Angle.

Sometimes it’s not as obvious to see how to use one of the special triangles. Watch Using a 30-60-90 Triangle for a 300° Angle to see how a similar process can be used for a 300° angle.

Special Angles Around a Circle

 

Recall the question posed in Focus. Historically, how did mathematicians and engineers solve trigonometric equations without calculators or other forms of modern technology? The answer is that they used special triangles!

 

These two special triangles (45-45-90 and 30-60-90) were used by trigonometric table makers to determine the exact trigonometric values for all the angles shown in the diagram.

 

 

Other techniques, including trigonometric identities, were used to add more angles to their tables. These developments will be discussed in Mathematics 30-1.

Finding Angles from a Point on the Terminal Arm

 

In Try This 1 you were given the coordinates of endpoints on terminal arms of 0°, 90°, 180°, 270°, and 360° angles and asked to determine the cosine, sine, and tangent of these angles. This process can be extended to other angles where a point on the terminal arm of the angle is known. Consider the following example.

 

Point P(−2, 3) is on the terminal arm of an unknown angle, θ. Determine the exact values for cos θ, sin θ, and tan θ.

 

The first step is to draw a picture.

 

 

Recalling the primary trigonometric ratio definitions, you can write the following ratios:

 

 

The only piece missing is r, the distance from the origin to P(−2, 3). This can be determined by imagining that the terminal arm is the hypotenuse of a right triangle.

 

 

The Pythagorean theorem can be used to determine r.

 

The primary trigonometric ratios of this angle can now be written in full.

 

 

Note that and are mathematically equivalent. By convention, the negative sign is normally written with the numerator.

 

Self-Check 2

Connect

 

Lesson 3 Assignment

Lesson 3 Summary

 

In this lesson you investigated the following question:

• How can the cosine, sine, or tangent of an angle be determined without technology?

You learned how to determine the side lengths of 30-60-90 and 45-45-90 special triangles and how using triangles can be used to determine the cosine, sine, and tangent for each of the following angles.

 

 

 

You also learned how to determine the exact values of cosine, sine, and tangent in situations where you only know the coordinates of a point on the terminal arm.

 

 

You don’t have to memorize the exact values for cosine, sine, and tangent for all sixteen angles shown in Figure 1.

• The values for 0°, 90°, 180°, 270°, and 360° can be easily determined using the coordinates of any point on their terminal arms.
  
  
• The values for 30°, 45°, and 60° can be determined using the special triangles shown above.
  
  
• All other values can be determined by using either 45-45-90 or 30-60-90 special triangles.

Another way to remember the ratios is to remember the red, blue, and green triangles.

 

The special triangles, 30-60-90 and 45-45-90, can be drawn in the unit circle, as shown in Figure 2. Knowing the ratios of the special triangles will determine the exact value of sin θ and cos θ and thus the x- and y-coordinates that lie on the edge of the circle.