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Learning Math: Measurement

Indirect Measurement and Trigonometry Part C: Steepness and Trigonometry (45 minutes)

Session 5, Part C

In This Part

  • Measuring Steepness
  • Examining Ratios and Angles
  • The Tangent

Measuring Steepness

Since early times, surveyors, navigators, and astronomers have employed triangles to measure distances that could not be measured directly. Trigonometry grew out of early astronomical observations, such as those of Hipparchus of Alexandria (140 B.C.E.). The word trigonometry comes from the ancient Greeks and literally means “triangle measurement.” In Part C, we explore right-triangle trigonometry, which provides us with another method of deriving angles and lengths when we can’t measure directly. Instead of using similar triangles, trigonometry is based on ratios of the sides of a right triangle that correspond to various angle measures. Note 6

While in this session we are only using the tangent ratio to measure indirectly, there are a total of six ratios associated with any angle in a right triangle: sine, cosine, tangent, cotangent, secant, and cosecant. Earlier mathematicians recorded these ratios and corresponding angles in tables they used for calculations, but today most of us use a scientific calculator to find this information.

The following drawings show two side views of the same ladder leaning against a wall:

 

 


Problem C1

Describe the differences between the two ways the ladder is positioned against the wall in the above drawings:

  1. What problems might occur if the ladder is very steep?
  2. What problems might occur if the ladder is not steep enough?

 


As the steepness of the ladder changes, the following measures also change:

  •  The height on the wall that is reached by the top of the ladder
  • The distance between the foot of the ladder and the wall
  • The angle between the ladder and the ground (often called the angle of elevation)

 


Problem C2

Let’s investigate different levels of steepness by using a ruler to represent a ladder, and an upright book or box to represent a wall, like this:

The angle between the height h and distance d must be 90 degrees.

Fill in the chart with five different sets of measurements:

 

Ratio (height:distance)

Angle of Elevation

 


Problem C3

What patterns do you notice between the height-to-distance ratios and the angles?

Look for general patterns. For example, what types of ratios result in small angles of elevation? What types result in large angles of elevation? When h = d, what is the angle of elevation?

 


As you’ve now seen, there are several ways to measure the steepness of a ladder. You can measure the angle ∠α or you can find the height-to-distance ratio, which is entirely dependent on ∠α and not the length of the ladder. This ratio can be expressed as a fraction, decimal, or percent. The ratio h:d is also called the tangent of ∠α, or tan α = h/d. It is a derived measurement rather than a direct measurement.

 


Problem C4

What happens to ∠α as the height-to-distance ratio increases?

 


Problem C5

Why must the angle between the height and the distance be 90 degrees?

 


Problem C6

Use the ratios below to determine the tangent of the various angles. Record the relationships in the table using tangent notation. For example, if you have an entry with a height-to-distance ratio of 3/3 and an angle measure of 45 degrees, you can record this relationship as tan 45° = 1.

Ratio (height:distance)

Angle
of Elevation

Tangent
Notation

5.5/9.5 30°
7.8/7.8 45°
9.5/5.5 60°
10.6/2.8 75°

 

Examining Ratios and Angles

Problem C7

Use a protractor or angle ruler and a ruler to make side-view scale drawings of a ladder leaning against a wall for each of the following situations. Label the measures of ∠α and the lengths h and d, and find the height-to-distance ratios. Record your answers in the table below, or print and use the Ladders Worksheet (PDF).

  1. α = 45°
  2. h = 2, d = 1
  3. α = 30°
  4. h = 1, d = 2
  5. α = 60°

 

Problem

Measure

h:d Ratio

Ratio as Decimal

a 45°
b 2:1
c 30°
d 1:2
e 60°

 

When given ∠a, start by drawing the angle. Next, choose an integer (a whole number) as the length for the distance from the wall. Draw the right angle and complete the triangle. Finally, measure the height and determine the h:d ratio.

 


Problem C8

Examine Problem C7 (b) and (d) above. What do you notice about the h:d ratio and the measure of ∠α?

 


Problem C9

We’ve slightly revised the table from Problem C7 to include the 15- and 75-degree angles:

Problem

Measure

Ratio as Decimal

a 15° 0.27
b 45° 1
c 30° 0.58
d 60° 1.72
e 75° 3.73

 

Using the data from this table, plot and connect the points on the Steepness Graph of the height-to-distance ratios for a ladder leaning against a wall at different angles. Use the Ladders Worksheet (PDF) to complete the solution.

  1. Steepness Graph:
  2. Examine the information in the Steepness Graph. What happens to ∠α as the h:d ratio increases?

 

Video Segment

In this video segment, participants explore the relationship between the angle of elevation and the height-to-distance ratio. They graph their data to see what happens to the ratio as the angle increases.

Were your findings similar? How would you explain this relationship in your own words?

You can find this segment on the session video approximately 15 minutes and 43 seconds after the Annenberg Media logo.

 


Problem C10

Suppose that it is safe to be on a ladder when the h:d ratio is larger than 2 and smaller than 3. Give a range of angles at which the ladder can be positioned safely.

 

The Tangent

The tangent is the ratio of vertical height to horizontal distance in any context that can be represented with a right triangle. The tangent ratio provides useful information about steepness and can help us determine the measure of ∠α, which is often referred to as the angle of elevation. Tables are available that show the relationship between the size of an angle and its tangent. Today, however, most people use the tangent key on a scientific calculator to obtain this information.

Experiment with your calculator, using the data in the table below for verification, to find the following:

  • The corresponding tangent, given an angle
  • The corresponding angle, given a tangent. Note 8

 

Angle
(in Degrees)

Tangent

45 1
46 1.036
47 1.072
48 1.111
49 1.15
50 1.192
51 1.235
52 1.28
53 1.327
54 1.376

 

We can use right-angle trigonometry to solve problems like those in Part A.

To determine the distance (b) of a tree (point C) across a river, first locate point B directly across the river (where the sighting line is perpendicular to the bank). Next we locate point A at some distance c on the same side of the river as B, and we physically measure the distance between the two (for example, 30 m). Standing at A, we sight C and measure ∠α (using a transit), which we determine to be 52 degrees:

d

Finally, we set up the tangent ratio to determine the distance between B and C:

tan 52° = b/c

Using this information, let’s calculate the distance b.

 


Problem C11

Find the width of the river at C (distance b) when ∠α is 52 degrees. Note 9

Use algebra to solve the equation tan 52° = b/30. First use the trig key on your calculator to find the value of tan 52°, and then multiply both sides by 30.

 


Other types of problems can also be solved using the tangent. For example, hang gliders are interested in the steepness of their glide paths. The angle that the hang glider makes with the ground as it descends is called a glide angle ( in the figure below):

 


Problem C12

When a hang glider travels a long distance, it is less likely to crash. Three gliders’ height-to-distance ratios (sometimes referred to as glide ratios) are given. Sketch the right triangles to show the glide paths. Which glider is the safest?

  • Glider 1 — 1:27
  • Glider 2 — 0.04
  • Glider 3 — 3/78

Rewrite the ratios for Gliders 2 and 3 as unit ratios (1:x). Then sketch the right triangles whose sides correspond to the unit ratios for each glider.

 


Problem C13

If the glide angle of a glider is 35 degrees, how much ground distance does a glider cover from a height of 100 m?

 

Notes

Note 6

When considering an angle and the height-to-distance ratio formed by that angle, it is common to name the angle with a Greek letter. The first letter in the Greek alphabet is α (alpha), the second letter is β (beta), and the third letter is γ (gamma). In this session, we frequently refer to ∠α and ∠β.


Note 7

Take your time working through the problems in order to make sense of the relationships, as future questions will build on what you learn here.

 


Note 8

If you have not used a scientific calculator recently, first obtain the information in the table given in this section. On some calculators, you enter the angle measure (e.g., 46) and then press the tangent button to get the tangent ratio (e.g., 1.035530314). (Note that the data in the table have been rounded, in this case to 1.036.)

What if you have the h/d tangent ratio and want to find the corresponding angle? In that case, you’d enter the ratio first (e.g., 1.036) and then press the inverse of the tangent key (e.g., tan-1), which is usually a “second” function, to obtain the angle measure (e.g., 46.01298288). This number can also be rounded.

 


Note 9

Notice that this problem is not very different from what you did in Part A. Using a tangent ratio is a kind of shortcut. When you drew similar triangles and set up the proportion between the corresponding sides, you were really calculating the tangent ratio of the triangle with known dimensions, and you then used that to find the unknown length on the other triangle. Now you can calculate the same ratio without the middle step of drawing a similar triangle — you just need to find the tangent of the angle and then use it in the equation.

 

Solutions

Problem C1

  1. If the ladder is too steep, it may be difficult to climb, and there is a good chance the ladder will fall over backward (say, in a strong wind).
  2. If the ladder is not steep enough, it may also be difficult to climb, it is likely to fall forward, and it may not reach high enough to be useful.

 


Problem C2

Answers will vary. You should find that as the angle between the ground and the ladder increases, the height that the ladder reaches on the wall increases while the distance from the base decreases.

 


Problem C3

When the angle is 45 degrees, the height and distance are equal. When the angle is larger than 45 degrees, the height-to-distance ratio is greater than 1, and when the angle is smaller than 45 degrees, the ratio is less than 1. Importantly, this ratio is based entirely on the angle, rather than on the length of the actual ladder used.

 


Problem C4

As the ratio increases, the angle increases, but it will always be less than 90 degrees.

 


Problem C5

By common definition, height is measured along a line that is perpendicular to the base, so the angle must be 90 degrees. If the angle were not 90 degrees, we would not be measuring the vertical height of the ladder against the

 


Problem C6

Answers will vary. (Tan 30° is about 0.58, while tan 60° is about 1.73.)

 


Problem C7

Answers may vary. Here are some possibilities:

 

Problem

Measure
∠α

h:d Ratio

Ratio as Decimal

a 45° 5:5 (or 1:1) 1
b 63° 6:3 (or 2:1) 2
c 30° 3.5:6 0.58
d 27° 3:6 (or 1:2) 0.5
e 60° 6:3.5 1.72

 


Problem C8

The ratios are reciprocals (2:1 and 1:2), while the angles are complementary (they sum to 90 degrees). One way to think about this is that if we “reversed” the triangle (switched h and d), we should also reverse the angles in the triangle. The 90-degree angle remains fixed, so the other two angles will switch. Since they are complementary, if one is 63 degrees, the other is 27 degrees, and vice versa. While it might be easy to see that the measures of both angles sum to 90 degrees, seeing that the h:d ratios are inverses may not be as obvious.

Notice that the angles in Problem C7 (c) and (e) are 30 and 60 degrees respectively, and are also complementary angles.

 


Problem C9

  1. Here is the completed Steepness Graph:
  2. The h:d ratio increases as the measure of increases, but the ratio is increasing at a greater rate. As the angle approaches 90 degrees, the ratio grows increasingly large (with no limit!). Try drawing a triangle with an 85-degree angle and then measure the h:d ratio. It will be very large! Notice how this is shown on the graph where the curve becomes steeper after the 45-degree mark.

 


Problem C10

This range can be determined by drawing triangles or by referring to a table of values for these ratios. The smallest safe angle is about 63 degrees (see Problem C7, part b), while the largest is about 72 degrees.

 


Problem C11

Tan 52° = b/30, or 1.28= b/30. Multiplying both sides by 30 yields the width of the river, which is 38.4 m.

 


Problem C12

One approach is to rewrite these ratios as unit ratios and then compare them. The ratios are as follows:

  • Glider 1 — 1:27
  • Glider 2 — 1:25
  • Glider 3 — 1:26

Glider 1 has the smallest glide ratio, so it can travel farther (27 m for every 1 m that it descends), and it descends at the slowest rate; therefore, it is the safest.

Another approach is to convert the ratios to decimals and then compare them (this time, looking for the smallest decimal).

 


Problem C13

Using a calculator, we see that tan 35° = 0.70. Since we know that tan 35° = h/d, we just plug in the numbers:

0.70 = 100/d

The distance is 100/0.7, or approximately 143 m.

 

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Learning Math: Measurement

Credits

"Steepness and Trigonometry" adapted from Looking at an Angle. Mathematics in Context. © 1998 by Encyclopedia Britannica Educational Educational Corporation. Used with permission. All rights reserved.

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