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:
Describe the differences between the two ways the ladder is positioned against the wall in the above drawings:
a.
What problems might occur if the ladder is very steep?
b.
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.
Use the Interactive Activity to investigate different levels of steepness, and fill in the chart below. Note 7
This activity requires the Flash plug-in, which you can download for free from Macromedia's Web site. A non-interactive version of this activity is available.
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? Close Tip
As you've now seen, there are several ways to measure the steepness of a ladder. You can measure the angle or you can find the height-to-distance ratio, which is entirely dependent on 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 , or tan = h/d. It is a derived measurement rather than a direct measurement.
Problem C4
What happens to 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 activity 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.
or you can find the height-to-distance ratio, which is entirely dependent on 
= h/d. It is a derived measurement rather than a direct measurement.
