Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 5, Part C:
Steepness and Trigonometry

In This Part: Measuring Steepness | Examining Ratios and Angles | 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 , 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 (using a transit), which we determine to be 52 degrees:

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 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.   Close Tip 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.    Close Tip 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?

Next > Homework

 Session 5: Index | Notes | Solutions | Video