Trigonometry Lesson 3:
Solving Problems involving Right Triangles
There are many applications of trigonometry and the Pythagorean theorem, especially in building construction,
engineering, and navigation. Since diagrams are often not provided, the key to solving many problems is to sketch
and accurately label a diagram.
Example:
A 12-foot ladder is leaning against a vertical wall. If it reaches 9.5 ft. up the
wall, what angle does it make with the level ground, to the nearest degree?
Solution:
First, draw and label a sketch of a representative diagram that includes a right
triangle.
The required angle is labeled θ. The measure of the angle θ can be determined
from the sine ratio.
The ladder makes an angle of approximately 52 with the ground.
Angles of Elevation and Depression
The ladder problem is an example of an angle of elevation.
An angle of elevation is measured between two rays that have a common
starting point called the vertex of the angle. One ray is horizontal, while the
other is above the horizontal.
Angles of elevation are also called angles of depression.
An angle of depression is similar to an angle of elevation, except that the second ray is below the horizontal ray.
Compass Directions and Headings
Compass directions may be given as north (N), south (S), east (E), and west (W), or halfway between these, such as
northwest (NW). Also, directions can be given as the number of degrees east or west, north or south. This is called
a heading. For headings, north or south is listed first. For example, a heading of N40ºE means 40º east of due
north, as shown in this illustration.
40°
Bearings
A bearing is a 3-digit number of degrees between 000º and 360º that indicates a direction
N
measured clockwise from north. For example, a bearing of 127º is in a direction 127º
measured clockwise from due north, as shown below.
Bearing of 127°
127°
Example:
From a vertical lighthouse 200 ft. above the surface of the water, a person sees a sailboat at an angle of
depression of 27º. How far is the sailboat from the water level of the lighthouse, to the nearest foot?
Solution:
Sketch and label a diagram:
The angle of depression from the person in the lighthouse to the sailboat is equal to the angle of elevation from
the sailboat to the person in the lighthouse.
Determine x by applying the tangent ratio.
The sailboat is approximately 393 ft. from the water level of the lighthouse.
Example:
A ship travels from a port at 30 km/h on a bearing of 137º. How far east of the port is the ship after three hours, to
the nearest tenth of a kilometer?
137°
Solution:
After three hours the ship has gone 3 x 30 = 90 km.
This is shown in the given labeled diagram.
90 km
Note that the acute angle between the vertical line
and the direction of travel is 1800 – 1370 = 430
The required distance east is labeled x.
x
The ship has travelled approximately 61.4 km east.
HOMEWORK BOOK:
VOCABULARY BOOK:
vertex
horizontal
clockwise
NAME: _________________________________
CLASS: 10
Word Problems Assignment
1)
A cat is watching a bird in a tree. From the position of the cat, the bird is at an angle of elevation of 40º. If
the cat is 7.1 m from the base of the tree, how high up in the tree is the bird, to the nearest tenth?
Diagram: Draw and label a sketch diagram that includes a right triangle. Let x be the height of the bird.
Solution:
-------------------------------------------------------------------------------------------------------------------------------------------------------2)
An airplane flies for 2.5 hours at 240 km/h on a heading of S25ºW. How far south has the plane flown
during this time, to the nearest km?
Diagram: Draw and label a sketch diagram that includes a right triangle. Let x be the unknown distance south.
Solution:
3)
A ladder is 6.5m long. It leans against a wall. The base of the ladder is 1.2m from the wall. What is the
angle of inclination of the ladder to the nearest tenth of a degree? Sketch a diagram and solve.
4)
A rope that supports a tent is 2.4m long. The rope is attached to the tent at a point that is 2.1m above the
ground. What is the angle of inclination of the rope to the nearest degree? Sketch a diagram and solve.
5)
An airplane approaches an airport. At a certain time, it is 939m high. Its angle of elevation measured from
the airport is 19.50. To the nearest meter, how far is the plane from the airport? Draw a diagram and solve.