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Trigonometry - Word Problems
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Trigonometry's connection to measurement places it in the learner's manuals for a wide variety o
professions. Carpenters, construction workers, designers, architects, and engineers, to name a
few, deal with measurements, and as such, they deal with triangle measures, or trigonometry.
Combining your skills with similar triangles, trigonometry and the Pythagorean Theorem, you ar
ready to tackle problems that are connected to more real world scenarios. The situations you will
be examining will be specifically related to right triangles, and you will be using our three main
trigonometric functions. Once a diagram is established, the mathematical solution will be the
same as those shown on Solving for Sides or Solving for Angles.
There are two new vocabulary terms that may appear in application problems.
Angle of Elevation:
The angle of elevation is always measured from the
ground up. It is an upward angle from a horizontal
line. It is always inside the triangle.
You can think of the angle of elevation in relation
to the movement of your eyes. You are looking
straight ahead and you must raise (elevate) your
eyes to see the top of a tree.
In this diagram, xº marks the
angle of elevation of the top of the tree
as seen from a point on the ground.
When trying to remember the
meaning of an angle of
elevation think of an elevator
that only goes up!
Angle of Depression:
The angle of depression is always OUTSIDE the
triangle. It is never inside the triangle. It is a
downward angle from a horizontal line.
You can think of the angle of depression in relation
to the movement of your eyes. You are standing at
the top of the lighthouse and you are looking
straight ahead. You must lower (depress) your eyes
to see the boat in the water.
In this diagram, xº marks the
angle of depression of the boat at sea
from the top of the lighthouse.
Notice how the horizontal line in the angle of depression diagram is PARALLEL to the ground
level. The fact that horizontal lines are always parallel guarantees that the alternate interior
angles are equal in measure. In the diagram, the angle marked xº is equal in measure to
m∠BAC. Simply stated, this means that ...
→ the angle of elevation = the angle of depression ←
When solving a problem with an angle of
depression you need to find the measure of an
angle INSIDE the triangle. There are two options:
Option 1: find the angle inside the triangle that
is adjacent (next door) to the angle of depression.
This adjacent angle will always be the complement
of the angle of depression, since the horizontal line
and the vertical line are perpendicular (90º). In the
diagram at the left, the adjacent angle is 52º.
Option 2: utilize the fact that the angle of
depression = the angle of elevation and label ∠BAC
as 38º inside the triangle.
Notice that both options, the answer is the same.
Let's see how to put these skills to work in word problems.
A nursery plants a new tree and attaches a
guy wire to help support the tree while its
roots take hold. An eight foot wire is
attached to the tree and to a stake in the
ground. From the stake in the ground the
SOLUTION:
• A "guy" wire is a support wire used to hold
a newly planted tree in place, preventing it
from bending or up-rooting during high
winds.
• The "angle of elevation" is from the ground
angle of elevation of the connection with the
tree is 42º. Find to the nearest tenth of a
foot, the height of the connection point on
the tree.
up.
• It is assumed that the tree is vertical, making
it perpendicular with the ground.
• This problem deals with "opposite" and
"hypotenuse" making it a sine problem.
•
From the top of a fire tower, a forest ranger
sees his partner on the ground at an angle of
depression of 40º. If the tower is 45 feet in
height, how far is the partner from the base
of the tower, to the nearest tenth of a foot?
SOLUTION:
• Remember that the "angle of depression" is
from a horizontal line of sight downward.
• It is assumed that the tower is vertical,
making it perpendicular with the ground.
• This solution will use alternate interior
angles from the parallel horizontal lines, so
place 40º inside the triangle by the partner
(bottom right).
• This solution deals with "opposite" and
"adjacent" making it a tangent problem.
•
Find the shadow cast by a 10 foot lamp post
when the angle of elevation of the sun is
58º. Find the length to the nearest tenth of a
foot.
SOLUTION:
• Remember that the "angle of elevation" is
from the horizontal ground line upward.
• It is assumed that the lamp post is vertical,
making it perpendicular with the ground.
• Shadows are on the ground! If you place the
"shadow" on the hypotenuse you have created
an apparition ( a "ghost"), not a shadow!
• This solution deals with "opposite" and
"adjacent" making it a tangent problem.
•
Not all trigonometry word problems will use the terms "angle of elevation" or "angle of
depression". You may need to read carefully to see where to indicate the angle in the problem.
A ladder leans against a brick wall. The foot
of the ladder is 6 feet from the wall. The
ladder reaches a height of 15 feet on the wall.
SOLUTION:
• In this problem place xº where the ladder
meets the wall. Do not assume that the angle
Find to the nearest degree, the angle the
ladder makes with the wall.
will always be at the ground level.
• It is assumed that the wall is vertical,
perpendicular with the ground.
• The foot of the ladder is the bottom of the
ladder, where it hits the ground.
• This solution deals with "opposite" and
"adjacent" making it a tangent problem.
•
It may be the case that a problem will be composed of two overlapping right triangles.
A radio station tower was built in
two sections. From a point 87 feet
from the base of the tower, the
angle of elevation of the top of the
first section is 25º, and the angle of
elevation of the top of the second
section is 40º. To the nearest foot,
what is the height of the top section
of the tower?
SOLUTION:
• Think of this problem as working with two separate triangles:
(1) the larger triangle with the 40º angle and a vertical side that represents the ENTIRE height,
b, of the tower, and
(2) the smaller triangle with the 25º angle and a vertical side, a, that represents the height of the
first (bottom) section of the tower.
• Solve for the vertical heights (b and a) in the two separate triangles.
• The needed height, x, of the second (top) section of the tower will be the difference between
the ENTIRE height, b, and the height of the first (bottom) section, a. You will need to subtract.
• In both triangles, the solution deals with "opposite" and "adjacent" making it a tangent
problem.
• Larger triangle with height b:
• Smaller triangle with height a:
• Difference (b - a): 73.00166791 - 40.56876626 = 32.43290165 ≈ 32 feet
Notice, in this problem, that the trigonometric functions could not work directly
on the side labeled "x" because that side was NOT the side of a right triangle.
Remember to set your graphing
calculator to
DEGREE MODE.
For help with
trig ratios on
your calculator,
click here.
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Terms of Use Contact Person: Donna Roberts
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