7.5 Applications of Right Triangle Trigonometry

Now that we can identify and solve for missing measures a right triangle with Sine, Cosine and Tangent to solve for missing measures in a right triangle.  We can look at some applications of right triangle trigonometry.

Angle Elevation or Depression.

When solving word problems, it is important to understand the terminology used to describe angles. In trigonometric word problems, the terms angle of elevation and angle of depression are commonly used. Both of these angles are always measured from a horizontal line as shown in the diagrams below.

For example, if we are standing on a hill and looking up at the Flying Pig, the angle of elevation would be the angle between our line of sight when looking strait ahead and the line of sight when looking at the Flying Pig.  Likewise, if we are looking down at the 30 Point Buck, the angle of depression would be the angle between our line of sight when looking straight ahead and the line of sight when looking at the 30 Point Buck.  We can also identify heights ( or drops ) and distances.

Note: Always draw a picture of the situation to help you create the right triangle.  ALWAYS!!!

Example A

An airplane approaching an airport spots the tower next to the runway at an angle of depression of 25o.   If the airplane is at an altitude of 15,000 ft, how far is the plane from the runway? Give your answer to the nearest mile.

Solution:

Step 1, Draw a diagram from the given information:
Step 2, Set-up a Right Triangle with the given information:

Step 3, Set-up Trigonometry Ratio(s) ( Sin, Cos, or Tan ) and solve:


Watch the video:

 

 

You Try It:

An eagle nest is on top of a high line electricity transmission tower which is 180 ft above the ground.
When you are looking at the eagle nest, the angle of elevation is 15 degrees.
How far away are you from the transmission tower?

 

Pitch, Slope, or Percent Grade

There are many applications of Right Triangle Trigonometry Ratios that can be found in construction as well.

The pitch of a roof :

Standard Building Code for stairs = 7in : 11in

ADA Ramp Specifications for new construction = 1:12 ramp slope ratio

Percent Grade of a steep hill = fall/rise. For a 6% grade you would drop down 6 ft for every 100 ft you go forward.

 

Example B

Rafters need to be custom made for a shed that is 45 feet wide with 2 foot wide overhang on each side of the shed.  Sheet steel panels are to be cut to length so that they fit from the eaves to the peak of the roof.  What will be the length of the 1-piece steel panels need to be, to the nearest half-inch, if the roof needs to have a 9 : 12 pitch.

Solution:

Step 1, Draw a diagram from the given information:
Step 2, Set-up a Right Triangle with the given information:
Step 3, Set-up Trigonometry Ratio(s) ( Sin, Cos, or Tan ) and solve:


Watch the video:

You Try It:

A handicap accessibility ramp is need to reach a deck on a house.  The deck is 4 feet above the ground.
The ramp needs to start exactly 20 feet from the deck.  What angle of elevation will this ramp have?

 

Directions and Bearings

The direction or bearing to a point is stated as the number of degrees east or west of north or south.

For Example:

 The direction from H to A is 30ºE of N.
The direction from H to B is 60ºW of N.
The direction from H to C is 70ºE of S.
The direction from H to D is 80ºW of S.

There will also be situations when you do not know which direction North is.  In this case, you can simply identify going forward ( ahead of you ) as the North position and determine the your direction or bearing from there.

Example C

In order to safely cross a river, a safety line needs to be strung between two trees on opposite sides of the river bank. When standing by a tree on the river bank and looking strait across the river, the tree on the opposite side is 30o W of N.  If you walk to your left 10 yards you will be able to look straight across the river to the tree on the opposite side.   If you need at least 2-yards of rope to go around each tree, what is the shortest length of rope you need?

Solution:

Step 1, Draw a diagram from the given information:

Step 2, Set-up a Right Triangle with the given information:

Step 3, Set-up Trigonometry Ratio(s) ( Sin, Cos, or Tan ) and solve:


Watch the video:

You Try It:

You want to use a drone to check on the water tank for the cows in a large pasture, 640 acres
( 1-square mile ).  You are parked at the gate located on the midpoint of the south side of the pasture. 
The pump and water tank are located on a bearing 40 degrees west of you position, a half-mile north
of the south fence.  How far is it from the where you are parked to the water tank?  If your drone has
a maximum distance of two-thirds of a mile from the transmitter before losing signal and returning to
the transmitter.  Will you be able to fly the drone so that it can hover over the water tank?

 

Practice Problems

Use Right Triangle Trigonometry and/or the Pythagorean Theorem to solve the following word problems.

A.  Suppose a flagpole casts a 12-ft shadow when the sun is at an angle of 49° with the ground.
      What is the height of the pole?  

B.  A north bound ship launches a rocket with a range of 200 km at sea, with a bearing of 30° East.
     a. How far north of its original position will the rocket land?
     b. How far east of its original position will the rocket land?

C.  A 20-ft ladder is placed against a wall at an angle of 50° with the ground. How far from the
      base of the wall is the bottom of the ladder?

D.  A surveyor on top of a building finds that there is a 28° angle of depression to the head of
      the 6-ft tall assistant. If the assistant is 40 ft from the building, how tall is the building?

 

Assignment:

Instructions: