When a 757 passenger jet begins its descent to the Ronald Reagan International Airport in Washington, D.C., it is 3900 feet from the ground.  Its typical angle of descent is about 6°.  Reagan International has a relatively short runway, so the stopping distance, angle of descent, and initial height all need to be precisely calculated.  If any of these factors is not measured accurately, the plane may not be able to safely touch the ground, or stop in time to avoid a collision.  Later in this lesson, question 

, you will check this given information using Trigonometry to determine when the pilot(s) must descend to this exact location.

When looking at any right triangle, a good approach towards gauging its usefulness is to identify known information about the figure.  In Trigonometry, all you need is one angle and one side.  If you have the ratios down, you can measure everything else.  First and foremost, identifying which side is the hypotenuse is very helpful.

The right angle in that diagram is drawn in for you, but can be assumed in most real world problem situations.  Structures are common examples, but these situations can include anything with a specific height.  Below, consider the next image.  Use a calculator or the table of trigonometric values.

🔑Try it - Decide which trigonometric function you would use and find the following:

Using Methods for Applications and Critical Thinking

🔑Take what you know about the sides of a right triangle, the trigonometric ratios, and the example above to guide you through these three problem-solving situations.  Use a calculator and the ratio values chart above to arrive at a solution - round to the nearest tenth.

a) Each base angle of an isosceles triangle measures 42° 30’.  The base is 14.6 meters long.  Find the altitude of the triangle, then find the area.

b) A regular (equiangular) pentagon is inscribed inside of a circle with a diameter of 8.34 cm.  The 

 of a regular polygon is the measure of a line segment from the center of the polygon to the midpoint of one of its sides.  Find the length of the apothem of the pentagon.

c) The main tent at a traveling circus requires lengths of rope to hold it up.  The tent has attachment points located 40 feet above the ground.  At each point, a rope is tied and then attached to a stake in the ground, creating an angle of 52° 15’.  How long is this length of rope?  How far away is the stake from the side of the tent?

There are many other applications that require trigonometric solutions.  For example, surveyors use special instruments to measure angles of 

.  The former is the angle between a horizontal line and the line of sight from an observer to an object at a higher level.  The latter is an angle made from a horizontal line and the line of sight from the observer to an object at a lower level.  These two angle measures, from an observer looking up and the object “looking” down, are equal because they form alternate interior angles (throwback) from the two parallel horizontal lines.

🔑Use this information to apply to your trigonometric knowledge to find solutions to the following prompts.  Draw a diagram for each example, providing an indication where the right angle is located, as well as the angle(s) of elevation and/or depression.  Round to the nearest tenth for your solution if necessary.

d) In the beginning of this lesson, you were introduced to a 757 passenger jet approaching Reagan International.  If you recall, its elevation should be 3900 feet above the ground to approach at a 6° angle of depression.  In this approach situation, what is the ground distance of the jet to the runway?  How far does the jet actually travel in the air?

e) The escalator at a capital city Metro station has a vertical rise of 195.8 feet.  If the angle of elevation of the escalator is 10° 21’ 36”, find the length of the escalator.

f) The longest truck-mounted ladder used by the Dallas Fire Department is 108 feet long and consists of four hydraulic sections.  The aerial expert for the department indicates that the optimum operating angle of this ladder is 60°.  The fire fighters find they need to reach the roof of an 84-foot burning building.  Assume the ladder is mounted 8 feet off of the ground.  How far from the building should the base of the ladder be placed to achieve the desired operating angle?  How far should the ladder be extended to reach the roof?

g) Cape Hatteras lighthouse on the North Carolina coast was built in 1870 and rises 208 feet above sea level.  From the top of the lighthouse, the keeper observes a yacht and a barge along the same line of sight.  The angle of depression for the yacht is 20° and 12° 30’ for the barge.  For safety purposes, the keeper thinks that the two vessels should be at least 300 feet apart.  If they are less than 300 feet she plans to sound the horn.  How far apart are the vessels, and does the keeper have to sound the horn?

🦄h) On May 18th, 1980, Mount Saint Helens, a volcano in Washington state, erupted with such force that the top of the mountain was blown off.  To determine the new height at the summit, a surveyor measured the angle of elevation to the top of the volcano to be 37°46’.  The surveyor then moved 1000 feet closer to the volcano and measured the new angle of elevation to be 40°30’.  Determine the new height of Mt. St. Helens.  (Hint: the only constant in solving this situation is the height of the volcanic peak)

Students will apply knowledge of trigonometric ratios in order to identify unknown elements of shapes and values for real world situations.

The Application Process