The Tangent Ratio

Canadian Airliner Ascending
Image Source: Photo by Jeff Wilson – www.warbirddepot.com

Trigonometry is used a lot in the work done by Pilots and Air Traffic Controllers.

However, a lot of the maths formulas are programmed into computer applications, and we do not see people sitting around pushing the sin, cos and tan buttons on calculators!

In this lesson we look at the Tangent Ratio.

Tangent Ratio 1
Original Plane Photo by Geoff Wilson

 
 

Definition of the Tangent Ratio

Tangent Ratio 2
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Like other Trig Ratios, the Tan Ratio works out the same value for any sized triangle that has the same Reference Angle in it.

Tangent Ratio 3
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Tangent Formulas Summary

Tangent Ratio 4
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Like for Cosine and Sine, we need to pick the best formula to suit our question.

For example, if we need to find an unknown Opposite, then we should use the second “OPP = ” formula.

But if we are asked to find an unknown Angle Value, we should use the last formula which contains “Tan-1″.

 
 

Using the Calulator for Tan Values

Tangent Ratio 5
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Tangent Ratio 6
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Steps for Tangent Questions

Follow these steps when doing any question which involves Opposite and Adjacent.

Tangent Ratio 7
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Tangent Examples

The following two examples show how to do a side length and Angle calculation for a Tangent type triangle.

Note that we would use SOH-CAH-TOA to work out that these triangle have Opposite and Adjacent larked on them, which means we have “OA” which fits into the “TOA” tan part of our method.

Tangent Ratio 8
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Tangent Ratio 9
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Practical Application of Tan Ratio

airliner crashed in amsterdam
Image Source: http://www.mirror.co.uk

At the start of this lesson, we looked at the Tan Ratio when a jet airliner takes off.

Sometimes planes also have to make emergency landings, or even crash landings.

When a plane comes down in an emergency situation, there is the force of it hitting the ground coming straight upwards at passengers.

There is also a sudden horizontal force as the plane dramatically slows down as it makes contact with the ground, which throws passengers forward, like when a car slams on its brakes.

This results in a Tan situation, where the combined forces will resolve into an effective downhill force along the hypotenuse of a right triangle.

 

In their episode on “The Brace Position” for airline crashes, The Mythbusters use a Triangle Rig and make use of the Tan Ratio to simulate a plane crash landing.

The following short video gives a great summary of this 30 minute episode.

The full 30 minute episode is available on Discovery Channel DVD, and is well worth watching during a Math class on Triginometry.

 
 

Tangent Ratio Worksheets

Do any of the following worksheets for practice with Tangent Ratio Questions

Trig Worksheet 1

Trig Worksheet 2

Trig Worksheet 3

 
 

Trigonometry Summary Sheet

Trigonometry Formulas
Image Copyright 2013 by Passy’s World of Mathematics

If you would like a free A4 Summary Sheet of the Sin Cos and Tan formulas that we use in Trig Ratios, then click the link below.

Trig Formulas Summary Sheet

 
 

SOH-CAH-TOA Pyramids

Trig Ratio Pyramids
Image Copyright 2013 by Passy’s World of Mathematics

We can use a set of three pyramids to get all of our Trig Ratio Formulas.

Some people might find it easier to set up the following SOH-CAH-TOA Formula Pyramids, and use these to obtain formulas.

Trig Ratio Pyramid Formulas
Image Copyright 2013 by Passy’s World of Mathematics

The Pyramids certainly provide a far more compact version of the full set of Trig Ratios Formulas.

Trig Ratio Pyramid Helper
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The Pyramids can be made in three simple steps:

Trig Ratio Pyramids 2
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In the second step we divide each pyramid in three, by ruling a horizontal line to form a smaller similar Triangle at the top

We then divide the bottom half Trapezoidal shape inh half.

In the Third step we write SOH CAH TOA into our pyramids, working from left to right.

The following diagram shows how things should look after completing steps 2 and 3.

Trig Ratio Pyramids 3
Image Copyright 2013 by Passy’s World of Mathematics

 

As much as we love Egypt and the Pyramids, we did NOT invent these SOH-CAH-TOA Pyramids here at Passy World.

We saw a maths teacher using them recently, and we also found pictures and explanations of them on the Internet.

 
 

Related Items

The Cosine Ratio
The Sine Ratio
Labeling Trigonometry Triangles
Trigonometric Ratios – Sin Cos and Tan
Classifying Triangles
Pythagoras and Right Triangles
Congruent Triangles
Tall Buildings and Large Dams
Similar Shapes and Similar Triangles
Geometry in the Animal Kingdom

 
 

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