Passy's World of Mathematics

The Tangent Ratio

Posted on August 23, 2013 by Passy

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.

Original Plane Photo by Geoff Wilson

Definition of the Tangent Ratio

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 Formulas Summary

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

Steps for Tangent Questions

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

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.

Practical Application of Tan Ratio

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

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

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.

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

The Pyramids can be made in three simple steps:

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.

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.

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Posted in Triangles, Trigonometry | Tagged adjacent formula, finding angle, finding tangent values, how to do tangent, how to do trig calculations, how to find hypotenuse, how to teach tangent ratio, how to teach trig ratios, hypotenuse, hypotenuse formula, inverse tan, inverse tangent, Online Trig Ratios Calculator, right triangles, tan ratio, tangent, tangent angle, tangent examples, tangent formula, tangent on calculator, tangent ratio, tangent triangle, triangle math, trig, trig calculations, trig calculator, trig formulas, trig funtion, trig ratio formulas sheet, trig ratios summary, trigonometry, what is the tangent | Leave a comment

The Cosine Ratio

Posted on August 11, 2013 by Passy

Image Source: http://rekkerd.org

In this lesson we are doing some more Trigonometry.

Image Source: Google Images

Shown above are some basic “sinusoidal” shaped sound waves.

Our ear detects these when music is played.

Various “Guitar Pedals” or “Stomp Boxes” can be used to modify these basic sound waves to create new and interesting sounds.

The mathematics of these curve shaped waves involves the trigonometry of right angled triangles.

These curved shaped waves are formed by graphing all of the decimal trigonometry values which can be formed by different shaped right angled triangles.

This is illustrated very nicely by the following diagram:

Image used with Permission from Russell Kightley

Distorted Heavy guitar sound occurs when smooth Sine Waves are mathematically transformed into Square shaped, Sawtooth, and Triangle shaped waves.

This is shown in the animation below:

Image Source: Google Images

The mathematics of wave re-shaping works to create a wide variety of electric guitar sounds.

Guitar Mathematics

Here at Passy’s World we love playing Guitar.

If you would like to find out more about the Mathematics of the Guitar, then check out our lesson on this at the following link:

http://passyworldofmathematics.com/guitar-mathematics/

Trigonometry is extremely important in areas involving waves such as Sound, Light, Electricity, and Oceanography.

In this lesson we look at the trigonometry function known as “Cosine”.

The Cosine Ratio

There are three main Trigonometry Ratios: Sine, Cosine, and Tangent.

It is difficult to try and learn all three of these at once, and so this lesson only covers the Cosine Ratio.

At Passy’s World, we have found that trying to learn all three Ratios at once, is like trying to learn how to Drive, Chip, and Putt, all in one Golf Lesson.

It is difficult, confusing, and frustrating.

We prefer to learn the Sine Ratio, and then the Cosine Ratio separately, before trying to deal with all three Trig Ratios.

Before doing this lesson on Cosine Ratios, it is important that you know how to label the sides of a Right Triangle as “Hypotenuse”, “Opposite”, and “Adjacent”.

If you need to learn how to label a Right Triangle, then click the link below:

http://passyworldofmathematics.com/the-sine-ratio/

It is also important that you have already studied the Sine Ratio in our previous lessone at the link below:

http://passyworldofmathematics.com/trigonometry-labeling-triangles/

The Cosine Ratio involves the Adjacent and Hypotenuse sides of a Right Triangle as follows:

The Cosine Ratio will be the same for any Right Triangle which has a particular Angle value contained in it.

The diagram below show this for three different sized 37 degree Right Triangles.

Cosine Triangle – Formulas Summary

Here are the four formulas we use when working with Cosine Triangles.

Using the Calculator for Cosine Triangles

If we are given an angle and we need to determine its decimal Cosine value, we can do this on a Calculator as shown below:

If we are given an Adjacent and a Hypotenuse, and we need to determine the reference Angle value, we can do this on a Calculator as shown below:

The above function is sometimes also called finding the “ArcCos” or the “ACos”.

Online Trigonometry Calculator

If you would like to use an online calculator to find “Cos” or Angle values, then there is one at the following link:

Online Trig Ratios Calculator

Note that you will need to set this calculator to 4 decimal places for Sin values.

Cosine Triangle – Working Out Steps

If we are working on a right triangle which involves an Adjacent and a Hypotenuse, then here are the steps we need to follow.

Follow the above steps for doing all questions for Cosine Triangles.

Cosine Triangle Examples

The following examples show how we apply our Cosine Triangle formulas to questions to work out unknown values on Right Triangles.

In this first example we are asked to find the value of the “Adjacent” side.

The next example shows how to find the value of the “Hypotenuse” for a Cosine Triangle.

In our next example we use the Inverse Cosine function to find an unknown Angle.

In this final example, we are given the Adjacent and Hypotenuse, and asked to fine the decimal value of Cosine.

It is important to read questions carefully, and not immediately assume that this is a Find the Angle example, like Example 3.

Cosine Ratio Worksheet

The following worksheet contains several type of Cosine Ratio questions, and includes Answers at the end of the sheet.

Click the following link to access this Cosine Ratio worksheet:

Cosine Ratio Worksheet

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Enjoy,
Passy

Posted in Triangles, Trigonometry | Tagged adjacent formula, cos ratio, cosine, cosine angle, cosine examples, cosine formula, cosine on calculator, cosine ratio, cosine ratio worksheet, cosine triangle, finding angle, finding cosine values, how to do cosine, how to do trig calculations, how to find hypotenuse, how to teach cosine ratio, how to teach trig ratios, hypotenuse, hypotenuse formula, inverse cosin, inverse cosine, Online Trig Ratios Calculator, right triangles, triangle math, trig, trig calculations, trig calculator, trig formulas, trig funtion, trig ratios worksheet, trigonometry, what is the cosine | 1 Comment

The Sine Ratio

Posted on August 9, 2013 by Passy

Image Source: PLane Landing at Maho Beach

Pilots need to use Trigonometry to make it onto the end of the runway when landing; particularly at Maho Beach in the Caribbean.

As the plane descends it has a speed at an angle, as well as a decreasing height, and its flight path forms the hypotenuse of a right triangle.

The pilot needs to use the correct angle and speed to make it onto the runway.

They also need to land at a point down the runway which still gives them enough length to stop the plane safely.

Here is a video showing just how difficult the landing at Maho Beach on San Maarten Island actually is.

Not only is it difficult to bring the plane down onto the runway, but in strong cross-winds the task is even made more difficult.

The pilot has to point the plane into the wind, so that it is not blown sideways off the runway.

The white bars that look like a Pedestrian Crossing at the end of the runway are used by incoming pilots for steering towards the runway at an angle to counteract the cross wind strength. This is very mathematical and the Pilot needs to get the exact answer!

There are some great examples of this in the following video.

Obviously the Pilot is far too busy during landings to do Trigonometry Calculations; however these calculations, along with many others, would be programmed into the computerised equipment that is giving status and warning indicators which are helping the pilot to successfully land the plane.

Trigonometry is also used in Navigation during the plane flight, in the form of “Bearings”, eg. fly for 200km at a bearing of North 20 degrees East.

Bearings are also used by ships and yachts setting their courses at sea.

Navigation and Bearings will be covered in a separate lesson, later in Trigonometry.

The Sine Ratio

There are three main Trigonometry Ratios: Sine, Cosine, and Tangent.

It is difficult to try and learn all three of these at once, and so this lesson only covers the Sine Ratio.

At Passy’s World, we have found that trying to learn all three Ratios at once, is like trying to learn how to Drive, Chip, and Putt, all in one Golf Lesson.

It is difficult, confusing, and frustrating.

We prefer to learn the Sine Ratio, and then the Cosine Ratio separately, before trying to deal with all three Trig Ratios.

Before doing this lesson on Sine Ratios, it is important that you know how to label the sides of a Right Triangle as “Hypotenuse”, “Opposite”, and “Adjacent”.

If you need to learn how to label a Right Triangle, then click the link below:

http://passyworldofmathematics.com/trigonometry-labeling-triangles/

The Sine Ratio involves the Opposite and Hypotenuse sides of a Right Triangle as follows:

The Sine Ratio will be the same for any Right Triangle which has a particular Angle value contained in it.

The diagram below show this for three different sized 37 degree Right Triangles.

Opposite Side Formula

If we need to find the value of an Opposite Side for a Sine Triangle, and we have values for the reference angle and the hypotenuse, then we can derive and use the following formula.

We use the above formula for finding unknown Opposite sides on Sine Triangles.

Note that we need to use a calculator when working with this formula; this is covered later in this lesson.

Hypotenuse Formula

If we take our previous Sine Triangle formula for the opposite, and do some rearranging, we can obtain a hypotenuse formula.

We use the above formula for finding unknown Hypotenuse values on Sine Triangles.

Note that we need to use a calculator when working with this formula; this is covered later in this lesson.

Angle Formula

The final Sine Triangle formula is for finding and unknown Angle, when we are given values for the opposite and the hypotenuse.

We use the above formula for finding unknown Angles on Sine Triangles.

Note that we need to use a calculator when working with this formula; this is covered later in this lesson.

Sine Triangle – Formulas Summary

Here are the four formulas we use when working with Sine Triangles.

Using the Calculator for Sine Triangles

If we are given an angle and we need to determine its decimal Sine value, we can do this on a Calculator as shown below:

If we are given an Opposite and a Hypotenuse, and we need to determine the reference Angle value, we can do this on a Calculator as shown below:

The above function is sometimes also called finding the “ArcSin” or the “ASin”.

Online Trigonometry Calculator

If you would like to use an online calculator to find “Sin” or Angle values, then there is one at the following link:

Online Trig Ratios Calculator

Note that you will need to set this calculator to 4 decimal places for Sin values.

Sine Triangle – Working Out Steps

If we are working on a right triangle which involves an Opposite and a Hypotenuse, then here are the steps we need to follow.

Follow the above steps for doing all questions for Sine Triangles.

Sine Triangle Examples

The following examples show how we apply our Sine Triangle formulas to questions to work out unknown values on Right Triangles.

In this first example we are asked to find the value of the “opposite” side.

The next example shows how to find the value of the “Hypotenuse” for a Sine Triangle.

In our next example we use the Inverse Sine function to find an unknown Angle.

In this final example, we are given the Opposite and Hypotenuse, and asked to fine the decimal value of Sine.

It is important to read questions carefully, and not immediately assume that this is a Find the Angle example, like Example 3.

Application Question Example

The following three minute video shows how to do a real world application question that involves using the Sine Ratio.

Sine Ratio Worksheet

The following worksheet contains several type of Sine Ratio questions, and includes Answers at the end of the sheet.

Click the following link to access this Sine Ratio worksheet:

Sine Ratio Worksheet

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Enjoy,
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Posted in Triangles, Trigonometry | Tagged arcsin, asin, finding angle, finding sine values, how to do sine, how to do trig calculations, how to find hypotenuse, how to teach sine ratio, how to teach trig ratios, hypotenuse, hypotenuse formula, inverse sin, inverse sine, Online Trig Ratios Calculator, opposite formula, right triangles, sin ratio, sine, sine angle, sine examples, sine formula, sine on calculator, sine ratio, sine ratio worksheet, sine triangle, triangle math, trig, trig calculations, trig calculator, trig formulas, trig funtion, trig ratios worksheet, trigonometry, what is the sine | 3 Comments

Trigonometric Ratios

Posted on August 5, 2013 by Passy

Image Source: http://www.fanpop.com

Pythagoras Theorem and Trigonometry were key mathematical methods that were used to help build the Pyramids.

Pythagoras looked at the Sides Relationship, and people like Hipparcus looked at the Relationship between Angles and Sides.

They named the Angles and Sides mathematics “Trigonometry”.

In this lesson we look at the “Trigonometric Ratios” associated with Right Angled Triangles.

This lesson assumes that people already know how to label the Trigonometry sides of a Right Triangle as Hopenuse, Opposite, and Adjacent.

If you do not know how to do this sides labeling, then go and do our previous lesson on this at the link below.

http://passyworldofmathematics.com/trigonometry-labeling-triangles/

Trigonometric Ratios Example

Hipparcus and other ancient mathematicians found that when we have Similar Right Triangles, (which all have the same base angle), we get their internal sides ratios being identical.

The above is only one example for a 37 degree base triangle; however it has been found that this concept works for any size base angle.

In addition, the above example only looks at the height versus the base of the triangles, but there are actually five other comparisons we can also do.

The full set of six Trigonometric Ratios is shown in the next section.

The Trigonometric Ratios

A Right Triangle has three sides: Hypotenuse, Opposite, and Adjacent.

If you do not know how to do this sides labeling, then go and do our previous lesson on this at the link below.

http://passyworldofmathematics.com/trigonometry-labeling-triangles/

The six Trigonometric ratios that we can make for a Right Triangle have special mathematical names as shown in the following Table.

We also have six math expressions which abbreviate these six names, and express the Trig Ratios in shorthand form.

In the remainder of this lesson, we will only be looking at three of these six ratios: Sine, Cosine, and Tangent.

SOH – CAH – TOA

To help memorize the three Trig Ratios for Sine, Cosine, and Tangent, the Acronym “SOH – CAH – TOA” is used.

Trigonometric Ratios Videos

Here is a short two and a half minute video which shows the Sine, Cosine, and Tangent Ratios.

This next seventeen minute video goes through the Trig Ratios, and does working out several example triangles.

Trig Ratio Examples

In this first example, we are given a Right Triangle with the sides labelled, and some number values for these sides.

We than use SOH-CAH-TOA to write the fraction and decimal values for Sin, Cos, and Tan for the 37 degree angle that is in the Right Triangle.

In this next example, the sides are not labeled and we are asked to find the Tan value of the unknow angle theta.

Using SOH-CAH-TOA, the Tan value is obtained by putting the Opposite side value over the Adjacent side value.

Our next example has the Cos value given to us, and we have to use it to work out the unknown Adjacent side on the Triangle.

This next example is similar to the previous example, but we are using the given Sine value to work out the Unknown “Opposite Side”.

This final example has a Cos value supplied, but it is for the top angle in the Triangle.

We need to label the Triangle, and then use Cos = Adjacent / Hypotenuse to work out the unknown Adjacent side “a”.

SOH-CAH-TOA Pyramids

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

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

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

The Pyramids can be made in three simple steps:

In the second step we divide each pyramid in three, by ruling a horzontal 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.

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.

Trigonometry Summary Sheet

If you would like a free A4 Summary Sheet which provides all 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 Music Video

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Enjoy,
Passy

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Trigonometry – Labeling Triangles

Posted on August 2, 2013 by Passy

Image Source: http://tributehomes.com.au

Triangles are a key feature of the Architecture of Buildings.

Here is a building which contains many triangles.

Image Source: http://www.infoteli.com

The building is in Eindhove Holland and is called ‘De Blob’.

It forms the Admirant Building Entrance.

The Admirant Building is a concrete structure of five storeys, surrounded by the skin of the bubble.

The bubble is made almost entirely of thick glass, held in place by many triangles attached to a complex steel structure .

The Bubble also contains two tunnel entrances, which lead to an underground parking area for about 1,700 bikes.

Triangles are used a lot in Architecture, but not usually in such spectacular from as in ‘De Blob’.

The usual use for triangles involves creating frames which make buildings rigid and strong.

Image Source: http://www.blogger.com

“Trigonometry” is a branch of mathematics which deals with measuring the sides and angles in Right Angled Triangles.

Architects use Trigonometry to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light.

Trigonometry allows architects to figure out measurements and angles so that their blueprints can be turned into real world structures from raw materials such as steel, wood, and concrete.

It is essential when designing a building to predetermine the geometrical patterns and how much material and labor will be required in order to erect the structure.

Thanks to Trigonometry, when the building is erected, it will be strong with accurate measurements and a budgeted dollar cost.

Definition of Trigonometry

The “Trigon” part of “Trigonometry” refers to a three sided geometrical shape, eg. a Triangle.

Trigon = 3 sides, Hexagon = 6 sides, Octagon = 8 sides, etc.

The “metry” part of “Trigonometry” refers to the activity of measuring.

So the word “Trigonometry” means measurements of the sides and angles in Triangles.

The full story on Trigonometry actually extends beyond Triangles:

Trigonometry is a branch of mathematics which deals with triangles, circles, waves and oscillations.

However, we will only be looking at Triangles in this particular lesson.

Trigonometry and Right Angled Triangles

The Trigonometry we will be covering in this lesson only applies to 90 degree Right Angled Triangles.

There is a standard way of labeling the angles and sides of a Right Triangle for Trigonometry.

Labeling Angles

Labeling The standard “reference angle” in a Right Triangle is shown below.

The Right Triangle has been drawn in standard position, which means that the “Reference Angle” (or slope angle) is at the bottom left, and the right angle is at the bottom right of the triangle.

Labeling Sides

The three sides of the Right Angled Triangle are labelled as “Hypotenuse”, “Opposite”, and “Adjacent”.

The “Hypotenuse” is the longest sloping side of the Triangle, just as it is in the “Pythagoras Theorem”.

The “Opposite” is the side that is directly across from the slope angle.

The “Adjacent” is the side that is closest to the slope angle.

When the Triangle is drawn in standard position, its full labeling for use in Trigonometry looks like this:

The angle “theta” at the bottom of the Triangle is called the standard Reference Angle.

We can also label the Triangle in the same way for the other angle at the top of the Triangle.

Video About Labelling Sides

The following five minute video goes through how to Label the Sides of a Triangle for using in Trigonometry.

It covers triangles drawn in all positions, as well as the basic standard position.

Here is another shorter video which shows how to label for both angles in the Triangle:

Labelling Sides Worksheets

This first worksheet labels sides using letters on the Triangle, and has Answers on Page 2 of the sheet.

Labeling Sides Worksheet 1

This second worksheet involves labelling sides using the numerical measurements which are on them, with Answers on Page 2 of the sheet

Labeling Sides Worksheet 2

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Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.

Help us to maintain this free service and keep it growing.

Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!

Enjoy,
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Passy's World of Mathematics

The Tangent Ratio

The Cosine Ratio

The Sine Ratio

Trigonometric Ratios

Trigonometry – Labeling Triangles

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