# The Sine Ratio

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.

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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:

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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.

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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.

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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.

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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.

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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.

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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:

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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:

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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.

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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.

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

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

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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.

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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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