The Cosine Ratio

Guitar Stomp Boxes
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In this lesson we are doing some more Trigonometry.

sine waves and music sound
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:

Cosine Ratio 2
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:

Fourier Transform on a Sine Wave
Image Source: Google Images

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

Cosine Ratio 3
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Guitar Mathematics

Passy Playing Guitar
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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:


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:

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


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

Cosine Ratio 4
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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 Ratio 5
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Cosine Triangle – Formulas Summary

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

Cosine Ratio 6
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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:

Cosine Ratio 7
Image Copyright 2013 by Passy’s World of Mathematics

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:

Cosine Ratio 8
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The above function is sometimes also called finding the “ArcCos” or the “ACos”.


Online Trigonometry Calculator

Online Trig 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.

Cosine Ratio 9
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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.

Cosine Ratio 10
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The next example shows how to find the value of the “Hypotenuse” for a Cosine Triangle.

Cosine Ratio 11
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In our next example we use the Inverse Cosine function to find an unknown Angle.

Sine Ratio 12
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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 13
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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


Related Items

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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One Response to The Cosine Ratio

  1. Pingback: The Tangent Ratio | Passy's World of Mathematics

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