# Trigonometric Ratios

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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”.

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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.

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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/

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

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

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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.

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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.

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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.

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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.

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

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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”.

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SOH-CAH-TOA Pyramids

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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.

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

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The Pyramids can be made in three simple steps:

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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.

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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

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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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