As we have been discovering, learning the study of Geometry is primarily about finding missing measurements, both lengths of sides and angle measures, in geometric figures.
If a figure has four or more sides, we often divide the figure into triangles by drawing diagonals, altitudes, medians, and/or angle bisectors.
The reason for doing this division into triangles is that we have several shortcuts for finding the missing measurements in certain triangles.
We have already looked at the 30-60 right and 45-right "special" triangles.
(These are sometimes referred to as 30-60-90 and 45-45-90 special triangles.
) These right triangles have relationships or ratios for the three sides that are always the same, and we can use these known ratios to shorten the work needed to find missing side measurements.
These special triangles are certainly helpful, but they only work with two types of right triangles.
What about all the other right triangles? To work with all those other right triangles, we use a relationship called SOHCAHTOA--pronounced sew-ka-toa.
I know this word sounds as if it might be an American Indian word, but it is really a mnemonic device for remembering the relationships of the sides and angles in a right triangle.
To understand everything in this mnemonic device we need to learn some new terms.
These terms are critical for success in both Geometry and Trigonometry, so it is important to get a firm handle on this information now.
You won't stop using this at the end of Geometry.
The letters in SOHCAHTOA stand for, in order from left to right, Sine, Opposite, Hypotenuse, Cosine, Adjacent, Hypotenuse, Tangent, Opposite, and Adjacent.
At this point in your studies, the words sine, cosine, and tangent may seem familiar to you from your graphing or scientific calculators, although the calculators use the abbreviations sin, cos, and tan; but these words likely have no meaning to you.
That is normal and OK.
Triangles have three sides so there are six ways that we could compare two sides together if we correctly understand that reciprocals are different.
The six ways we can compare two sides together form the six trigonometric ratios.
Sine, cosine, and tangent are the three most commonly used of the six trig ratios.
As you remember, a ratio is simply a comparison of two numbers.
A ratio can be written as decimals, fractions, and per cents.
For working with right triangles, the numbers we are comparing are the lengths of two of the sides of the triangle.
To fully understand SOHCAHTOA, we need a diagram.
On a piece of paper--the one you keep handy when reading math articles--draw a backwards capital letter "L.
" Make the legs visibly different lengths.
Now, draw the line segment connecting the far endpoints of the legs.
Label the lower left angle with the letter A outside but close to the angle.
Label the upper angle as B, and label the 90 degree angle as C.
Now we need to label the sides with the terms adjacent, opposite, and hypotenuse.
The hypotenuse is always the side opposite the right angle, but the other two labels are "relative.
" This means that they are different if we are considering angle A rather than angle B.
For example, in our triangle, the side opposite angle B is segment AC, but the side opposite angle A is segment BC.
Thus, labeling is impossible until we know which angle is to be used.
We are almost ready to explain what SOHCAHTOA actually represents, but there is one point I want to stress that is missed by most Geometry students.
When we write in the short cut version sin = opp/hyp, we are leaving out a very important part of the statement.
These ratios are dependent on the angle being used.
The short cut version sin = opp/hyp stands for the longer sentence, "The sine ratio for a given angle X is the ratio of the side opposite X to the hypotenuse of the triangle.
You must always remember that the words sin, cos, and tan should be read sine of A or cosine of B or tangent of X.
NEVER FORGET THE ANGLES! Using X to represent the angle, SOHCAHTOA stands for the following ratios: sine x = opposite/hypotenuse, cosine X = adjacent/hypotenuse, and tangent X = opposite/adjacent.
These are often written in short form as: sin = opp/hyp, cos = adj/hyp, and tan = opp/adj.
In another article we will look at how to actually use SOHCAHTOA to find missing sides and angles, but as a quick check of what we have just discussed here, let's use some specific sides.
Let's use a 3, 4, 5 right triangle and the picture we drew earlier.
Label the hypotenuse with 5, the base side with 3, and the vertical side with 4 and we'll use the angle names A and B and C from before.
Using these numbers, sin A = 4/5, cos A = 3/5, and tan A = 4/3.
If you agree with these numbers, then you have a good understanding of this material.
If these numbers do not yet make sense, re-read this article and re-draw the diagram as many times as it takes to make these ratios understandable.
In the next articles, we will attach meaning and purpose to the process we are introducing.
here.
For now, it is important that you remember the trig functions are nothing more that taking the ratio of two sides of a right triangle.
In another article we will use these ratios to actually find missing angle, and in another article we will look at how to give these visual images meaning in you head so that you can estimate answers.
We will always have calculators and computers to do the hard work for us; but often we just need to have a quick ballpark estimate.
We can learn that skill as well.
SOHCAHTOA is a very powerful tool--one you want to master as quickly as possible.
Besides, it makes you seem REALLY SMART!!!!!! That itself is worth a great deal!
If a figure has four or more sides, we often divide the figure into triangles by drawing diagonals, altitudes, medians, and/or angle bisectors.
The reason for doing this division into triangles is that we have several shortcuts for finding the missing measurements in certain triangles.
We have already looked at the 30-60 right and 45-right "special" triangles.
(These are sometimes referred to as 30-60-90 and 45-45-90 special triangles.
) These right triangles have relationships or ratios for the three sides that are always the same, and we can use these known ratios to shorten the work needed to find missing side measurements.
These special triangles are certainly helpful, but they only work with two types of right triangles.
What about all the other right triangles? To work with all those other right triangles, we use a relationship called SOHCAHTOA--pronounced sew-ka-toa.
I know this word sounds as if it might be an American Indian word, but it is really a mnemonic device for remembering the relationships of the sides and angles in a right triangle.
To understand everything in this mnemonic device we need to learn some new terms.
These terms are critical for success in both Geometry and Trigonometry, so it is important to get a firm handle on this information now.
You won't stop using this at the end of Geometry.
The letters in SOHCAHTOA stand for, in order from left to right, Sine, Opposite, Hypotenuse, Cosine, Adjacent, Hypotenuse, Tangent, Opposite, and Adjacent.
At this point in your studies, the words sine, cosine, and tangent may seem familiar to you from your graphing or scientific calculators, although the calculators use the abbreviations sin, cos, and tan; but these words likely have no meaning to you.
That is normal and OK.
Triangles have three sides so there are six ways that we could compare two sides together if we correctly understand that reciprocals are different.
The six ways we can compare two sides together form the six trigonometric ratios.
Sine, cosine, and tangent are the three most commonly used of the six trig ratios.
As you remember, a ratio is simply a comparison of two numbers.
A ratio can be written as decimals, fractions, and per cents.
For working with right triangles, the numbers we are comparing are the lengths of two of the sides of the triangle.
To fully understand SOHCAHTOA, we need a diagram.
On a piece of paper--the one you keep handy when reading math articles--draw a backwards capital letter "L.
" Make the legs visibly different lengths.
Now, draw the line segment connecting the far endpoints of the legs.
Label the lower left angle with the letter A outside but close to the angle.
Label the upper angle as B, and label the 90 degree angle as C.
Now we need to label the sides with the terms adjacent, opposite, and hypotenuse.
The hypotenuse is always the side opposite the right angle, but the other two labels are "relative.
" This means that they are different if we are considering angle A rather than angle B.
For example, in our triangle, the side opposite angle B is segment AC, but the side opposite angle A is segment BC.
Thus, labeling is impossible until we know which angle is to be used.
We are almost ready to explain what SOHCAHTOA actually represents, but there is one point I want to stress that is missed by most Geometry students.
When we write in the short cut version sin = opp/hyp, we are leaving out a very important part of the statement.
These ratios are dependent on the angle being used.
The short cut version sin = opp/hyp stands for the longer sentence, "The sine ratio for a given angle X is the ratio of the side opposite X to the hypotenuse of the triangle.
You must always remember that the words sin, cos, and tan should be read sine of A or cosine of B or tangent of X.
NEVER FORGET THE ANGLES! Using X to represent the angle, SOHCAHTOA stands for the following ratios: sine x = opposite/hypotenuse, cosine X = adjacent/hypotenuse, and tangent X = opposite/adjacent.
These are often written in short form as: sin = opp/hyp, cos = adj/hyp, and tan = opp/adj.
In another article we will look at how to actually use SOHCAHTOA to find missing sides and angles, but as a quick check of what we have just discussed here, let's use some specific sides.
Let's use a 3, 4, 5 right triangle and the picture we drew earlier.
Label the hypotenuse with 5, the base side with 3, and the vertical side with 4 and we'll use the angle names A and B and C from before.
Using these numbers, sin A = 4/5, cos A = 3/5, and tan A = 4/3.
If you agree with these numbers, then you have a good understanding of this material.
If these numbers do not yet make sense, re-read this article and re-draw the diagram as many times as it takes to make these ratios understandable.
In the next articles, we will attach meaning and purpose to the process we are introducing.
here.
For now, it is important that you remember the trig functions are nothing more that taking the ratio of two sides of a right triangle.
In another article we will use these ratios to actually find missing angle, and in another article we will look at how to give these visual images meaning in you head so that you can estimate answers.
We will always have calculators and computers to do the hard work for us; but often we just need to have a quick ballpark estimate.
We can learn that skill as well.
SOHCAHTOA is a very powerful tool--one you want to master as quickly as possible.
Besides, it makes you seem REALLY SMART!!!!!! That itself is worth a great deal!
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