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Geometry For Beginners – What Is Meant By SOHCAHTOA?
As we have discovered, learning the study of geometry is mainly about finding missing measurements, both side lengths and angle measurements, 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 bisectors. The reason for this division into triangles is that we have several shortcuts to find missing measurements in certain triangles.
We have already looked at the “special” 30-60 straight and 45 straight triangles. (These are sometimes called 30-60-90 and 45-45-90 special triangles.) These right triangles have relations or ratios for the three sides that are always the same, and we can use these known ratios to shorten work. necessary to find the missing side measurements. These special triangles are certainly useful, but they only work with two types of right triangles. What about all the other right triangles? To work with all of these other right triangles, we use a relationship called SOHCAHTOA–pronounced sew-ka-toa.
I know that word sounds like it’s a Native American word, but it’s really a mnemonic device for remembering the relationships between the sides and angles of a right triangle. To understand everything in this mnemonic device, we must learn new terms. These terms are essential for success in both geometry and trigonometry, so it’s important to have a good grasp of this information now. You won’t stop using it at the end of the geometry.
The letters in SOHCAHTOA mean, in order from left to right, Sine, Oopposite, Hypotent, VSosin, Aadjacent, Hypotent, Jagent, Oopposite, and Aadjacent. At this point in your studies, the words sine, cosine, and tangent may sound familiar to you from your graphing or scientific calculators, although calculators use the abbreviations sin, cos, and tan; but those words probably don’t make sense to you. It’s normal and OK.
Triangles have three sides, so there are six ways to compare two sides if we correctly understand that the 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 trigonometric ratios. As you remember, a ratio is simply a comparison of two numbers. A ratio can be written in the form of decimals, fractions and percentages. To work 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 an upside-down capital letter “L”. Make the legs visibly different lengths. Now draw the line segment connecting the far ends of the legs. Label the bottom left corner with the letter A on the outside but near the corner. Label the top 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 still the side opposite the right angle, but the other two labels are “relative”. This means that they are different if we consider 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. So, labeling is impossible until we know which angle should be used.
We’re almost ready to explain what SOHCAHTOA actually stands for, but there’s one point I want to make that most geometry students miss. When we write in the abbreviated version sin = opp/hyp, we omit a very important part of the statement. These ratios depend on the angle used. The abbreviated version sin = opp/hyp represents the longest sentence, “The ratio of the sine for a given angle X is the ratio of the side opposite X to the hypotenuse of the triangle. You should always remember that the words sin, cos and tan should be read as sine of one or cosine from B or tangent of X. NEVER FORGET THE ANGLES!
Using X to represent angle, SOHCAHTOA represents the following ratios: sine x = opposite/hypotenuse, cosine X = adjacent/hypotenuse, and tangent X = opposite/adjacent. These are often written in abbreviated form as follows: sin = opp/hyp, cos = adj/hyp, and tan = opp/adj.
In another article, we’ll look at how to use SOHCAHTOA to find missing sides and angles, but to quickly check what we just discussed here, let’s use some specific sides. Let’s use a 3, 4, 5 right triangle and the image 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 still don’t make sense, read this article again and redraw the diagram as many times as necessary to make these ratios understandable.
In future articles, we’ll give meaning and purpose to the process we’re introducing here. For now, it’s important for you to remember that trigonometric functions are nothing more than taking the ratio of two sides of a right triangle. In another article, we’ll use these ratios to find the missing angle, and in another article, we’ll look at how to make sense of those visual images in your head so you can estimate the answers. We will always have calculators and computers to do the hard work for us; but often we just need to have a rough estimate. We can also learn this skill.
SOHCAHTOA is a very powerful tool that you want to master as quickly as possible. Plus, it makes you look REALLY SMART!!!!!! That in itself is worth a lot!
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