As we have been discovering, learning the study of Geometry is mainly about finding the missing measures, both the lengths of the sides and the measures of the angles, 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 to find the missing measures in certain triangles.
We have already seen the “special” triangles of 30-60 rectangles and 45 rectangles. (These are sometimes called special 30-60-90 and 45-45-90 triangles.) These right triangles have ratios or proportions for the three sides that are always the same, and we can use these known proportions to shorten the work. needed to find the measurements of the missing sides. 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 those other right triangles, we use a relationship called SOHCAHTOA, which is pronounced sew-ka-toa.
I know this word sounds like a Native American word, but it’s actually a mnemonic to remember 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 to success in both Geometry and Trigonometry, so it is important to have a firm grip on this information now. You won’t stop using this at the end of Geometry.
The letters in SOHCAHTOA represent, in order from left to right, yesine, EITHERpostulate, Hypotent, againstbear, HASunderlying, Hypotent, TAngel, EITHERopposite, and HASadjacent. At this point in his studies, the words sine, cosine, and tangent may seem familiar to you from your graphing or scientific calculators, even though the calculators use the abbreviations sin, cos, and tan; but these words probably have no meaning to you. That’s normal and okay.
Triangles have three sides, so there are six ways we can compare two sides if we correctly understand that the reciprocals are different. The six ways we can compare two sides together make up the six trigonometric ratios. Sine, cosine, and tangent are the three most commonly used trigonometric ratios of the six. As you may remember, a ratio is simply a comparison of two numbers. A ratio can be written as decimals, fractions, and percents. To work with right triangles, the numbers we are comparing are the lengths of two of the triangle’s sides.
To fully understand SOHCAHTOA, we need a diagram. On a piece of paper, the one you have on hand 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 outside but close to the corner. Label the top angle B and label the 90-degree angle 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 instead of 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 what angle to use.
We are almost ready to explain what SOHCAHTOA really stands for, but there is one point I want to emphasize that most Geometry students miss. When we write in the shorthand version sin = opp/hyp, we are missing a very important part of the declaration. These proportions depend on the angle used. The shortened version sin = opp/hyp represents the longer 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 sine. from A or cosine from B golden tangent of X. NEVER FORGET THE ANGLES!
Using X to represent the angle, SOHCAHTOA represents the following ratios: sine x = opposite/hypotenuse, cosine X = adjacent/hypotenuse, and tangent X = opposite/adjacent. They are often written in abbreviated form as: sin = opp/hyp, cos = adj/hyp, and tan = opp/adj.
We’ll see how to use SOHCAHTOA to find missing sides and angles in another article, but as a quick review of what we’ve just discussed here, let’s use some specific sides. Let’s use a 3, 4, 5 right triangle and the drawing we made earlier. Label the hypotenuse 5, the base side 3, and the vertical side 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 will give meaning and purpose to the process we are introducing. For now, it’s important to remember that the 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 actually find the missing angle, and in another article we’ll see how to give meaning to these 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 quick ballpark estimate. We can learn that skill too.
SOHCAHTOA is a very powerful tool, one that you want to master as quickly as possible. Also, it makes you look VERY SMART!!!!!! That in itself is worth a lot!
