As we have been discovering, learning the study of Geometry is primarily about finding the missing measures, both side lengths 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 to find the missing measurements in certain triangles.

We have already seen the “special” triangles of 30-60 rectangles and 45 rectangles. (These are sometimes called the special 30-60-90 and 45-45-90 triangles.) These right triangles have ratios or ratios for all three sides that are always the same, and we can use these familiar ratios to shorten the job. needed to find the measures 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, pronounced sew-ka-toa.

I know this word sounds like a Native American word, but it’s actually a mnemonic device 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 grasp on this information now. You will not stop using this at the end of Geometry.

The letters in SOHCAHTOA represent, in order from left to right, Sine, EITHERpostulate, hand powerful, againstbear, INunderlying, hand powerful, youangel, EITHERopposite, and INadjacent. At this point in your study of it, 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 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 form the six trigonometric ratios. Sine, cosine, and tangent are the three most commonly used trigonometric ratios of the six. As you 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 keep handy when reading math articles, draw a capital letter “L” upside down. 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. Therefore, labeling is impossible until we know which angle will be used.

We’re almost ready to explain what SOHCAHTOA really stands for, but there’s one point I want to make that most Geometry students miss. When we write in the shortened version sin = opp/hyp, we are omitting a very important part of the statement. These proportions depend on the angle that is used. The abbreviated 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 must be read sine from to or cosine from B golden 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. They are often written in shorthand as: sin = opp/hyp, cos = adj/hyp, and tan = opp/adj.

We’ll look at how to use SOHCAHTOA to find the missing sides and angles in another article, but as a quick review of what we 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 side of the base 3, and the vertical side 4, and we’ll use the angle names A and B and C from earlier. 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, reread this article 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 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 look at 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 rough 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 SO SMART!!!!!! That in itself is worth a lot!