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Geometry For Beginners – How To Use SOHCAHTOA To Find Missing Measurements In A Right Triangle
As has been discussed in several articles in this series, the primary focus of Geometry is to find missing measurements–both side lengths and angle measures–in geometric figures. We have already shown how the 36-60 right and 45-right special triangles can help. In addition, we started looking at another potential shortcut, SOHCAHTOA. This is a mnemonic device for remembering the trigonometric ratios; and in a previous article, we discussed this device at length from the standpoint of what the letters stand for and what the trig ratios actually represent. In this article, we will put this information to work as a tool to find the missing measurements in any right triangle.
Remember that SOHCAHTOA is telling us which two sides of a right triangle form the ratio of each trig function. It stands for: sine = opposite side/ hypotenuse, cosine = adjacent side/ hypotenuse, and tangent = opposite side/ adjacent side. You must remember how to spell and pronounce this “word” correctly. SOHCAHTOA is pronounced sew-ka-toa; and you must emphasize to yourself out loud the ‘o’ sound of SOH and the ‘ah’ sound of CAH.
To begin working with SOHCAHTOA to find missing measurements–usually angles–let’s draw our visual image. Draw a backwards capital “L” and then draw in the segment connecting the endpoints of the legs. Label the lower left corner as angle X. Let’s also pretend we have a 3, 4, 5 right triangle. Thus, the hypotenuse has to be the 5 side, and let’s make the base leg the 3 leg and the vertical leg the 4 leg. There is nothing special about this triangle. It just helps if we are all picturing the same thing. I chose to use a Pythagorean triple of 3, 4, 5 because everyone already knows the sides really do form a right triangle. I also chose it because so many students make an assumption they shouldn’t! For some unknown reason, many Geometry students believe that a 3, 4, 5 right triangle is also a 30-60 right triangle. Of course, this can’t be since in a 30-60 right triangle, one side is half the hypotenuse, and we don’t have that. But we are going to use SOHCAHTOA to find the actual angle measures and, hopefully, convince people the angles are not 30 and 60.
If we only knew two sides of the triangle, then we would have to use whichever trig function uses those two sides. For example, if we only knew the adjacent side and the hypotenuse for angle X, then we would be forced to used the CAH part of SOHCAHTOA. Fortunately, we know all three sides of the triangle, so we can choose whichever trig function we prefer. Over time and with practice, you will develop favorites.
In order to find the angles these trig ratios will determine, we need either a scientific or graphing calculator; and we will be using the “second” on “inverse” key. My personal preference is to use the tangent function when possible, and since we know both the opposite and adjacent sides, the tangent function can be used. We can now write the equation tan X = 4/3. However, to solve this equation we need to use that inverse key on our calculator. This key basically instructs the calculator to tell us what angle produces that 4/3 ratio of sides. Type into your calculator the following sequence, including the parentheses: 2nd tan (4/3) ENTER. Your calculator should produce the answer 53.1 degrees. If, instead, you got 0.927, your calculator is set to give you answers in radian measure and not degrees. Reset your angle settings.
Now, let’s see what happens if we use different sides. Using the SOH part of the formula gives use the equation sin X = 4/5 or X = inverse sin (4/5). Surprise! We still find out that X = 53.1 degrees. Doing likewise with the CAH part, gives use cos X = 3/5 or X = inv cos (3/5), and…TA DAH…53.1 degrees again. I hope you get the point here, that if you are given all three sides, which trig function you use makes no difference.
As you can see, SOHCAHTOA is a very powerful tool for finding missing angles in right triangles. It can also be used to find a missing side if an angle and one side are known. In the practice problem we have used, we knew we had sides 3, 4, and 5, and a right angle. We just used SOHCAHTOA to find ONE of our missing angles. How do we find the other missing angle? By far the quickest way to find the missing angle is to use the fact that the total of the angles of a triangle must be 180 degrees. We can find the missing angle by subtracting the 53.1 degrees from 90 degrees for 36.9 degrees.
Caution! Using this simple method seems like a good idea, but because it is dependent on our work for another answer, if we made a mistake on the first answer, the second is guaranteed to be wrong as well. When accuracy is more important than speed, it is best to use SOHCAHTOA again for the second angle, and then check your answers by verifying the three angles total 180 degrees. This method guarantees your answers are correct.
I also hope you understand that a 3, 4, 5 right triangle is NOT a 30-60 right triangle. It is close, with angles of 36.9 and 53.1 degrees, but definitely not the same!
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