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Geometry For Beginners – How To Use SOHCAHTOA To Find Missing Measurements In A Right Triangle

As discussed in several articles in this series, the primary purpose of Geometry is to find missing measures—both side lengths and angle measures—in geometric figures. We have already shown how the special 36-60 right and 45-right triangles can help. Additionally, we started looking at another potential acronym, SOHCAHTOA. This is a mnemonic tool to remember trigonometric ratios; and in a previous article, we discussed this tool at length, what the letters stand for and what the trig ratios actually represent. In this article, we will use this information as a tool to find the missing dimensions in any right triangle.

Remember that SOHCAHTOA tells us which two sides of a right triangle make up the ratio of any triangular function. It is for: sis = oopposite side hhypotenuse, cosin = oneneighboring hhypotenuse, and tagent = oopposite side oneneighboring side. You must remember how to spell and pronounce this “word” correctly. SOHCAHTOA is pronounced sew-ka-toa; and you must loudly pronounce the ‘o’ sound of SOH and the ‘ah’ sound of CAH.

To start working with SOHCAHTOA to find the missing dimensions – usually angles – let’s draw our visual image. Draw a large backwards “L” and then draw the part that connects the end points of the legs. Label the lower left angle as the X angle. Let’s pretend that there are 3, 4, 5 right triangles. So, the hypotenuse must be side 5, and let’s make the base leg 3 and the vertical leg 4. There’s nothing special about this triangle. It just helps that we all picture the same thing. I chose to use the Pythagorean triad of 3, 4, 5 because everyone already knows that the sides actually form a right triangle. I also chose it because many students make an assumption that 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 doesn’t happen because in a 30-60 right triangle, a side is half the hypotenuse, and we don’t have that. But we’ll use SOHCAHTOA to find the exact angle measures and, hopefully, convince people that the angles are not 30 and 60.

If we only knew two sides of the triangle, then we would have to use whatever trigonometry function used those two sides. For example, if we only knew the adjacent side and hypotenuse for angle X, then we would have to use the CAH part of SOHCAHTOA. Fortunately, we know all three sides of the triangle, so we can choose whichever function of the triangle we prefer. With time and practice, you will develop a preference.

To find the angles that will determine these trig ratios, we need either a scientific or a graphing calculator; and we will use “second” on the “reverse” key. My personal preference is to use the tangent function whenever possible, and since we know both opposite and adjacent sides, the tangent function can be used. Now we can 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 allows the calculator to tell us which angle makes the 4/3 side ratio. Enter the following line, including the parentheses, into your calculator: 2nd tan (4/3) ENTER. Your calculator should give the answer 53.1 degrees. If, instead, you got 0.927, your calculator is set to give you answers in radians rather than degrees. Reset your angle settings.

Now, let’s see what happens if we use different aspects. Using the SOH part of the formula uses the equation sin X = 4/5 or X = inverse sin (4/5). Surprise! We still learn that X = 53.1 degrees. Similarly with the CAH section, using cos X = 3/5 or X = inv cos (3/5), and … TA DAH … again gives 53.1 degrees. I hope you get the point here, that if you are given all three aspects, it doesn’t matter which function you use.

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 the angle and side are known. In the practice problem we 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? The fastest way to find the missing angle is to use the fact that all angles of a triangle must be 180 degrees. We can find the missing angle by subtracting 53.1 degrees from 90 degrees to 36.9 degrees.

Pay attention! Using this simple method seems like a good idea, but since it depends on our work for another answer, if we made a mistake in the first answer, the second one is guaranteed to be wrong. When accuracy is more important than speed, it’s best to use SOHCAHTOA again for the second angle, and then check your answers by adjusting the three angles to a total of 180 degrees. This method guarantees that your answers are correct.

I also hope you understand that the 3, 4, 5 right triangle is NOT the 30-60 right triangle. It’s close, with angles of 36.9 and 53.1 degrees, but definitely not the same!

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