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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 main goal of geometry is to find missing measurements – both side lengths and angle measurements – in geometric figures. We have already shown how the special 36-60 right and 45 right triangles can help. Also, we started looking for another potential shortcut, SOHCAHTOA. This is a mnemonic device for memorizing trigonometric ratios; and in a previous article we discussed this device at length from the point of view of what the letters represent and what the trigonometric ratios actually represent. In this article, we’ll use this information as a tool to find the missing measurements in any right triangle.

Recall that SOHCAHTOA tells us which two sides of a right triangle form the ratio of each trigonometric function. This means: **s**ine =** oh**opposite side/ **h**ypotent, **vs**osin = **a**adjacent side/ **h**ypotent, and **you**agent =** oh**opposite side/ **a**adjacent side. You need to remember how to spell and pronounce this “word” correctly. SOHCAHTOA is pronounced sew-ka-toa; and you must underline out loud the “o” sound of SOH and the “ah” sound of CAH.

To start working with SOHCAHTOA to find the missing measurements – usually angles – let’s draw our visual image. Draw an upside-down capital “L”, then draw the segment connecting the ends of the legs. Label the lower left corner as angle X. Also suppose we have a right triangle 3, 4, 5. So the hypotenuse must be side 5, and let’s make the base leg leg 3 and the vertical leg leg 4. There is nothing special about this triangle. It just helps if we all imagine the same thing. I chose to use a Pythagorean triple of 3, 4, 5 because everyone already knows that the sides really form a right triangle. I also chose it because so many students assume 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, that can’t be the case since in a 30-60 right triangle, one side is half the hypotenuse, and we don’t have that. But we’re going to use SOHCAHTOA to find the actual angle measurements and hopefully convince people that the angles aren’t 30 and 60.

If we only knew two sides of the triangle, we would have to use the trigonometric function which uses those two sides. For example, if we only knew the adjacent side and the hypotenuse for the angle X, we would be forced to use the CAH part of SOHCAHTOA. Fortunately, we know the three sides of the triangle, so we can choose the trigonometric function we prefer. Over time and with practice, you will develop favorites.

In order to find the angles that these trigonometric ratios will determine, we need a scientific or graphing calculator; and we will use the “second” key on “reverse”. My personal preference is to use the tangent function when possible, and since we know both 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 this inverse key on our calculator. This key basically tells the calculator to tell us what angle produces this aspect ratio of 4/3. Type the following sequence, including the parentheses, into your calculator: 2nd tan (4/3) ENTER. Your calculator should produce the answer 53.1 degrees. If you got 0.927 instead, your calculator is set to give you answers in radians, not degrees. Reset your angle settings.

Now let’s see what happens if we use different sides. Using the SOH part of the formula results in using the equation sin X = 4/5 or X = inverse sin (4/5). Surprise! We always find that X = 53.1 degrees. Doing the same with the CAH part, we use cos X = 3/5 or X = inv cos (3/5), and…TA DAH…53.1 degrees again. I hope you understand here that if you have all three sides, the trigonometric 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 a 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 to find the other missing angle? By far the fastest way to find the missing angle is to use the fact that the total angles of a triangle must be 180 degrees. We can find the missing angle by subtracting 53.1 degrees from 90 degrees for 36.9 degrees.

Caution! Using this simple method seems like a good idea, but since it depends on our work for another answer, if we made a mistake on the first answer, the second one is guaranteed to be wrong as well. When accuracy is more important than speed, it’s best to use SOHCAHTOA again for the second angle, then check your answers by checking that all three angles add up to 180 degrees. This method ensures that your answers are correct.

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

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