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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 focus of geometry is finding missing measures — 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. Additionally, we started looking at another potential shortcut, SOHCAHTOA. It is a mnemonic device for memorizing trigonometric ratios; And in the previous article, we discussed this device in detail in terms of what the letters stand for and what the trig ratio represents. In this article, we will put this information to work as a tool for finding the missing measure of any right triangle.

Note that SOHCAHTOA is telling us which two sides of the right triangle form the ratio of each trig function. This means: **s**ine =** o**opposite side/ **h**hypotenuse, **c**Ocean = **a**near side/ **h**ypotenuse, and **T**agent =** o**opposite side/ **a**near side You must remember how to spell and pronounce this “word” correctly. SOHCAHTOA is pronounced sew-ka-toa; And you should say the ‘o’ sound of SOH and the ‘ah’ sound of CAH out loud.

To begin working with SOHCAHTOA to find a missing measure–usually an angle–we’ll image our scene. Draw the capital “L” on the back and then draw the section connecting the end points of the legs. Label the lower left corner as the X angle. Let’s also pretend we have 3, 4, 5 right triangles. Thus, the hypotenuse must have 5 sides, making the base leg 3 legs and the vertical leg 4 legs. There is nothing special about this triangle. It helps if you’re all picturing the same thing. I chose to use the Pythagorean triple 3, 4, 5 because everyone already knows that the sides actually make a right triangle. I also chose it because 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, this cannot be because in a 30-60 right triangle, one side is half the hypotenuse and we don’t have it. But we’re going to use SOHCAHTOA to find actual angle measures and, hopefully, convince people that angles aren’t 30 and 60.

If we only know two sides of a triangle, we need to use any trig function that uses those two sides. For example, if we only knew the adjacent side and the hypotenuse for angle X, we would be forced to use the CAH part of SOHCAHTOA. Fortunately, we know all three sides of the triangle, so we can choose any trig function we like. With time and practice, you will develop interests.

To figure out what angles these trig ratios will determine, we need either a scientific or graphing calculator; And we’re going to use “seconds” on the “inverse” 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, you need to use that inverse key on your calculator to solve this equation. This key basically tells the calculator to tell you which angle forms the 4/3 side ratio. Type the following sequence with parentheses into your calculator: 2nd tan (4/3) ENTER. Your calculator should produce the answer 53.1 degrees. Instead, if you get 0.927, 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, use the equation sin X = 4/5 or X = inverse sin (4/5). Surprise! We still know that X = 53.1 degrees. Doing the same with part CAH, 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 given all three sides, it doesn’t matter which trig 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 the missing side if one angle and one side are known. In the practice problem we used, we knew we had 3, 4, and 5 sides and a right angle. We just used SOHCAHTOA to find one of our missing angles. How to find the second missing angle? The quickest way to find the missing angle is to find that the total number of 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.

Beware! Using this simple method seems like a good idea, but since it depends on your work for the second answer, if you get the first answer wrong, the second answer 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 three angles totaling 180 degrees. This method guarantees that your answers are correct.

I hope you are not 3, 4, 5 right triangles 30-60 right triangles. It’s close, with angles of 36.9 and 53.1 degrees, but definitely not the same!

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