Geometry for Beginners: How to Use SOHCAHTOA to Find Missing Measurements in a Right Triangle

As discussed in various articles in this series, the main focus of geometry is to find missing measurements, both side lengths and angle measurements, in geometric figures. We have already shown how the special triangles 36-60 rectangles and 45 rectangles can help. Also, we started looking for another potential shortcut, SOHCAHTOA. This is a mnemonic device for remembering trigonometric ratios; And in a previous article, we discussed this device in depth from the point of view of what the letters represent and what the trigonometric relationships 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 tells us which two sides of a right triangle form the ratio of each trigonometric function. What represents: sine = oropposite side / hypotenusa, vsosine = Foradjacent side / hypotenusa, and tangent = oropposite side / Foradjacent side. You must remember how to spell and pronounce this “word” correctly. SOHCAHTOA is pronounced sew-ka-toa; and should emphasize loudly 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 a capital “L” backwards and then draw the segment connecting the ends of the legs. Label the lower left corner angle X. Suppose also that we have a 3, 4, 5 right triangle. Therefore, the hypotenuse has to be side 5, and let the base leg be leg 3 and the vertical leg the 4th leg. 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 actually form 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 can’t be because in a 30-60 right triangle, one side is half the hypotenuse and we don’t have that. But we will use SOHCAHTOA to find the actual 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 any trigonometric function that uses those two sides. For example, if we only knew the adjacent side and hypotenuse for angle X, then 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. With time and practice, you will develop favorites.

To find the angles that will determine these trigonometric relationships, we need a scientific or graphing calculator; and we will use the “second” key in “backspace”. 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. 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 tells the calculator to tell us what angle produces that 4/3 side ratio. Type the following sequence on your calculator, including 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 up to give you answers in radians 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, the equation sin X = 4/5 or X = inverse sin (4/5) is used. Surprise! We still find that X = 53.1 degrees. Doing the same with the CAH part, you 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, 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 that we had sides 3, 4, and 5, and a right angle. We only use 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 in a triangle must be 180 degrees. We can find the missing angle by subtracting 53.1 degrees from 90 degrees to 36.9 degrees.

Bail! Using this simple method seems like a good idea, but because it depends on our work to get another answer, if we made a mistake in the first answer, 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 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 it’s definitely not the same!