Explaining How SohCahToa Works
Date: 07/26/2004 at 14:04:33 From: Justin Subject: Sohcahwhat I don't do very well in math, and I'm really confused about trigonometry. Can you please explain what "sohcahtoa" is all about so that I can understand it? I'd really appreciate it.
Date: 07/26/2004 at 14:24:27 From: Doctor Achilles Subject: Re: Sohcahwhat Hi Justin, Thanks for writing to Dr. Math. SohCahToa is a memory trick that many students learn to help them remember certain ratios in trigonometry. I'll start by talking about the ratios, then show you how SohCahToa can help you remember them. Let's say we have a right triangle ABC with sides a, b, and c (the side is opposite the angle and uses a small letter, so side "a" is opposite angle "A"): B /| / | c / | / | a / | / | ------- A b C There are six possible ratios of the three sides: a/c b/c a/b b/a c/b c/a Each of these six ratios has a name in trigonometry. Three of them are the most commonly used, so let's look at those. Imagine we're standing at angle A and looking at the three sides of the triangle from there. The side opposite angle A is side a. The side adjacent (touching) angle A is side b. The hypotenuse of the triangle is side c (remember, the hypotenuse is the side opposite the right angle, and always the longest of the three sides). One of the six possible side ratios is called the sine, and it's defined as the side opposite the angle divided by the hypotenuse. So, from angle A: sine of angle A = sin(A) = opposite side/hypotenuse = a/c The cosine ratio is defined as the side adjacent to the angle divided by the hypotenuse. So, from angle A: cosine of angle A = cos(A) = adjacent side/hypotenuse = b/c Finally, the tangent ratio is defined as the side opposite the angle divided by the side adjacent to the angle. So, from angle A: tangent of angle A = tan(A) = opposite side/adjacent side = a/b Now, let's look at sohcahtoa again. Break it into three parts: soh cah toa In each part, the first letter stands for the ratio name (sine, cosine, tangent) and the next two letters are the sides of that ratio (opposite, adjacent, hypotenuse). soh means Sine = Opposite/Hypotenuse cah means Cosine = Adjacent/Hypotenuse toa means Tangent = Opposite/Adjacent Again, sohcahtoa is just a memory trick to help you remember these three ratios. The other three possible side ratios are the reciprocals of these three, and that's how people remember them. Let's go back to our triangle again and stand at angle B instead of A: B /| / | c / | / | a / | / | ------- A b C Note that from angle B, b is the opposite side and a is the adjacent side, while c is still the hypotenuse. Let's use sohcahtoa to find the three main ratios from angle B: soh means sin(B) = opposite/hypotenuse = b/c cah means cos(B) = adjacent/hypotenuse = a/c toa means tan(B) = opposite/adjacent = b/a As you can see, which of the possible side ratios you get depends both on which function (sine, cosine, tangent) you do and also on which angle you are using. These three basic ratios are the cornerstone of most of the work you will do in trigonometry, so understanding them and remembering them is important. SohCahToa helps you do both! If you're still stuck or you'd like to discuss some specific problems using trig, please write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/
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