Trigonometry: SOHCAHTOA and Special Triangles

In our last trigonometry video, we discussed SOHCAHTOA. Now we move on to a discussion of SOHCAHTOA and special triangles.

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Transcript: SOHCAHTOA and Special Triangles

Now that we’ve introduced SOHCAHTOA, we can talk about the very important topic of SOHCAHTOA and special triangles.

So this lesson will assume, first of all, that you’re familiar with SOHCAHTOA, which we covered in the previous lesson. And it will also assume that you’re familiar with the idea of the special right triangles.

Special Right Triangles

These special right triangles are the 45-45-90 triangle and the 30-60-90 triangle. So, if you don’t know much about these triangles or it’s really fuzzy, it might help to go back to that lesson in the geometry module, special right triangles, which explains the two triangles in-depth. So here, we’ll be assuming that you already understand the basics of those triangles.

First of all, the 45-45-90 triangle. As you may remember, these are the angles and proportions. Sometimes people call this the 1-1- radical 2 triangle, that denotes the sides of the triangle. Sometimes it’s also called the Isosceles Right Triangle. This is the only possible triangle that is an Isosceles Right Triangle.

Finding the Value of SOHCAHTOA for Special Triangles

Because we know all the proportions for the sides, we can find the value of the three SOHCAHTOA functions for 45 degrees. So we’ll do that right now.

Sine, of course, if opposite over hypotenuse, so starting at either 45 degree angle the opposite is 1 and the hypotenuse is radical 2. Now we have to rationalize this fraction which means that we’re gonna multiply root 2 over root 2.

And that will give us root 2 over 2. That is the rationalized fraction, the rationalized form of the reciprocal of root 2. And that is the sign of 45 degrees. Well of course, the cosine of 45 degrees is going to be exactly the same thing. It’s gonna be 1/root 2 and we’re gonna rationalize it and get root 2/2.

So the sine and the cosine are equal precisely because the opposite and the adjacent are equal. The tangent opposite over adjacent, that is going to be 1/1, which is just 1.

45 Degrees

By the way, notice, if an angle is less than 45 degree, then it has a tangent of less than 1. If an angle is greater than 45, then it has a tangent greater than 1.

So that’s an important point. And of course, when it’s exactly 45 degrees, the tangent is exactly 1. Now we’ll talk about the 30-60-90 triangle. As you may remember, we get this from cutting an equilateral triangle in half, these are the proportions. This is sometimes known as the 1-root 3-2 triangle that denotes its sides.

Sometimes, we also call it the half equilateral triangle to help us remember that, in fact, we got it from cutting an equilateral triangle in half. We can use SOHCAHTOA ratios for both the 30 degree and 60 degree angle using this triangle, because we know all the sides. So first of all, let’s start with 30 degrees. From 30 degrees the opposite is 1 the hypotenuse is 2.

So the sine is just one half. For the cosine, the adjacent is root 3, and my hypotenuse is still 2, so that’s root 3/2. For the tangent, the opposite is 1, and the adjacent is root 3. So that’s 1/root 3, again we have to rationalize this, multiply by root 3/root 3, and we get root 3/3.

And that is the rationalized form of the tangent of 30. So a couple of things to notice. Notice that 30 degrees is an angle less than 45 degrees. So the cosine has to be bigger than the sine precisely because for an angle less than 45 degrees, the adjacent has to be bigger than the opposite. And for much the same way, the tangent has to be less than 1.

Now, for the sine of 60 degrees. So the opposite over the hypotenuse now is root 3/2. The adjacent over the hypotenuse now is one over two, or one half. And the tangent now, opposite over adjacent is root 3/1, which is just root 3. So for 60 degrees, which is an angle greater than 45 degrees, notice that the sine has to be greater than the cosine and the tangent has to be greater than 1.

Both of those precisely because for an angle greater than 45 degrees, the opposite is bigger than the adjacent. Conceivably, the test could expect you to know any of those nine values. Most often, the test supplies these values but it is still good to understand their basis.

Practice Problem

Here’s the practice problem. Pause the video and we’ll talk about this.

sohcahtoa and special triangles, pause - magoosh

Image By Denis Mikheev

Okay, so let’s think about this. We can find the value of each one of those. And so the sine of 30 degrees is one-half, the cosine is root 3/2. We’re gonna square that. And by squaring it we’re square the first one, twice the product of the terms and then the square of the second one.

Notice that we canceled the 2 and the one-half, so that means that we just have 1/4- root 3/2 + 3/4. The 1/4 plus 3/4 is just 1. So it’s 1- root 3/2. We go back to our answer choices and we select that as our answer. In summary, here are the values that are good to know.

For the sine of 45 equals to cosine of 45 that’s root 2/2. The tangent is 1, the sine of 30 is one-half. The cosine of 60 is also one-half. The cosine of 30 is root 3/2, so is the sine of 60. The tangent of 30 is root 3/3 and the tangent of 60 is root 3. These are nine very good values to know.

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  • Mike MᶜGarry

    Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as "member of the month" for over two years at GMAT Club. Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test.