Special Triangles

Just like multiplication is a short-cut for repeated addition and we have tables of values that show us multiplication values (times tables), similar tables of values exist for trigonometry.  

Our goal with this section is to learn about where all of those values originate.  It would be a lot of information to memorize, but if understood, is easily recalled.

All of the values in the table come from repositioning two triangles on a coordinate plane.  They are the isosceles right triangle (45° – 45° – 90°), and the 30° – 60° – 90°.

There are some conventions (or rules) that guide how these triangles are positioned on a coordinate plane, but we’ll get into that in the next section.  For now, our goal is to really know and understand these two right triangles, inside and out.

While trigonometry can be used for any sides triangle, or any angle for that matter, these two triangles make the foundation.  Their values can be calculated without use of ugly tables, or the use of calculators.  As such, these are about as important to understanding trigonometry as anything other than the fundamentals and basics of what trigonometry is.

Let’s get to it.

Our first triangle is the isosceles right triangle.  Let’s apply the Pythagorean Theorem to this triangle and see what happens.  (Keep in mind that because it is isosceles, the two legs are the same length as are the two angles, both 45°.)

The reason we do not multiply 2 and 64 is because our next step is to take the square root.  When simplifying a square root we want to factor to find the largest square number.  So, it is easier to write 2 × 64, than it is to write 128, and then factor it.

What would happen if this triangle didn’t have sides of 8 (unknown units), and instead had sides of 19?

Here’s the pay-off.  No matter the size of the triangle, if it an isosceles right triangle, it is similar to both of these.  That means the ratio of the sides is the same.  So, the leg : hypotenuse is the same for the triangle with sides of 8 and the triangle with the sides of 19.  Let’s look at it to see it is true.

Now, with trigonometry we use special notation to signify these ratios.  We would say sin(45°), or cos(45°).  In either case, they’re equal to, “one divided by the square root of two.”

We can say that for an isosceles right triangle, the hypotenuse is equal to the leg times the square root of two.

Rationalize the Denominator

When using a calculator, rationalizing the denominator isn’t necessary.  But, this level of trigonometry is done without a calculator.  In this case, doing math with a rational denominator is a whole lot simpler than with an irrational denominator.  (The square root of two is irrational, it cannot be written as a ratio of integers, cannot be written as a decimal.  Our most accurate way of writing it is with the radical symbol.)

In case you forgot how to rationalize the denominator, let’s review. 

We had one triangle with a hypotenuse of 6, and found the sides were “three – root two.”  The other had a hypotenuse of 14, and the sides ended up being “seven – root two.”

No matter what the hypotenuse is, the sides are half of that times the square root of two.  Condensing what we’ve learned about the isosceles right triangle, we have the following:

Let’s look at our other triangle.  It isn’t a clean and pretty, but still easy enough to remember.  We’ll not go through the exploration of finding the sides, but instead just verifying the relationship between the sides.  We are talking about a 30°, 60°, 90° triangle, so the shortest side will be opposite of the 30° angle, while the longer side (not the hypotenuse), will be opposite of the 60° angle.

Because 30° < 60°, the sides opposite of each respectively, must also hold the same size relationship.  This is true because 1 <  The short leg is one-half of the hypotenuse, while the long leg is one-half times the square root of three, times the hypotenuse.  Since the square root of three is less than 2 (square root of four is two), “root – three divided by two,” is still less than one.  This is important because they hypotenuse is the longest side of the triangle.

So, the numbers are plausible, now let’s verify with the Pythagorean Theorem.  Keep in mind that the number we say is x could be any positive integer.  So this works for any whole number you can imagine.

Let’s see what happens with the trigonometric functions.  We will use the hypotenuse of x, because it could be any number.  We don’t need to explore specific values when we can explore all possible values at once!

This is ugly, but it can be rewritten with a division symbol like this:

Division is multiplication by the reciprocal. 

So, the sine of thirty degrees, regardless of the size of the triangle, is ½.  That means, the short leg, divided by the hypotenuse of any 30°, 60°, 90°, is ½.

And the cosine of 30°, will be shown below.

Summary:   These two special right triangles are the most important triangles to know when learning trigonometry.  While trigonometry can be used for any angle, the fundamentals will be developed with angles from these two triangles.  It is imperative that these are understood fully.

The first lesson (PowerPoint) shows why the relationships exist between the sides of the special right triangles.  To download the PowerPoint, please click here.

The second PowerPoint lesson shows, in greater detail, how to use the properties of the special right triangles to find missing sides and angles of triangles.  To download the PowerPoint, please click here.

This was recorded in a Google Meeting where students were asking questions and we reviewed the topic.  

Assignment Review