So far, with the help of SOHCAHTOA, we have seen that:

Sine = opposite / hypotenuse

Cosine = adjacent / hypotenuse

Tangent = opposite / adjacent

These new functions are related to the originals because they represent the inverse ratios. (Of course, inverse means you swap the top and the bottom…)

Cosecant = hypotenuse / opposite… (compare to Sine)

Secant = hypotenuse / adjacent…… (compare to Cosine)

Cotangent = adjacent / opposite…… (compare to Tangent)

Also, these functions can be abbreviated: Cosecant = Csc, Secant = Sec, Cotangent = Cot.

At the middle school and high school math level, you will rarely have a need to use these functions, but it is good for you to know what they are. However, most trig problems at this stage can easily be solved with the original three functions.  Just in case, though, it’s always good to know all the trig functions: sine, cosine, tangent, secant, cosecant, cotangent. If you’d be interested in additional material, a good place to start would be these links that explain secant, cosecant, and cotangent. You can also search that site for many other math concept explanations (just remember to come back here!)

Thanks for reading this quick “cheat sheet” version of these additional trig functions. Please remember to bookmark my site to come back again!

If you’ve been following along up to this point, the tangent definition probably doesn’t even need to be explained to you, since it is so similar to its sine and cosine brothers. I’m sure you have figured it out already! However, for those who came directly to this page without going through those lessons, I’m going to give the full explanation anyways.

The tangent trigonometry function’s definition is another simple one. The tangent of an angle in a right angle triangle is the ratio of its opposite side length divided by its adjacent side length. This is as easy as it gets! To calculate the tangent of the angle, divide one side length by the other side length, and you’ve got your answer! If you want to see a cool interactive tangent calculator, take a look at Math Open Reference to see that demonstration. (Make sure you read the rest of my post first, though!)

In more general terms, you will find this relationship expressed as follows: TanB = Opposite / Adjacent. Tangent is often shortened to just tan, and in this case, TanB is saying “tangent of angle B”. ou could have TanA or TanZ, or even use Greek letters to designate the angle (commonly theta for an angle).

So, once again going back to our basic right angle triangle, recall the triangle naming system, where the angles are shown by capital letters, and the sides directly opposite them are the lower case letters (e.g. the vertical side here should be name “b”). More importantly, recall the relative naming system, where the side names depend on what angle you are talking about. They hypotenuse is always the longest side, directly opposite to the right angle. Then, you have the side names “adjacent” and “opposite” to apply to the sides as they relate to the angle in question. For example, for angle B, the vertical side (AC) can be called “b” or “opposite” (because it is opposite the angle), where as the bottom horizontal side (BC) will be called “a” or “adjacent” (relative to angle B). It is important to realize that these names are relative, be the opposite and adjacent sides to angle B are NOT the same as the opposite and adjacent sides to A. If you don’t pay attention, you will quickly make errors in your trig calculations. This is a very basic concept that is easy to mix up, so it pays to understand it.

So then, we have TanB = b/c. We also have Tan = Opposite / Adjacent, which is easily remembered by our SOHCAHTOA trick. The last part is the key for tangent: TOA means “Tangent is Opposite over Adjacent.”

Working with the tangent trigonometry function is just as easy as doing calculations with sine or cosine. All you need is the basic side lengths are angles to be able to work out the simple relationship. Combining the tangent function with sine and cosine functions, you now have a solid understanding of the basic trig functions that will allow you to figure out any missing side length in a triangle. Furthermore, if you specifically have all of the triangle sides and are looking to calculate the missing angles, you can use the inverse trig functions (usually combine the ‘shift’ key on your calculator with the trig buttons) to fill in those gaps. (In this case, finding the angle when you already know the opposite and adjacent side lengths, you can use arctan.)

I’ll maybe make a small note here to also say that you will often see the term “tangent” when working with circles as well. At first glance, you may think that tangent actually has two separate meanings now (as I mistakenly implied originally). Whereas it can be used to describe the ratio of opposite and adjacent sides in triangles (as I have demonstrated in this post), the tangent of a circle denotes a line that touches a circle at precisely a single point (i.e. it doesn’t intersect or cross the circle at more than one point). The tangent line of a circle always perpendicular to the radius of the circle where it touches. You can connect these two thoughts with a circle. First, draw a circle, and then draw a radius from its center to its circumference. Then draw a “tangent line” (right angle) from the point where the radius meets the circle, and then connect this line back to the center. You have now created a triangle consisting of a radius, the tangent line, and the adjoining hypotenuse. If you consider the angle the the hypotenuse makes with the radius, say A, then by the definition of the trigonometric function, you have TanA = opposite / adjacent. The “opposite” in this case, is what we have already identified as the tangent line to the circle. So, hopefully this demonstrates to you that while we may talk about a “tangent line” to a circle, it all really comes down to the basic trig identity again.

The cosine definition is actually just as simple as the sine definition, though you need to keep straight which is which. Let me first show you what cosine means (the abbreviation for cosine is just “cos”), and then I will reiterate the trick so that you don’t confuse these similar math functions!

The cosine of an angle in a right angle triangle is quite simply “the ratio of the length of the side adjacent to your angle and the length of the hypotenuse.” That’s all there is to it: a ratio of two numbers. Of course, the trick is remember which two numbers are specific to cosine. The good thing is that in a triangle, there are only three sides to choose from. But more importantly, there is a very easy memory trick so that you will never mix the ratio up with the sine ratio.

As I mentioned in my last post, this great memory trick I’m talking about is to just remember this completely made-up, nonsense word: SOHCAHTOA.

(MathsRevision has a pretty good trigonometry page that explains this concept similarly, with a different example and a short video).

If you break the word into its three components, you will understand quickly why this word is so great. Remember how the word sounds, then just put a space after every three letters, which gives you three words made up of three letters each: SOH CAH TOA.

The beauty of this word is that it tells you all you need to know about the three basic trig angles. The first letter of each component refers to the trigonometric function, and the following two letters represent the names of the sides that are involved in that ratio.

The key to understanding this is to remember the triangle naming rules, and what is meant by “opposite” side, “adjacent” side, and “hypotenuse.”

So what does SOHCAHTOA tell you? It tells you that “Sine is Opposite over Hypotenuse” (see the first letters? S.O.H. = Sine Opposite Hypotenuse). Similarly, it tells you “Cosine is Adjacent over Hypotenuse.” I’ll leave the last part for my last post in this three part series, but I’m sure you can figure it out pretty easily now.

So, there you have it! Now you understand what is meant by the cosine definition, how it applies to your trigonometry homework, and how to use it. For more information, or some great examples to practice with, check out this post at mathisfun.com. Also, the Wolfram Alpha page for cosine is also very helpful.

As I’ target=”_blank”m sure you’ve figured out already, my next post will be on the third of the three basic trig functions. (The suspense must be killing you!)

The sine definition relates an angle of a triangle to the ratio of its opposite side and the hypotenuse. So, if we look at angle B in the picture below, (and keeping in mind the notation for sides), we can see that SinB = b/c.

In more general terms, SinB = opposite/hypotenuse. We write “Sin” as the shorthand form of Sine. (Sounds like mathematicians are kind of lazy… since it appears that writing 4 letters was too much, they had to shorten it to 3!)

Similarly, SinA = a/c. Again, it is opposite/hypotenuse… remember how I said it was important to understand the RELATIVE notation scheme! If you need a refresher, take a look at my prior post that goes over triangle naming conventions.

SOHCAHTOA is the trig acronym that describes the three basic functions and their ratios. It may be easier to remember if you look at it with spaces: SOH CAH TOA. The one that we are interested in this post is the SOH term, which is short for “Sine is Opposite over Hypotenuse.” There are many other mnemonics that can be used to remember the order of these relationships, and to help make this a more fun post, I encourage everyone to leave their favourite memory trick for SOHCAHTOA in the comments below. There is no one right way to have soh cah toa explained. Whatever works for you to help you remember is all that is important.

Working with the Sine function is fairly straightforward, and usually just a matter of plugging the appropriate numbers into the ratio.

For example, you can imagine a triangle with known side lengths, and be asked to find the angles. This is a very common and simple question that you will get. In this case, you would say Sin(B) = opposite/hypotenuse (where you substitute in the known values for the sides.) This will give you Sin(B) = “some value.” And now, just as in working with addition or multiplication, when you want to solve for a specific variable, you have to isolate it… and to isolate it, you must do the same thing to both sides. Therefore, to get rid of the Sine, you must do ‘inverse sine‘ to each side (which is usually the same calculator button, but pushing SHIFT to access it). Then you’ll get B = inverse sine of (some value), which is your answer. Technically, you are taking the arcsine of the ratio.

Similarly, if you know a mixtures of some of the sides and some angles, you may be asked to find the unknowns. You can then say Sin(known angle) = opposite/hypotenuse (where one of these sides is known and the other unknown), and then just simply solve for the unknown side length.

Here’s a quick 3-4-5 triangle example:

So from this triangle, using the sine definition, we can tell that:

SinA = (4/5)

SinA = 0.8 (now push inverse sin on your calculator…)

A = 53.13 degrees

Also:

SinB = (3/5)

SinB = 0.6

B = 36.87 degrees

A useful trick for quickly solving triangles is to understand that if you sum up the three angles, they will always total 180 degrees. So for this triangle, after solving angle A, you could subtract it and 90 from 180 to find B. Of course, this defeats the purpose of practicing Sine in this example.

On the other hand, if we already knew that angle B = 36.87 degrees, and we wanted the length of the unknown hypotenuse, then we do:

Sin(36.87 degrees) = 3/hypotenuse

0.6 = 3/hypotenuse

hypotenuse = 3/0.6 = 5

As you can see, there really isn’t anything complicated about performing these steps. For the most part, it’s simple arithmetic with a few things about triangles thrown in for good measure. I hope this explains the sine definition so that it is understandable, and I also hope that now that you have had sohcahtoa explained, that makes more sense too. As always, please don’t hesitate to comment if you’re unsure or if you would like additional help. I’ll discuss cosine and tangent in the next few posts.

You may be taught these in school slightly differently. Instead of using the names arcsine, arccosine, and arctangent (arcfunctions!), it is also very common to see these represented as the basic trig notation with a -1 exponent, like these:

arcsine = sin^-1

arccosine = cos^-1

arctangent = tan^-1

(Interestingly, SparkNotes indicates that these functions are typically written with a capital letter, as in Arcsin, though I don’t seem to see that convention commonly used elsewhere. Leave me a comment below if you know more about this.)

These are likely the symbols that you will see on your calculator. Typically, they are the shifted function of the regular sin, cos, and tan buttons. It is important to note the distinction between the basic trigonometric functions and these new inverse trig functions. When using the basic trig functions, the value you are obtaining is the ratio of the two relevant sides of the triangle for the given angle. When using the inverse trig functions, what you are solving for is the actual angle that produces the given ratio of sides. So, make sure you push the right button on the calculator! Also, it is important to realize that these -1 exponent notations are NOT actual exponents indicating to take 1/sine, etc. They are simply a notation to imply the inverse trig functions.

Now, follow along with an easy trick that I explain below to help you work with inverse trig functions, and then go check out Math Warehouse, where there is a really good explanation, as well as a bunch of problems and a video for working with inverse sine, cosine and tangent.

The most basic way of finding an inverse function, in general, is a trick I was taught long ago, where you take your given function, substitute y for f(x), switch the x and y, and then rearrange.

f(x) = x² + 5

y = x² + 5…. now switch x and y

x = y² +5…. and rearrange

y² = x – 5

y = sqrt(x-5)

And there you have determined the inverse function. This is the same strategy that is being applied when we are talking about inverse trig functions. However, having the inverse trig buttons on our calculators really take all of this extensive and possibly difficult rearranging and calculating out of the picture.

Here is a basic example of one of these inverse trigonometric functions. Hopefully you will see that they are extremely easy to work with.

Find the angle for the given trig ratio:

sin(θ) = 1 /√2

θ = sin^-1(1 /√2)

θ = 45°

Hopefully this introduction to the inverse trig functions has been useful for you. There is a lot more information about this math concept that is probably quite beyond the scope of what is necessary to actually solve most of the basic trigonometry questions you will find in your maths homework. I have found several good resources, if you would like to learn more about inverse trigonometric functions. They can be found on The Math Page (among many other great math resources out there). Please leave a comment below if you know of any other great sites that you would like to share with other readers.

The Pythagorean Identities get their name because they are based on the famous Theorem of Pythagoras. You are very likely already familiar with it. Simply, for a right angle triangle, it says “the square of the hypotenuse is the sum of the squares of the other two sides.” Mathematically, you have seen this represented as:

a² + b² = c², where a and b are sides and c is the hypotenuse. 

Now, I will show you how to derive these special trig identities, using this theorem as our starting point. To do this, we need to start with a right triangle, created by the radius of a unit circle and the axis:

We can say that the right triangle formed by dropping a line from the point that the radius touches the circle (anywhere in quadrant I is sufficient for this demonstration) down to the axis has a base of x units long and y units high. (The actual numbers are not important, but they will depend on the specific angle, if you did need to calculate them for whatever reason. You don’t here.) The radius in a unit circle, by definition, is 1. Now, let’s apply the definitions of sine and cosine to our triangle. Recall:

sin(ɵ) = opposite / hypotenuse = y / 1 = y

cos(ɵ) = adjacent / hypotenuse = x / 1 = x

So, now we can relabel our diagram by substituting in these basic trig identities.

With the triangle now correctly labeled for our derivation, we can apply the Theorem of Pythagoras to arrive at one of the Pythagorean Identities. Since a² + b² = c², we can therefore equate the sides of our triangle to these terms to give us our first of the trig Pythagorean Identities:

sin²(ɵ) + cos²(ɵ) = 1

If you have followed along up till now and understood everything I’ve done, then you are well on your way to remembering this trigonometric identity. If you can remember how to derive it, you don’t even have to memorize it (though it always helps!) For the next Pythagorean Identity, you start with this first identity, and you apply some basic algebra and trigonometry to it to derive the second and third identities. Recall the definitions of secant, cosecant, and cotangent:

sec(ɵ) = 1 / cos(ɵ)

csc(ɵ) = 1 / sin(ɵ)

cot(ɵ) = 1 / tan(ɵ) = cos(ɵ) / sin(ɵ)

With those inverse trig functions in mind, let’s take the first Pythagorean Identity and divide all of its terms by cos²(ɵ). That gives you: 

1 / cos²(ɵ) = sin²(ɵ) / cos²(ɵ) + cos²(ɵ) / cos²(ɵ)

sec²(ɵ) = tan²(ɵ) + 1

And this is the second Pythagorean Identity! Using the same strategy we just used to derive that one, go back to the first one and divide everything by sin²(ɵ), to arrive at the third Pythagorean Identity!

csc²(ɵ) = 1 + cot²(ɵ)

I hope that from this tutorial, you now understand how these identities get their name, how you can derive them, and how to use this knowledge to help you to memorize or recall them. Using the fundamental trigonometry identities and trig relations, it is easy to come up with more advanced trigonometric formulas. If you need to refer back to this Pythagorean Identities list, please bookmark this page and come back again.

Special angles are great to know because their trigonometric functions equate to very specific and known ratios, so if you can memorize these it will save you a lot of time in doing trigonometry homework! To make things a bit easier, if you can’t remember these exact values, it is even easier to memorize the special angle triangles that these angles are based off of! And there are only two triangles, so you will find that it is very easy to derive the trig functions if you can’t remember them.

Specifically, the trig functions are easy to find for these special angles, which are: 0, 30, 45, 60, and 90 degrees.

45-45-90 Triangle

This will hopefully make sense after looking at the triangles I mentioned. Here’s another site that also talks about remembering the patterns of these triangles instead of specifically remembering the math. Create a right angle triangle with two 45 degree angles, and with two sides of 1 unit length. By using the Theorem of Pythagoras, you can find that the hypotenuse of this triangle is easy to calculate to be length √2. This is what this triangle looks like:

So then, from these values and using the memorization trick of SOHCAHTOA, you can obtain the trigonometric values for this special angle of 45 degrees. You can work out that:

Don’t worry if you can’t remember these values and ratios. The easiest way to remember them is to memorize how to construct the special angle triangle. And as you can see, this triangle is very simple: a right angle triangle with a 45 degree angle and 2 sides of length 1, and you can easily fill in the rest and then work out the ratios yourself.

30-60-90 Triangle

The second of the special angle triangles, which describes the remainder of the special angles, is slightly more complex, but not by much. Create a right angle triangle with angles of 30, 60, and 90 degrees. The lengths of the sides of this triangle are 1, 2, √3 (with 2 being the longest side, the hypotenuse. Make sure you don’t put the √3 as the hypotenuse!). FreeMathHelp also has a good explanation of this particular triangle. This triangle looks like this:

Here are the trig ratios that you can easily find:

Once again, just remember the triangle, and the ratios are easy to derive!

For 0 and 90 degrees, there isn’t a triangle to remember (although please feel free to correct me if I am wrong!), so you will actually have to memorize these values. However, these aren’t complex. I usually just remember the pattern of the following list:

If you can’t memorize the actual trigonometric ratios for the special angles, the key is to recall the special angle triangles that describe them. Make sure that you know how to construct the triangles, and then you can solve the trig ratios of the trigonometry special angles. You will quickly find that doing trigonometry questions that use these special angles are easy!