Creating a Unit Circle Part 3 of 4

Creating a Unit Circle Part 3 of 4 - SOH-CAH-TOA

SOH-CAH-TOA

You must memorize for the rest of your life this nonsense phrase SOH-CAH-TOA (said "so" "KAH" "TOE-ah" Note: I teach math, not English, so give the pronunciation your best)

If you are wise, from this point onward in your Trigonometry career you will write SOH-CAH-TOA at the top of every worksheet, test, and scratch paper until it has become a part of you.

Here is what it stands for:

For the purposes of this lesson, we're just reviewing this information and we're not going to go too far in depth.

Which Side is Which?

It all has to do with how the side is placed relative to 𝚹 (Theta, we use Greek symbols to be variables for angle measurements). The Leg that is attached to or "Adjacent" to 𝚹 is the "Adjacent" side. The side not touching 𝚹 at all or the side that is "Opposite" 𝚹 is the "Opposite" side. the Hypotenuse is still the Hypotenuse, as always.

So, sin(𝚹) = the measurement of the side opposite to 𝚹 divided by the measurement of the hypotenuse

cos(𝚹) = the measurement of the side adjacent to 𝚹 divided by the measurement of the hypotenuse

Here's an example using the 30-60-90 triangle:

Practice Finding the Sin and Cos of Right Triangles

Remember that you can't leave a radical in the denominator!

Here is a worksheet with answers at the bottom to help you practice.

I also found this worksheet for more practice problems with the answers attached at the bottom.

I Still Need More Help!

Check out the Math is Fun Website's explanation

Here is a Khan Academy article that you might find useful

Watch this video to see Nancy, our Math BFF, walk through Trig Ratios

What Should I Have Filled Out In My Notes?

You should now be able to fill out the rest of the table with all of the answers for the sine and cosine of all of the angles in the table on your notes page.

Creating a Unit Circle Part 2 of 4

Welcome to Part 2 Changing Degrees to Radians and Radians to Degrees

Changing Degrees to Radians

The deal is this, you have probably only ever seen angles in degrees to this point in your math career. In fact, you may not have even known there was another way to measure angles, but yes, yes there is, and it is called the radian. 

From here on out, radians are going to be SUPER important, because degrees don't really translate well into working with other units, like feet and things, but radians do. In the end, we will end up using radians more in practical studies and uses of trigonometry than we will be using degrees.

The Degrees to Radians Proportion

We use proportion problems to change degrees to radians. It all starts with the fact that 180 degrees (a straight line) equals pi radians

 

To solve for radians or degrees I just set up the following proportion:

On the right side of the proportion, I have the ratio that I know, on the left side I would fill in the number that I know and solve for the one that I want.

For example, if I wanted to change 20 degrees into radians, here is how I would go about it:

So, we find that 20 degrees = 𝛑/9 radians

NOTE: YOU NEVER TURN 𝛑 INTO A DECIMAL!!!

With radians we always solve leaving 𝛑 as 𝛑. We just simplify all of the numbers around 𝛑, much like when we have a variable we are working around.

Now let's look at an example where we're solving to change 2𝛑/3 radians into degrees:

Though these efforts we see that 2𝛑/3 radians = 120 degrees

What Should I Have Filled Out In My Notes After This Section?

After you have finished this section, you should be able to fill in the  "Degrees to Radians" proportion section of your notes.

ALSO, on the table just below that section, you should be able to correctly convert all of the degrees to radians, and fill in the radians section.

Time to Practice!!

Check out this quiz, where you again get to race for your best time, but the one has some multiple choice questions!

Here is a link to the Khan Academy Practice page. The great thing about their page is it has links to their videos and other helpful information along with the practice.

This link will bring you to a worksheet you can print out and then check your answers using the attached answer key.

I NEED MORE!!!

Check out this Math BFF Video! I love my gal Nancy the MIT grad, she's got some awesome videos out there! (Full disclosure, we've never met, but I think she's awesome!)

Want to see another way to convert? Here's a website that explains another way to look at converting between the two.

The Math is Fun page will take you more in depth on what a radian is, talks more again about converting and will even give you practice problems at the bottom.

Creating a Unit Circle Part 1 of 4

Trigonometry all starts with the Unit Circle.
WHAT'S A UNIT CIRCLE?
It would be easy enough for you search "unit circle" and find one and copy it, but the best way to memorize and use one is to learn to create your own from scratch.

Basically, this is me teaching you how to fish, instead of treating you like a dumb fish

Start with These Notes

You're going to want to click the link above or this link right here and either print off or get these doodle notes somewhere you can draw on them on your computer.

Now for Some Special Right Triangles!

There are two special right triangles in math that are SO special that we call them THE Special Right Triangles. You need to know them and understand them before you can make your Unit Circle.

The 45-45-90 Triangle

Let us discover the 45-45-90 Triangle. (NOTE: if you are more of a visual learner, then feel free to skip down to the video below)

This triangle starts out life as a square

 Now we know some cool things about squares

1. All sides are the same length

2. all of the angles are 90 degrees

Another cool thing is that if you cut a square in half from corner to corner you get two triangles. (FYI, that's why the formula for the area of a square is base times height and the area of a triangle is 1/2 times base times height, I'll let you think on that)

Now, if I did my job right, and I cut this exactly in half, then I cut those two 90 degree angles in half too, so what do they measure now?

hmmm.... 90 divided by 2 = 45, so they are now both 45 degree angles!

Now, wait a minute, so you're telling me this triangle, has a 45 degree, a 45 degree and a 90 degree angle?

Oh yes, and that is why, clever mathematicians that we are, we call this special triangle the 45-45-90 triangle.

FYI, if you wonder or have forgotten, that little square in the triangle, where the 90 degrees should be is a symbol that means 90 degrees. It really doesn't matter how poorly I draw an angle or a triangle, if I put that little square symbol there, then no matter what anyone else says, it's a 90 degree angle.

Next, I'm going to take advantage of the fact that every side of the square has the same length, and call both sides of my new triangle a length of 1. For those of you well versed in math you are now gnashing your teeth at me that I have not given you units, but that's just how these special triangles go, they just have 1"unit" to each leg of the right triangle in this case

Now I've also made the other triangle lighter in color, so you can pretty much ignore it better. 

Now, we are going to use our old friend the Pythagorean Theorem from our days studying right triangles to find the length of hypotenuse of this triangle (or the green line. The purple lines remember are called "legs")

So, you can see that I first plugged in the lengths of the legs that I knew, so I could solve for the hypotenuse that I didn't know.

At the end there, I square root both sides to get right of the exponent or "squared" on the C, now can you guess why in the end I rejected the negative square root of 2??

YOUR RIGHT! Well, that is assuming you guessed that I can't have a negative length. We're not in the business of creating triangles that constantly suck away space, so triangles and other shapes in these cases can only have positive side lengths.

Now, I know what you're saying, wow, special....hmm.... What was the point of all of this?

The point!

All 45-45-90 triangles are proportional to each other. For example, if I have a 45-45-90 triangle and I know one of the legs is 5 feet long, then I can say "oh, if I multiply '1 unit' from my standard triangle above by 5 feet, then I would have a triangle like this one" Then I already know that the other leg of the triangle will also be 5 feet long, and I know that all I have to do to find the length of the hypotenuse is to multiply the square root of 2 by 5 or write down "5 square root of 2."

After I explain the next Special Triangle I'll give you some places to go and practice this, so it will make more sense and you can feel even happier about these awesome sauce triangles. First, stand up and stretch and get a drink of water, and get ready for.......

The 30-60-90 Triangle

Are you guessing that this one is probably named after its angles too? Because you'd be right. Oh those cooky Mathematicians, we do love patterns, don't we?

For this Special Triangle, we will be starting with an equal lateral triangle

BOOM! There my friends is our equal lateral triangle! (no it's not perfect, but we're going to pretend, okay?)

Let's review what we know about equal lateral triangles:

1. All of the sides are the same length

2. all of the angles are 60 degrees

This time, instead of calling every side 1 "unit" long, I'm going to call every side 2 "units" long. Also, I am again going to cut this shape in half to create two equal right triangles.

Ta Da! Now, if I again did my job right, then I should still have one 60 degree angle in my right triangle, which is the angle from the corner of the triangle that I left untouched, because remember in the equal lateral triangle, all of the angles started out at 60 degrees.

I should also  have cut that top angle exactly in half, so 60 degrees divided by 2 = 30 degrees now for that angle.

Lastly, by cutting that top angle in half, when I drew my straight line down from the top, I have also cut the bottom side of the triangle in half. 2 units divided by 2 = 1 unit

Meaning my new right triangle should look like this:

I made the other half lighter again, so as to make it easier to ignore from here on out.

And check that out! We do indeed have a triangle with 30 degrees, 60 degrees and 90 degrees, or a 30-60-90 triangle.

Once again, we need to solve for the missing side. The difference is that this time, we know the hypotenuse. This time, we need to solve for the missing leg length.

If you look above, you will see where I plugged the leg length I knew and the hypotenuse into the Pythagorean Theorem I was then able to solve to find that the other leg length is the square root of 3.

Once again, I was able to ignore the negative answer, because I can only have a positive side length.

How Do I Remember Which Side Length Goes with Which Side??

Once again, like with the 45-45-90 triangle, all 30-60-90 triangles are proportional, so we can use this standard one to solve for all others.

UNLIKE, the 45-45-90 triangle, then 30-60-90 triangle has inconveniently decided to have three different side lengths to memorize, instead of just two.

Remember this rule: The larger angle is opposite the larger side or if you prefer The smaller angle is opposite the shorter side.

This rule is ALWAYS true, not just for these special triangles.

How does that help? Well, let's say I remember the sides are 1, 2, and the square root of 3. I also remember it's a 30-60-90 triangle. The only other thing I need to remember is that the square root of 3 is bigger than 1, but smaller than 2 (which makes sense because the square root of 1 = 1 and the square root of 4 = 2, so square root of 3 is between the square root of 1 and the square root of 4, easier to see in the picture below)

Okay, yes, I could have also just put the square root of 3 into a calculator and seen that it was 1.73... but where's the fun in that?

Anyway, I also know that 30 < 60 < 90 , so that means that 1 goes across from 30 degrees, square root of 3 goes across from 60 degrees and 2 go across from 90 degrees.

What Should I Have in My Notes So Far?

Now, you should have filled out under the section heading "Special Triangles" the angles inside the triangles, and the corresponding side lengths. You should also have written down any other helpful reminders that you feel will help you best remember these concepts later, so you don't have to read through this whole article every time you want to know about Special Triangles again, you can simply look back at your notes

Practice! Practice! Practice!

It's important to stop and practice each concept before we move on.

Click here for a worksheet that has an answer key at the bottom, so you can first try the worksheet, then check to see if you are getting your answers right.

 Click here for an instant feedback quiz. You can even race to see if you can beat your best time or beat the best time of your friends. (Oh yes, this is going to be the next awesome app :-))

Not ready for the open ended questions? Click here for a multiple-choice test to get yourself started. 

Yeah, So, I Still Don't Get It, What Else Ya Got?

Check out the Purple Math Article on this topic

You might also want to Check out this YouTube Video, which you might find helpful. (FYI, that's not me, but I do enjoy the music at the beginning.) He also solves example problems in this video.

Leave Me Some Comments If You Still Have Questions or Comments