Hey, everyone. We just learned how trig functions relate angles to their corresponding point on the unit circle, where x and y values are cosine and sine values of that angle, respectively. And that's where memorization comes in. Now, memorization and math can be really tricky, but here I'm going to walk you through 2 different ways to memorize the trig values of the 3 most common angles, 30, 45, and 60 degrees. With this knowledge, you'll be able to solve pretty much any trig problem that gets thrown your way. So let's go ahead and get started here with what you may hear referred to as the 1, 2, 3 rule.

No matter how you choose to memorize these values, you're always going to start in the same way with the square root over 2. Because all of these trig values are the square root of something over 2, and our job is just to memorize what that something is. So memorizing that with the 1, 2, 3 rule, you may be wondering why it's called that. And it's because we're going to start with our x values and count 1, 2, 3 going clockwise. Then for our y values, we're going to count 1, 2, 3 going counterclockwise.

Now, what exactly do I mean by that? Well, we're going to start in this upper left corner with the x value of 60 degrees, and we're going to start counting from 1, going 1, 2, 3 clockwise around our unit circle for our x values. Then we're going to go back counterclockwise and count 1, 2, 3 back up for our y values. These are all of our trig values. We've gone 1, 2, 3 clockwise, 1, 2, 3 counterclockwise, and we're done. Now we can do a bit more simplification here because we know that the square root of 1 is just 1. So this x value or this cosine of 60 is really just 1/2, and the sine value of 30 degrees or the y value is also just 1/2.

Now, remember that these values, these x and y values, also represent the base and height of the corresponding triangle. And we also don't want to forget about our tangent value. Remember that the tangent of any angle can be found by simply dividing sine by cosine, so once we have those sine and cosine values, we can find our tangent pretty easily. So looking at 30 degrees here, if I take my sine value, 1/2, and divide it by my cosine value to get tangent, here I'm really just effectively dividing those numerators because they have the same exact denominators. So for my tangent, I get one over the square root of 3. Or with my denominator rationalized, I get 33 as my tangent.

Now you can find the tangent value of these other two angles the same exact way, and you can feel free to pause here and try that on your own. Now this was the 1, 2, 3 method, but remember that I promised you 2 different methods of memorizing this. So let's take a look at one more, a bit more visual hands-on approach to memorizing these values referred to as the left hand rule.

So for our left-hand rule, you're going to take, you guessed it, your left hand and put it in front of your face like this. Now to you, your hand will look something like this, and I want you to consider your pinky as being at 0 degrees and your thumb as being 90 degrees, effectively making your left hand the first quadrant of the unit circle. So your other three fingers are going to represent those three common angles. So 30 degrees, 45 degrees, and 60 degrees.

Now really imagine your left hand as being that first quadrant of the unit circle. And from here, we can find our trig values by simply counting on our fingers. So let's go ahead and focus in on the angle 30 degrees. Now to do that, you're going to look at your hand and you're going to take that finger closest to your pinky that represents 30 degrees and you're going to fold it inward. Now we're going to count the number of fingers are above that folded-in finger and below that folded-in finger in order to find our trig values.

So for our sine and cosine of 30 degrees, we're going to start in that same way. Remember, the square root of something over 2, and our number of fingers is going to tell us what that something is. So with our finger folded in here, we're going to count the number of fingers that are above that folded-in 30 degree finger and put that under our square root in order to get the cosine of 30 degrees. So the cosine of 30 degrees here, counting those fingers, I have 3 fingers above. So this is the square root of 3 over 2. The cosine of any angle is the square root of your fingers above divided by 2. Now for our sine, we're instead going to look at the number of fingers below, in this case, just one, our pinky. So we get the sine of 30 degrees as being the square root of 1 over 2 or just 1/2.

Now for our tangent, remember that we can always just take the sine divided by the cosine. Or here, we can also rely on counting our fingers again. So for the tangent of an angle, we're going to take the square root of the fingers below that folded-in finger and divide it by the square root of the fingers above. So here the fingers below again were 1. So the square root of 1 over the square root of 3. So the tangent of 30 degrees, we get 1 over the square root of 3. Or with that denominator rationalized, we end up with 33. This left-hand rule will work for any angle of the first quadrant, any finger you can fold in, and use this to find your trig values.

Now that we've seen these trig values of these common angles, let's get a bit more practice in that first quadrant. Thanks for watching, and I'll see you in the next one.