Common Values of Sine, Cosine, & Tangent - Video Tutorials & Practice Problems

Sine, Cosine, & Tangent of 30°, 45°, & 60°

Hey, 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. Now, sometimes you won't be given this ordered pair and you'll just be asked to find the sign or cosine of some angle with no other information. And that's where memorization comes in. Now, memorization and math can be really tricky. But here I'm going walk you through two different ways to memorize the trig values of the three most common angles, 3045 and 60 degrees. And with this knowledge, you'll be able to solve pretty much any trig problem that gets done your way. So let's go ahead and get started here with what you may hear referred to as the 123 rule. Now, no matter how you choose to memorize these values, you're always going to start in the same way with the square root over two because all of these trig values are the square root of something over two. And our job is just to memorize what that something is. So memorizing that with a 123 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 123 going clockwise. Then for our Y values we're going to count 123, going at counter clockwise. Now, what exactly do I mean by that? Well, we're going to start this upper left corner with the X value of 60 degrees. And we're going to start counting from one, going 123 clockwise around our unit circle for X values. Then we're going to go back counterclockwise and count 123 back up for our Y values. Now, we're done here. These are all of our trig values. We've gone 123 clockwise, 123 counterclockwise. And we're done. Now, we can do a bit more simplification here because we know that the square root of one is just one. So this X value or this cosine of 60 is really just one half and the sine value of 30 degrees or the Y value is also just one half. 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 the my sine value one half 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 three or with my denominator rationalized, I get the square root of 3/3. As my tangent, 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 123 method. But remember that I promised you two 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 at 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 zero 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 of 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 that are up above that folded and finger and below that folded and finger in order to find our trait 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 two. 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 three fingers above. So this is the square root of 3/2. The cosine of any angle is the square root of your fingers above divided by two. Now, for our sign, we're instead going to look at the number of fingers below. In this case, just one are pinky. So we get the sign of 30 degrees as being the square root of 1/2 or just one half. Now, for a tangent, remember that we can always just take the sign 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 one. So the square root of one over the square root of three. So the tangent of 30 degrees, we get one over the square root of three or with that denominator rationalized, we end up with the square root of 3/3. Now, this left hand rule will work for any angle of the first quadrant. Any finger you can fold in and use this in order 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.

Example 1

A common problem you may be faced with is filling in an entirely blank unit circle and being able to do that starts with the first quadrant. So here we're going to fill in all of the missing information in the first quadrant of this unit circle. Now, the way that I do this might not be the way that you choose to do this. And that's totally OK, always find what works best for you. And here I'm going to walk you through my thought process of filling in this first quadrant. Now, feel free to pause here and try this on your own before jumping back in with me. Now, the way that I like to start with this unit circle is to start with my, in my Y axis because this is the easiest information for me to remember. So starting with zero degrees, I know that this point is right here on my x axis at 10, then going up to this other axis, my y axis. This is at 90 degrees, which I also know is high over two radiance and is located at the point on the y axis 01. So from here. Now that I have my information on each axis, I like to move on to my degree angle measures because this is also something that's rather easy for me to remember. I know that I start with 30 degrees, then 45 degrees and then 60 degrees. Now here is where it usually gets a bit trickier because I'm going to move on to our radiant angle measures, which can be a bit more difficult to remember because you're probably more familiar with degrees than radiance. Now, you could choose to use a formula to convert these degrees into radiance or you can kind of reason it out using the information you already have on your unit circle, which is what I'm going to do here. So here I'm going to start 45 degrees. Now, I know that 45 degrees is halfway between zero and 90 degrees. So that means that that radiant angle measure needs to be halfway between zero and pi over 2.5 of pi over two. If I multiply those together, that gives me a value of pi over four. So that radiant angle measure for 45 degrees is pi over four, then moving on to 30 degrees, 30 degrees I know is one third of 90 degrees. So I know that that radiant angle measure has to be one third of pi over two. Now multiplying that out, that gives me a value of pi over six, which is what my radiant angle measure is here pi over six. Now moving on to 60 degrees, I know that 60 degrees is two times my 30 degree angle measure. So that means my radiant angle measure is going to be two times pi over six. So multiplying pi over six times two, gives me two pi over six or simplifying that fraction pi over three. So that radiant measure for 60 degrees is pi over three. Now, you can also choose to just memorize what these radiant values are if that's more your style. So here we would want to think about what are the denominators because all of the numerators are the same. They're all pi so we would think about the denominators. OK? I have zero radiance here that I have 6432. So you can kind of think about it as counting down 6432 for your denominators. Of course, skipping five there. So it can be a little bit tricky to memorize. But remember you can always reason it out as we did here. Now that we have all of those angle measures filled in, we can move on to really the meat of the unit circle, the cosine, sine and tangent values, all of our trig values or in this case, also our X and our Y values. So let's start there with our X and Y values or our cosine and sine values. Now to do this, remember we're always going to start the same exact way with the square root over two for every single one of these values. It's always the square root of something over two, not the square root of two, the square root of something over two. And I'm going to do this using the 123 method because that's the easiest for me to remember but feel free to use the left hand method or the left hand rule or whatever else works. So let's go ahead and get started here. Now, remember with the 123 rule, we're going to start with this X value up here and count 123 clockwise and then go back 123 counterclockwise and those are cosine and sine values. Now remember we can simplify a bit here because the square root of one is just one. So for both of these instances of square root of one that just becomes one. Now that we have our cosine and sine values, we can go ahead and find our tangent. Now remember the tangent of any angle, we can just use our information that we already know because the tangent of an angle is the sine of that angle divided by the cosine of that angle. Now knowing what we know about our X and Y values, this is also just equal to Y over X which we can find by looking at our unit circle right here. Now looking at this first value 30 degrees, identifying the tangent of 30 degrees. I have one half and root 3/2 as my s and my cosine value. So I want to take my S one half and divide it by my cosine root 3/2. Now, because these have the same exact denominator, I am effectively just dividing those numerators. So real, this just gives me a value of one over route three. Now rationalizing this denominator, this gives me a value of route 3/3 as that tangent. Now moving on to 45 degrees, I have route 2/2 and route 2/2, the same exact values for sine and for cosine. So when I divide them, I'm simply going to get a value of one as that tangent. Then looking up here at 60 degrees, I have one half and 3/2. And remember that when they had the same denominator, we're effectively just dividing those numerators. So I can take root three, divide it by one to get me my tangent. So route 3/1, which is really just equal to route three. Now, we don't want to forget those values on our axes here. Remember we still need to identify these tangents. So here down at zero degrees, I would take a zero divided by one which is just zero. So the tangent of zero is zero. And then up here at 90 degrees, I would take one divided by zero, which is not a number at all. And that leaves me with an undefined value up here. And now we have completely filled in all of that information in the first quadrant. Now, when dealing with something like the unit circle repetition can be really helpful. So feel free to try this problem over and over again and check in with me as needed. Thanks for watching and of course, let me know if you have any questions.