Common Values of Sine, Cosine, & Tangent - Online Tutor, Practice Problems & Exam Prep

Trigonometric functions connect angles to points on the unit circle, with cosine and sine values represented as x and y coordinates. To memorize key angles (30°, 45°, 60°), use the 1, 2, 3 rule or the left hand rule. The 1, 2, 3 rule involves counting values clockwise and counterclockwise, while the left hand rule uses fingers to represent angles. Tangent values can be derived by dividing sine by cosine. Key equations include ${\tan }^{\theta }=\frac{\sin }{\cos }$.

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

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.

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 okay. Always find what works best for you. And here, I am 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 x and my y axes because this is the easiest information for me to remember. Starting with 0 degrees, I know that this point is right here on my x axis at 1.0. Then going up to this other axis, my y axis, which is at 90 degrees π12 radians and is located at the point on the y-axis 1. 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 radian angle measures, which can be a bit more difficult to remember because you're probably more familiar with degrees than radians. You could choose to use a formula to convert these degrees into radians, 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 with 45 degrees. I know that 45 degrees is halfway between 0 and 90 degrees, so that means that that radian angle measure needs to be halfway between 0 and π12. And half of π12, if I multiply those together, that gives me a value of π14. So that radian angle measure for 45 degrees is π14.

Then moving on to 30 degrees. I know 30 degrees is 1 third of 90 degrees so I know that that radian angle measure has to be 1 third of π12. Now multiplying that out, that gives me a value of π16, which is what my radian angle measure is here, π16.

Now moving on to 60 degrees. I know that 60 degrees is 2 times my 30-degree angle measure. So that means my radian angle measure is going to be 2 times π16. So multiplying π16 times 2 gives me 2 π16 or simplifying that fraction, π13. So that radian angle measure for 60 degrees is π13.

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 of something over 2, not the square root of 2, the square root of something over 2. 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 1, 2, 3 clockwise, and then go back 1, 2, 3 counterclockwise, and those are our cosine and sine values. Remember we can simplify a bit here because the square root of 1 is just 1. So for both of these instances of square root of 1, that just becomes 1. 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 our 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 the square root 3 over 2 as my sine and my cosine value. So I want to take my sine one half and divide it by my cosine the square root of 3 over 2. Now because these have the same exact denominator, I'm effectively just dividing those numerators. So really, this just gives me a value of 13. Now rationalizing this denominator, this gives me a value of 33 as that tangent.

Then moving on to 45 degrees. I have the square root of 2 over 2 and the square root of 2 over 2, the same exact values for sine and for cosine. So when I divide them, I'm simply going to get a value of 1 as that tangent. Then looking up here at 60 degrees, I have the square root of 3 over 2 and square root of 1 over 2. And remember that when they have the same denominator, we're effectively just dividing those numerators. So I can take the square root of 3 divided by 1 to get me my tangent. So the square root of 3 over 1 which is really just equal to the square root of 3.

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 0 degrees, I would take a 0 divided by 1, which is just a 0. So the tangent of 0 degrees is 0. And then up here at 90 degrees, I would take 1 divided by 0 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.

Here’s what students ask on this topic:

What are the sine, cosine, and tangent values for 30 degrees?

For 30 degrees, the sine, cosine, and tangent values are as follows:

$\sin (30°)=\frac{1}{2}$

$\cos (30°)=\frac{\sqrt{3}}{2}$

$\tan (30°)=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$

How can I use the 1, 2, 3 rule to memorize trigonometric values?

The 1, 2, 3 rule helps you memorize trig values for 30°, 45°, and 60° by counting values on the unit circle. Start with the square root over 2. For cosine (x-values), count 1, 2, 3 clockwise starting from 60°. For sine (y-values), count 1, 2, 3 counterclockwise starting from 30°. Simplify the square roots: cos(60°) = 1/2, sin(30°) = 1/2, etc. For tangent, divide sine by cosine. For example, tan(30°) = sin(30°)/cos(30°) = (1/2)/(√3/2) = 1/√3 = √3/3.

What is the left hand rule for memorizing trigonometric values?

The left hand rule uses your left hand to represent the first quadrant of the unit circle. Your pinky is 0°, and your thumb is 90°. Fold the finger representing the angle (e.g., 30°, 45°, 60°). Count fingers above and below the folded finger. For cosine, use the square root of the number of fingers above divided by 2. For sine, use the square root of the number of fingers below divided by 2. For tangent, divide the square root of fingers below by the square root of fingers above. For example, for 30°, cos(30°) = √3/2, sin(30°) = 1/2, tan(30°) = 1/√3 = √3/3.

How do you find the tangent value of an angle using sine and cosine?

The tangent of an angle can be found by dividing the sine value by the cosine value. The formula is:

$\tan \theta =\frac{\sin }{\theta }$

For example, for 45°, sin(45°) = √2/2 and cos(45°) = √2/2. Therefore, tan(45°) = (√2/2) / (√2/2) = 1.

What are the sine, cosine, and tangent values for 45 degrees?

For 45 degrees, the sine, cosine, and tangent values are as follows:

$\sin (45°)=\frac{\sqrt{2}}{2}$

$\cos (45°)=\frac{\sqrt{2}}{2}$

$\tan (45°)=\frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2}=1$