Introduction to Trigonometric Functions - Online Tutor, Practice Problems & Exam Prep

Converting between Degrees & Radians

Hey, everyone. Welcome back. So as we've been talking about angles, we've gotten really used to the idea of working with degrees. If you go around a circle, full circle, you'll go full 360 degrees. And if I asked you to draw me something like a 120, you'd be able to draw me something that looks about like this.

What we're going to look at in this video is another unit of measuring angles called radians. It's kind of like how we use inches and centimeters to measure distance. We can use degrees and radians to measure angles. Now, we'll be using radians a lot in later videos when we talk about functions and graphing, but for now, I just want to understand the basic difference between degrees and radians, how we draw them, and more importantly, also how we convert between them.

Let's go ahead and get started here. A radian really is just a different unit for measuring angles based on the idea of going around the circumference of a circle. So what we saw is that if you go a full circle, that's 360 degrees.

So if I cut up the circle into 360 tiny little pieces, each one of those things would be a degree. So this tiny little sliver over here, that's what one degree is. What a radian is, is it's basically just based on the idea of going around the circumference of a circle. If you go around the circumference of a circle, that's 2πr, that's a formula for geometry. So the whole idea of a radian is if you grab the radius of a circle and you take that distance and then start going around the circumference of a circle, that same distance, but now is curved, the angle that you're making there, that little pizza slice that you've just made is 1 radian.

And it's about 57 degrees. That's about, like, approximate. Like, it's a little bit of a decimal. So if you go around a full circle, that's a full 2π radians.

So a full circle is 360 degrees and 2π radians. But usually when we work with radians, we're going to work with them in multiples or fractions of pi. So just as we cut up 360 into 4 quarters, we can cut up 2π into 4 quarters. If you go halfway around, half of 2π is π radian. If you go half of that, that's going to be π/2 radians.

If you go three quarters of the way around the circle, that's going to be 3π/2 radians. So that's what our sort of axes become in radian form. So how do we go back and forth between degrees and radians? Well, it really just comes down to these two formulas over here. To convert back and forth between degrees and radians or radians and degrees, we're just going to multiply whatever angle that we have by π/180 or 180/π.

We're going to do a couple of these examples of this. In fact, our first example is going to be, how do we convert a 120 degrees? This angle that we have over here, what does that end up being in terms of radians?

So what we're going to do here is this is an angle that's in terms of degrees. That's theta degrees. So that means we're going to use this top equation over here. So to convert this to radians, what we're going to have to do is take our 120 degrees, and we're going to have to multiply it by something.

All you have to do is just memorize that the two numbers involved are pi and 180. How do you figure out which one goes on top? It really just comes down to what unit you're trying to get rid of.

I'm trying to get rid of degrees, so that means I want degrees on the bottom. I want the 180 to be on the bottom. Because now what happens is you'll see the degrees will cancel, and then you'll end up with pi radians being on top. So then how does this simplify?

Well, what happens is 120 over 180, you can knock a 0 off, and 12 over 18 just simplifies to a fraction. So what happens is you're still left with a pi over here, but now this simplifies to a fraction of 2/3. So this is 2π/3 radians. So this is exactly what 120 degrees is. It's just written in a different unit. These two angles here mean the exact same thing. They both mean an angle that looks like this. It's just in different units.

So that's how you convert from degrees to radians. Let's do one more example over here, because now we're actually going to do the opposite. Now we're going to convert this angle that's given to us in radians, and we're going to convert it to degrees. So this is an angle that's going to be theta radians.

And so what we're going to do is we're going to convert this to theta in degrees. So how do we do this? Well, we take what the given angle is, 6π/5, and we know we're going to have to multiply by π/180 or 180/&π. How do we know which one? Well, it just depends on what we're trying to get rid of. We're trying to get rid of radians and be left with degrees. So that means radians, we want to be on the bottom and 180 degrees on top. Usually, you can tell that this is going to be the case because you want the pi to cancel, so you want to divide by pi to get rid of it. That's a good way to sort of double-check that.

So we're going to cancel out the pies over here. And, basically, when you actually sort of multiply these numbers out, what you're going to get here is you're going to get 1080 degrees divided by 5. And if you work this out, which is what you'll actually end up getting here is you'll end up getting 216 degrees. So that's what 6π/5 is going to become 216 degrees.

Hopefully, this made sense, folks. Thanks for watching, and I'll see you in the next one.

Convert the angle $540°$ from degrees to radians.

• A. $3\pi$ rad
• B. $\frac{3\pi}{2}$ rad
• C. $\frac{5\pi}{2}$ rad
• D. $4\pi$ rad

Convert the angle $-\frac{5\pi}{6}$ from radians to degrees.

• A. $-120\degree$
• B. $-150\degree$
• C. $-210\degree$
• D. $-240\degree$

Introduction to Trigonometric Functions

Hey, everyone, and welcome back. So, in earlier videos, we've brushed up on this idea of angles and triangles. And what we're going to be looking at in this video is arguably one of the most critical topics, both in this course and in future math courses, that you will likely take, which is the idea of trigonometric functions, which we also call trig functions for short. Now, if you're not sure what trig functions are, all they do is they relate angles to the side lengths of a right triangle. Now in this video, there are 3 main types of trig functions that we're going to talk about, which are sine, cosine, and tangent.

Now that might sound a little confusing or like it's a bunch of gibberish, but don't sweat it because all we're going to learn is that these 3 trigonometric functions are just ratios. An example of a ratio would be something like 3 5ths or 4 5ths. So these types of fractions or ratios are all that these trigonometric functions are. So without further ado, let's get into this video and see how we can solve some problems where we need to find these specific ratios based on a right triangle. Now how can we go about finding the sine of some angle in this triangle that we'll call theta?

How can we find the sine of this? Well, sine is opposite over hypotenuse. So, if you wanted to find the sine of this angle theta, what you do is look at the angle that you have and go to the opposite side of the triangle, which in this case is 3, and then divide that by the hypotenuse, which is always the long side of the triangle, or 5. So 3/5 is going to be the sine of theta. As you can see, solving for these trigonometric functions is pretty straightforward.

So, let's now say we wanted to find the cosine of theta. Well, if we're dealing with the cosine this time, what we need to do is look at our angle theta and find the adjacent side and divide it by the hypotenuse. Now the adjacent side of this triangle is the side next to the angle, which in this case is 4, and then this is divided by the hypotenuse or long side of the triangle which is 5. So the cosine of our angle theta is 4/5. But now let's say we wanted to find the tangent of our angle theta.

If we're dealing with the tangent this time, what we need to do is take the opposite side of the triangle and divide it by the adjacent side. So the opposite side to our angle theta is 3, and then the adjacent side is 4. So 3/4 would be the tangent of our angle theta. Now, you might feel like this is a lot to remember. Like, how are we just supposed to know that sine is opposite over hypotenuse, or that tangent is opposite over adjacent?

Well, it turns out there is a memory tool that we use to remember this, and that is SOHCAHTOA. Because this memory tool, SOHCAHTOA, tells us how each of the trigonometric functions relates to the sides of the right triangle. So we have that sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. Now one more thing I want to mention about the tangent is that the tangent of theta is also equal to the sine of theta divided by the cosine of theta. So if we were to take the sine of theta, which we figured out was 3/5 and divide that by the cosine of theta, which we figured out was 4/5, notice how the 5ths would just cancel in this fraction.

So all we would end up with is 3 over 4, and 3/4 is exactly what we calculated the tangent to be. So this is another relationship to be familiar with when dealing with these trigonometric functions. Now to really make sure we have these trig functions and this SOHCAHTOA memory tool down, let's try some examples where we have to solve some problems based on a right triangle. Now in this problem, we are asked to find the values of the trig function indicated given the right triangle that we have over here. Now we're going to start with example a.

And before I do anything, what I'm going to do is write down the memory tool SOHCAHTOA. Now looking at example a, we have the sine. And sine is opposite over hypotenuse according to SOHCAHTOA. So we'll have the opposite side of our right triangle divided by the hypotenuse. Now we're dealing with angle x in this example, so if we go to angle x in our triangle, we're going to have the side opposite to angle x, which is 12 divided by the hypotenuse or the long side, which is 13.

So 12/13 is the ratio we're looking for and the answer for example a. Now let's try solving example b where we are asked to find the tangent of x. Well, if we once again go to our angle x here, we can find the tangent by looking at SOHCAHTOA, which says that tangent is opposite over adjacent. So if we take this ratio, we can go to our angle x, and we can find the side opposite to x, which in this case is 12, and then divide it by the adjacent side or the side next to x, which in this case is 5. So 12/5 is the ratio that we're looking for for example b.

But now let's try solving example c, where we are asked to find the cosine of our angle y. Now when dealing with the cosine, cosine is adjacent over hypotenuse based on this memory tool SOHCAHTOA. Now what you might think we need to do is look at our angle and then find the adjacent side which would be 5 and divide it by the hypotenuse or the long side which is 13. Now if 5/13ths is what you're thinking the correct ratio is, this is actually not correct, and here’s the reason why. Notice in this example we are dealing with angle y, and because we're dealing with angle y, we can no longer use x as a reference.

We have to use y as the reference angle. So rather than having 5/13ths, we're going to have the ratios based on the adjacent and hypotenuse of y. So if we go to y, the adjacent side to y is actually 12, and then the hypotenuse or the long side is still 13. So 12/13 would be the ratio for the cosine of y. So this is something you need to watch out for when dealing with these types of problems.

Because notice in the first two examples, we were dealing with angle x, so we used x as a reference. Where in this example, we were dealing with y, so we used angle y as a reference. So that's the main idea behind solving trigonometric functions and using this SOHCAHTOA memory tool. Hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.

Given the right triangle below, evaluate $\cos\theta$.

• A. $\cos\theta=\frac{8}{17}$
• B. $\cos\theta=\frac{8}{15}$
• C. $\cos\theta=\frac{15}{17}$
• D. $\cos\theta=\frac{15}{8}$

Given the right triangle below, evaluate $\tan\theta$.

• A. $\tan\theta=\frac35$
• B. $\tan\theta=\frac45$
• C. $\tan\theta=\frac43$
• D. $\tan\theta=\frac34$

Trig Values in Quadrants II, III, & IV

Hey, everyone. We now know all of the trig values of our 3 common angles in quadrant 1. And we also know that all of our other angles have a reference angle equal to one of those common angles. Now all that's left to find is the trig values of all of these angles. And learning that many more new trig values does not sound like fun.

But luckily, you don't have to do that at all because the trig values of all of these angles are the exact same as those of their reference angle with just one tiny difference that I'm going to walk you through right now. So let's go ahead and get started. Now I mentioned one tiny difference, and that difference is the sign. So the sine, cosine, and tangent of each of these angles is the exact same value as that of their reference angle, but just with a different sign depending on what quadrant our angle is located in. So let's go ahead and just come right down here and look at our unit circle.

So in quadrant 1, we know all of these trig values, the cosine, sine, and tangent of each of these angles. And we see that being in quadrant 1, these are all positive values. But what about as we move over to quadrant 2? Let's specifically take a look at our 150 degree angle here. Now remember that this has a reference angle equal to 30 degrees because that's the angle it makes with the nearest x-axis.

So with that reference angle of 30 degrees, we can further visualize this by seeing that they form the exact same triangle just flipped. So it makes sense that the base and height of these triangles would be the exact same, which are cosine and sine values. So I end up with 32 for that cosine or that x value, one half for that y value, that sine, and 33 for tangent. Now the only thing that remains to consider here is the sign of each of these values because we're no longer located in quadrant 1.

Now looking at our relationship to the x and y axes, I see that my x values over here should be negative, whereas my y values remain positive. So I want to reflect that in these trig values. So my x value is negative, negative root three over 2. My y value is positive. Now when looking at the tangent, we just consider how we actually find the tangent.

Right? So the tangent is the sine over the cosine or y over x. So knowing that our y value is positive and our x value is negative, if I were to divide a positive value by a negative value, that would give me a negative value. So here my tangent should also be negative. Now looking at these values altogether, only one of them is positive.

And this will be true for all of our trig values in quadrant 2. Only the sine value, our y value here, is going to be positive for all trig values in quadrant 2. Now let's move on to quadrant 3 and specifically 225 degrees. Now I know that the reference angle of 225 degrees is 45 degrees in quadrant 1, so I can take my trig values here and simply copy them over here for the trig values of 225 degrees. So here I can copy these in with the cosine and sine both being root two over 2, my x and y values, and then the tangent as 1.

Now all that we need to consider here is remember the sign. So based on where we are in the x and y axes, we see that these are both these both should be negative values. My x and my y value are both negative. Now thinking about the tangent, in order to find the tangent, I would be dividing a negative by a negative, which would actually give me a positive value. So my tangent here in quadrant 3 remains positive.

Now looking at these values altogether, the only one that's positive is tangent, which will, of course, be true of all of my trig values in that quadrant 3. So here in quadrant 3, the tangent will always be positive and it's the only positive one. Now finally, let's take a look over here at quadrant 4. Now in quadrant 4, we're going to look specifically at 300 degrees. Now we know that 300 degrees has a reference angle of 60, so I can go ahead and take my 60 degree trig values and copy them right down here for the trig values of 300 degrees.

Now doing that, my cosine and sine, I get 1 half and a root three over 2, literally just copying those values, then for the tangent root three. Now we need to consider, of course, the sign. Now the sign here, looking in this quadrant, quadrant 4, we know that we are in the negative y values and positive x values, so we need to reflect that on these trig values. So negative y values, this negative root three over 2. And then for our tangent, we need to consider what we're doing.

Of course, here we would be dividing a negative by a positive, which would give us a negative value. So our tangent here will also be negative. Now finally looking at all 3 of these trig values as a whole, the only one that's positive is this cosine, this x value, 1 half. So here in quadrant 4, the cosine will always be positive. Okay.

We've looked at all of our quadrants here and we've seen the sign difference in each of them. All of our trig values are the same, just with a little variation in sign depending on our location on the unit circle. Now how can we remember this? We can remember this using a mnemonic device using the first letter of each function that's positive. So in quadrant 1, all of them are positive.

Quadrant 2, the sine is positive. Quadrant 3, the tangent. And quadrant 4, the cosine, ASTC. All students take calculus. Now this might not always be a true statement, but it's one that can help us remember which trig function is positive in our unit circle.

So now we have all of the information we need in order to completely fill in an entirely blank unit circle. So let's continue getting some more practice together. Thanks for watching, and I'll see you in the next one.

Trig Values in Quadrants II, III, & IV Example 1

Hey, everyone. In this problem, we're asked to use reference angles to complete the missing trig values in quadrants 2, 3, and 4. And now that we know that all of these trig values are going to be the exact same as those in quadrant 1, with just varying sign based on the quadrant, we can do this really quickly. So remember our mnemonic device here of how we remember which function is positive in each quadrant. All students take calculus telling us the first letter of each function that's positive within each quadrant.

Now let's take first take a look at quadrant 2. And looking at 135 degrees and 120 degrees. Now these have reference angles of 45 degrees and 60 degrees respectively, so we could just go ahead and copy those trig values over to this quadrant. So for 135 degrees, I have √22, √22, and 1. Then for 120 degrees, I have 1232, and 32 for that tangent.

Now in this quadrant, only the sine is positive based on that mnemonic device, so that means that both my cosine and tangent need to be negative here. So my x values of both of these should be negative as well as this tangent here on the end. Now we're done with quadrant 2. Let's move on to quadrant 3 down here. Now looking at quadrant 3, I have 210 degrees and 240 degrees as those missing trig values.

So looking back up to quadrant 1, I know that 210 degrees has a reference angle of 30 degrees and 240 degrees has a reference angle of 60 degrees, so I can copy those trig values over. So for 210 degrees, I have 322 for my cosine, 12 for my sine, and 333 for that tangent. Then for 240 degrees or 4 pi over 3 radians, I have 12 for the cosine, 322 for the sine, and 33 for that tangent. Now in this quadrant, only the tangent is positive, so that means both my sine and cosine need to be negative. So x and y here are both negative when looking at these coordinates.

Now finally, let's take a look at quadrant 4 over here. Now looking in this quadrant, my missing information is for 330 and 315 degrees, which have reference angles of 30 degrees and 45 degrees, respectively. So let's go ahead and copy those trig values over here in this quadrant 4. So for 330 degrees, I have √32, 12, and √33. Then for that 315 degrees, I have √22, √22, and finally, one for that last tangent value.

Now in quadrant 4, only the cosine is positive here, so that tells me that I need to go ahead and make both my sine and my tangent negative. So my sine value and my tangent negative in both of these angles. Now we have completely filled in all of our information for the unit circle. Now continue to practice this on your own and check-in with me as needed. Thanks for watching, and I'll see you in the next one.

Identify what angle, $\theta$ , satisfies the following conditions.

$\sin\theta=\frac12$; $\tan\theta$ < $0$

• A. 30°
• B. 150°
• C. 60°
• D. 300°

Identify what angle, $\theta$ , satisfies the following conditions.

$\cos\theta=\frac{\sqrt3}{2}$; $\sin\theta$ < $0$

• A. 30°
• B. 60°
• C. 120°
• D. 330°

Identify what angle, $\theta$ , satisfies the following conditions.

$\tan\theta=-1$; $\cos\theta$ > $0$

• A. 45°
• B. 135°
• C. 315°
• D. 330°

Trig Values in Quadrants II, III, & IV Example 2

In this problem, we're given a completely blank unit circle and asked to fill in all of the missing information. Now, this is a specific type of problem that you'll get better at with repetition, and it's definitely something that you should try on your own before jumping back in with me. Now, here I'm going to walk you through my thought process in filling in the unit circle, and what works best for me might not be what works best for you. So, of course, try this on your own and figure out exactly what method works best for you for filling in the entire unit circle. Let's go ahead and get started here.

Now, looking at our unit circle, the first thing that I like to do is fill in all of the information for my 4 quadrantal angles because to me, it's the simplest information on here. Looking at this first angle measure, I have 0 degrees or 0 radians, and I'm going to go ahead and fill in all of my angle measures for my 3 other quadrantal angles. Up here, I have 90 degrees, 180 degrees, 270 degrees, and remember coming back around to a full rotation is 360 degrees. Now, oftentimes a blank unit circle will only have one blank there, but it's important to remember that this is also a full rotation around. Let's also fill in those radian angle measures.

We have π2 radians, π radians, and 3π2 radians. Now we have all those angle measures filled in both degrees and radians, so let's consider our trig values of our quadrantal angles as well. Now, remember on the coordinate system, we know that this point is located right here at one 0 and then we find our tangent value by simply dividing y over x. So doing that here, I have 0 over 1 which gives me a value of 0. Then up here for 90 degrees, this is located at the point 1, and dividing y over x here gives me 1 over 0, an undefined value.

Now over here with π radians, we're at negative one zero and we again get 0 for our tangent. And then down here at 2 and 70 degrees, we're at 0 and negative one, which is also an undefined tangent value. Okay. We have all of the information for our 4 quadrantal angles. So now let's focus on all of the other angle measures for all of our other quadrants.

Let's start with quadrant 1 here because these are likely the easiest for you to remember as they are for me. So here, we're looking at our 3 common angles. We have 30 degrees, 45 degrees, and 60 degrees. Now I'm also going to go ahead and fill in my radian angle measures here. In this first quadrant, we have π6, π4, and π3.

Now here, let's go ahead and focus on all of our reference angles. Now all of these angles here have a reference angle of 30, so that's going to really help me to figure out what these angle measures are. So let's start here in quadrant 2. In quadrant 2, I look at this reference angle in reference to the nearest part of the x-axis, which is at 180 degrees. This is 30 degrees away from that 180, which tells me I can find this angle measure by simply taking 180 minus 30 degrees.

Now this gives me an angle measure of 150 degrees. Now for that radian angle measure, I can actually do the same exact thing but just with pi. This is π6 radians away, so I take pi minus π6 here and that gives me my radian angle measure of 5π6. This will work for any angle in this quadrant because we can just do that for any angle. We just subtract our angle from 180 and from pi.

Now let's look in our quadrant 3 here and do the same exact thing. This is in the opposite direction of 180, so here I can actually take 180 and add my angle to it instead to get that angle measure. This is 30 degrees away because we're still looking at our 30 degree reference angle here. So if I take 180 and I add 30 degrees, that's going to give me an(angle measure) of 210 degrees. Now I can do the same thing with pi, of course.

I add pi to my angle here, which is π6, so pi+ix gives me an angle measure of 7π6. Now let's move on to quadrant 4. Here, we look at this in reference to our final angle, our full rotation of 360 degrees, and this is, of course, still a 30-degree reference angle, so here I can take 360 degrees and I can subtract my angle here, so 360 minus theta, my angle. Now here, this is 30 degrees, so 360 minus 30 gives me 330 degrees. Now, of course, this is also 2 pi radians, so I can also take 2 pi and subtract my angle in radians to get my radian angle measure here.

So here, 2 pi minus pi6 will give me 11π6. Now, looking at all of these radian angle measures, you can also choose to memorize these sorts of patterns that happen with your radian angle measures. So, looking at all of these, I have π6, 5π6, 7π6, and 11π6. Now, for angles that have the same reference angle, they always have the same denominator in that radian angle measure. These all have a denominator of 6.

And they count from 1, 5, 7, 11. So you can remember that pattern if it's easier for you to memorize here. Now here looking at this, I can apply this same line of thinking to all of my other reference angles. So I want you to go ahead and try that on your own and then check back in with me. Okay.

Now that we have all of those angle measures in there, we can move on to trig values. Let's start in quadrant 1 because we have some memory tools that will help us fill this in rather easily. Now remember, in quadrant 1, we always start the same way regardless of how we choose to memorize this. We always start with the square root of 2 for every single one of these values. So I can go ahead and fill that in for every trig value in quadrant 1.

Now once I do that, I can go ahead and use a memory tool. Here, I'm going to use the 123 memory tool, but you can, of course, use whatever works for you. So here, using the 123 memory tool, I'm going to start here with this value and I'm going to count 123 in the clockwise direction for those x values and then 1, 2, 3 back in the counterclockwise direction for those y values. So these are my trig values in quadrant 1. Now we can, of course, simplify these square roots of 1 because they're just 1.

Now, from here, we want to fill in our tangent values. You can choose to memorize your tangent values, but it's easier for me to just remember that this is y over x. So I can take these values that I already have and simply divide them. Now because they have the same denominator, we're effectively just dividing those numerators. So here in this first one, I divide my numerators of y over x, one over the square root of 3.

Now, this is a correct value, but remember, it's going to be better if we rationalize our denominator here. So this will give me the square root of 3 over 3 for the tangent of 30 degrees. Now, for 45 degrees, if I take y over x, those numerators, I get a value of 1. And then for 60 degrees, I get the square root of 3 over 1, which is just the square root of 3. Now we have all of our trig values in quadrant 1, but we have these 3 other quadrants.

Now, remember, all of our trig values are the exact same as those of their reference angles, and we only have to worry about the sign. So from here, I'm just going to go ahead and fill in all of these trig values, copying them over from quadrant 1 to their respective reference angle in every single quadrant, and then we can worry about the sign after. So I'm going to go ahead and fill all of those values in now. Okay. All of our trig values are copied over to quadrants 2, 3, and 4.

But remember, the sign of these values may be different depending on what quadrant they're in. So that's where we consider the sign. Now, remember our mnemonic device to remember the signs of these values. All students take calculus, telling us the first letter of whatever trig function is positive in that quadrant. So in quadrant 2, we have s.

That tells us that only the sign is positive. So, only my y values here, the square root of 3 over 2, the square root of 2 over 2, and 1 half should be positive. So all of my other values are negative, and I can go ahead and fill that in for all of my other values, putting a negative sign on each of them. Now we can do the same exact thing in quadrants 3 and 4. Now in quadrant 3, only the tangent is positive, and in quadrant 4, only the cosine is positive, so all of my other trig values must be negative.

I'm going to go ahead and copy those negative signs in now based on that, and then we'll rejoin together. Okay. You should have all of your trig values filled in now with negative signs on the correct values based on our mnemonic device all students take calculus. So take a chance to double-check all of your values here with mine. Now we have completely filled in all of the missing information on our unit circle.

Now, again, this is a problem that gets easier with repetition, so feel free to try this as many times as you need until you have it down. Thanks for watching, and, of course, let me know if you have any questions.