Hey everyone! If I gave you a unit circle and I asked you to find the sine of 30 degrees, you would probably locate 30 degrees on your unit circle and simply identify your y value. What if instead I asked you to find the sine of 390 degrees? It doesn't sound quite so simple. But if we look at our circle here and we go a full rotation around from 0 to 360 degrees and then just go an additional 30 degrees, I'm actually right here at 390 degrees. So the trig values are going to be the exact same as those of 30 degrees, an angle that I already know. And that's because these angles are coterminal, so their trig values are going to be the exact same.
Now the first time that you see an angle that's either really large or maybe even negative, it might seem a little bit strange. But here I'm going to walk you through exactly how to identify a coterminal angle on the unit circle and use that to really quickly and easily identify your trig values. So let's go ahead and get started here. Now you might remember coterminal angles more formally as an angle with the same terminal side. And you might also remember finding them by adding or subtracting multiples of 360 degrees or 2π radians. So with my 390 degree angle, if I were to subtract a full rotation around a circle of 360 degrees, I would end up with my coterminal angle of 30 degrees.
Now we're going to do basically the same thing here, but it's going to be even easier to visualize on our unit circle because we already have all of these angles here. So we can do this without having to set up any algebraic equation and just visually finding these angles.
So let's go ahead and jump right into this first example and find the tangent of 3π. Now the tangent of 3π. The first thing we want to do here is find the coterminal angle of 3π so that we can easily find our trig value that's already on our unit circle. So let's consider this by looking at our unit circle. If I go around my circle once from 0 to 2π radians and then I go another additional π radians, I've really gone 3π radians around my circle, but I've ended right back here at π, which is my coterminal angle. So the coterminal angle of 3π π is simply π π. So to find the tangent, I just need to look at the tangent value of π π here, which happens to be 0.
The tangent of 3π is equal to 0, and we're done here. Now let's move on to our next example here. Here, we're asked to find the cosine of negative π/4. Now remember, for a negative angle, that just tells us that we're going clockwise around our circle rather than counterclockwise. So let's again visualize this on our unit circle here. Now if I start at 0 and I go negative π/4 radians, so this way, clockwise, negative π/4, I'm gonna end up here at 7π/4 radians. So the coterminal angle of negative π/4 is 7π/4.
Now that I know that, I can identify the cosine by just finding the cosine of that coterminal angle 7π/4, which is of course my x value here. So the cosine of negative π/4 is 22 and I'm good to go here.
Now we have one final example, the sine of 390 degrees. So we want to go ahead and identify this on our unit circle. Now you might remember this from earlier, but let's just visualize this one more time. So 390 degrees. If I go around my circle once, that's 360 degrees. And if I go another 30 degrees, that puts me at 390 degrees. So the coterminal angle here is, of course, 30 degrees. Now to find the sine of this angle, we're just gonna look at the sine of 30 degrees which is of course our y value, in this case, 1/2. So the sine of 390 degrees is 1/2.
Now that we know how to find trig values using coterminal angles, let's get some more practice together. Thanks for watching and let me know if you have questions.