 Welcome back. Today what we're going to do is show how to find the values of the six circular functions given information about one of the circular functions and the quadrant in which the arc, the terminal point of the arc, lies. So here's what we're going to be looking at. We're going to be finding the values of all six circular functions of x. If we know that cosine of x equals 3 sevenths and the terminal point of x is in the fourth quadrant. So what I would like you to do now is just pause the screencast and think about what types of tools, i.e. definitions, identities, and so forth, that we have to help us solve this problem. The other thing you might want to think about is given the fact that this is in quadrant 4, what are the signs of the other five circular functions? So here are the tools that we have for this. One of which, again, once we know that, if we know the value of sine or cosine, we can at least get the value of the squared of the other one of those two. In particularly, what we use is the Pythagorean identity. With that, we will be able to determine a value now of sine squared. And since we know x is in quadrant 4, the sine will be negative. And once we have both the sine and the cosine, then we can use these definitions to determine the values of the other four circular functions. So here we go. As you can see, we've got this started here. Cosine of x is 3 sevenths, and we basically substitute into the Pythagorean identity. And now we continue with some computations and solving that equation for the square of the sine. So what we would have is 9 over 49 plus sine squared of x equals 1. And again, now what we want to do is isolate sine squared of x on one side of the equation. So what we will do is subtract that fraction, 9 over 49, from both sides of the equation. And we get that. And now of course, what we want to do is combine that into a single fraction. And in particular, what we will do is write the number 1 is 49 over 49. And of course, we're subtracting 9 over 49. So what we get is sine squared of x is equal to 40 over 49. And now what we of course do is take the square root. If we didn't know anything further about x, basically what we could conclude was that sine, I'm sorry, sine of x is equal to plus or minus the square root of 40 over 7. And we put in a 7 and that's the square root of 49. But we do know that x is in quadrant 4. So sine of x is negative. So we now have a specific value for sine of x, namely minus the square root of 40 over 7. We have both values for sine and cosine. We can now get the other four circular functions. And that's what we will do here. In particular, we can use a reciprocal definition. So secant of x is equal to 1 over cosine of x. So that is equal to 1 over fraction 3 7. So here we have to be careful with our work with fractions. When we divide, we invert and multiply. And so we get secant of x equals 7 thirds. So this invert and multiply will be a frequent thing we use when we take the reciprocal of a fraction. We do the same thing for cosecant of x, which is 1 over the sine of x, which is 1 over minus the square root of 40 over 7. So we're going to do this a little more rapidly. We basically, again, invert and multiply. And we get minus 7 over the square root of 40 for cosecant of x. And now we go after tangent and cotangent. And again, the tangent function is sine over cosine. And again, we have a little work to do with fractions. The sine is minus the square root of 40 over 7. Cosine is 3 7. So again, we invert and multiply. And we end up, as we can see, the 7's will cancel. And we end up with minus the square root of 40 over 3. And of course, the last one, which we'll try to fit in the corner here, is cotangent. And again, we use that reciprocal relationship. And again, what we will get is just the reciprocal of that fraction. And we'll get minus 3 over the square root of 40. So we now have all six values for the circular functions in this case in the summary of the results that we just obtained. One thing I want to do again is look for some little consistency in this. Remember, this is in quadrant 4. So cosine and hence secant are both positive. And the other four circular functions are negative. We can always get decimal approximations for these values by using our calculator. But what is shown here are the exact values for the circular functions of this arc x. And there you have it. We'll have another screen cast that will show another example of this, which you can view if you choose. Thank you for watching.