 Hello once again as we continue our journey through trigonometry. In this screencast we're going to once again be trying to find the values of circular functions given certain information about one of the circular functions and hopefully enough information to determine in which quadrant the arc lies. So here's our problem. And it is somewhat similar to the other one but just a little bit different. We're given a value of sine of x equal to minus point seven one and the tangent of x being greater than zero. So again, you should perhaps pause the screencast and stop and think about how you might approach this problem and what tools you might have at your disposal to help solve it. And always remember I'm going to present one way but there are often times more than one way to solve a particular problem. Okay, so here again we have the tools that we have to solve this and down below here we have the particular problem stated that we are going to try to solve. And again given a value of the sine function we will quite quickly turn to the Pythagorean identity and at least be able to determine cosine squared. And that's where we have to know the quadrant in which we have here. And what we're given here is a value for sine and so we do know that the sine of x is negative and that tells us that we're in either quadrant three or quadrant four but we're also given tangent is greater than zero and therefore we can say we're in quadrant three so the rx lies in quadrant three and that will then be able to help us determine the sine that we use with our square root for cosine. So here's the setup then for determining the value of cosine. And again we've started with the Pythagorean theorem and have substituted in the value for sine of x. And as we go through this computation we will see that cosine squared of x will be equal to one minus zero point five zero four one which is equal to point four nine five nine. Because we've already determined that x is in the third quadrant we do know that cosine of x is negative and so we can conclude cosine of x equals minus square root of point four nine five nine. Now we can always haul out our calculator and get a decimal approximation for this value but in reality it's not really necessary. We can leave at least for the time being the cosine has minus this square root. And now of course our task is to determine the other four functions. And that's what we will be doing here. And we'll basically be using the reciprocal identities and some definitions of tangent and cotangent. So our first one is secant of x which we know is one over the cosine of x and since we have the value for cosine of x we can simply substitute that in. And again that's a negative square root with the negative in front of the fraction and we get our value for secant of x. And again just leave the answer in that form. And the next one is cosecant of x which again we know is one over sine of x and now I can make the substitution and again this is a negative number negative one over point seven one. And again we'll use our calculator a little bit later but for right now we're just going to leave the answer in that form. And now we can attack tangent of x and cotangent of x. And tangent of x is sine of x over cosine of x. And again we have those values. Notice that we have sine of x is minus point seven one and cosine is minus a square root. So if we substitute those in again we've got a negative divided by a negative. So tangent of x will be zero point seven one divided by the square root point four nine five nine. And finally the cotangent of x we will just complete as one over the tangent of x. And again that will be a positive number and will be the square root of point four nine five nine divided by point seven one. And there we have now determined the six values of the circular functions. Here is a summary of the results that we have. And again at this stage I did use my calculator and calculated these decimal approximations for five of the six circular functions. We were given an exact value for sine of x is minus point seven one. So there's another example of how to use basically the Pythagorean identity and the definitions of the other circular functions determine all values of the circular function. So long for now.