 The Pythagorean identities allow us to find the values of all the trigonometric functions from just one up to the sine of the function itself. For example, suppose we know that sine of theta is three-fifths, let's find cosine of theta. So our Pythagorean identity gives us sine squared theta plus cosine squared theta equals one. We're given the value of sine of theta, so we can replace it, and solve for cosine of theta, and because we're taking a square root, we have to indicate that the value could be plus or minus the square root. And here's the problem, unless we know more about theta, we don't know the sine of the cosine. Well okay, suppose we do know more about theta. For example, suppose sine of theta is three-fifths and theta is a second quadrant angle, let's find the cosine of theta. So we'll start by drawing a unit circle centered at the origin with an angle theta in the second quadrant. Since we can always use an identity in any problem, we know that sine squared theta plus cosine squared theta must equal one. We know the value of sine of theta it's given to us, so we can substitute in and solve for cosine, and we have two possible values of cosine theta plus or minus four-fifths. Now remember that the cosine of theta is going to be our x-coordinate, and since we do a picture, we see that the x-coordinate of our point must be negative. And so, since theta is an angle in the second quadrant, cosine of theta must be negative. And that means the cosine of theta must be negative four-fifths. So suppose tangent theta equals twelve over five and theta is an angle in the third quadrant. Find the values of sine and cosine. Since we solved every other problem in trigonometry by drawing a picture, we should draw a picture. But you know what? It's boring always getting the answer, so why should we do something that works every time? Let's not draw a picture. From our Pythagorean identity, sine squared theta plus cosine squared theta equals one, or tangent squared theta plus one equals secant squared theta, since we know the tangent of theta is twelve-fifths we can substitute, solve for secant theta, and since we're looking for sine and cosine, we found neither of them. Remember that by definition, secant theta is one of our cosine theta. So since secant theta is one of our cosine theta, then cosine theta is one of our secant theta. And so that means that cosine of theta is plus or minus five-thirteenths. But there can be only one. Cosine of theta must be either five-thirteenths or negative five-thirteenths. So which one is it? Well, it looks like we're going to have to draw that picture anyway. So we'll draw a unit circle centered at the origin with an angle in the third quadrant. Remember the x-coordinate corresponds to the cosine of the angle. And since theta is an angle of the third quadrant, x must be negative, so cosine of theta must be negative five-thirteenths. To find sine of theta, we can do a number of different things, but probably the easiest is to use our relationship tangent theta equals sine theta over cosine theta. So sine theta is equal to tangent theta times cosine theta. And we know the value of tangent theta and the value of cosine theta, so we can calculate the value of sine theta.