 The trigonometric identities allow us to find all trigonometric functions from the value of any one of them provided we know the quadrant. We used the Pythagorean identities earlier. We can use the double-angle identities as well. For example, suppose cosine theta is one-fifth and theta is a quadrant for angle. Let's find sine of 2 theta. So our double-angle identity for cosine and for sine are... And so we know that sine of 2 theta is 2 sine theta cosine theta. So let's sketch a picture of a quadrant for angle. So we'll draw a unit circle centered at the origin, then draw an angle whose terminal side is in quadrant 4. Quatrant 4. And remember the cosine value is the x-coordinate and the sine value is the y-coordinate. And so here, notice that cosine is positive, but sine will be negative. So by the Pythagorean identity, sine squared theta plus cosine squared theta is equal to 1, and we know the value of cosine theta. So we can substitute and solve for sine. And since we know that the sine is negative, our actual value for sine of theta is negative square root of 24-fifths. So we can substitute that into our formula, and that gives us the sine of 2 theta. Now, if we don't know anything about theta, we end up with a bunch of possibilities for the trigonometric function values. So sine of 2 theta is 4-fifths. Let's find all possible values of cosine theta and sine theta. So for a variety of reasons, it's actually helpful to find the associated cosine value. So from the Pythagorean identity, we know that the sine squared of 2 theta plus the cosine squared of 2 theta is equal to 1. We know the sine of 2 theta, so we'll replace and solve. And this time, we don't know anything about the quadrant that 2 theta is in, so cosine of 2 theta is plus or minus 3-fifths. Now, while we do have two double angle identities for sine of 2 theta and cosine of 2 theta, it's actually easier to use cosine of 2 theta because the Pythagorean identity allows us to write this in terms of one function only. So we know that cosine 2 theta is cosine squared minus sine squared theta, but our Pythagorean identity tells us that sine squared theta is 1 minus cosine squared theta, and that simplifies to 2 cosine squared theta minus 1. But wait, there's more. If we replace the cosine squared theta with 1 minus sine squared theta, we'd get, and that gets us a different value for cosine 2 theta. So remember we have two possible values for cosine 2 theta. So cosine of 2 theta is plus or minus 3-fifths. If we take cosine 2 theta equal to 3-fifths and replace in our formula, we can then solve for cosine of theta. And since we don't know anything about the quadrant, we have to leave cosine theta as plus or minus the square root of 4-fifths. We might reduce the square root, but that's about all we can do. The second form of our cosine identity also allows us to find the values for sine of theta. So again, we're taking cosine equal to 3-fifths and solving for sine. Again, not knowing the quadrant that theta is in means that we have to allow for a sine to be both plus or minus square root 1-fifth. But wait, remember that cosine of 2 theta could have been negative 3-fifths. So using that value, we can solve for cosine and get two more values for cosine. And similarly, we can solve for the value of sine and get two more values for the sine of theta.