 As a general rule, the only way to find the value of a trigonometric function is to use a table or some calculating device. But every educated person in a free society should know a few things, like how to distinguish between evidence and anecdotes, how to distinguish between coherent, logical arguments and meaningless 140-character tweets, and the exact values of sine, cosine, and tangent for certain angles. And so we have the following results, the values of sine, cosine, and tangent for angles of 0, 30 degrees, 45 degrees, 60 degrees, and 90 degrees. So the sine of 0 is 0, and the cosine of 0 is 1. The sine and cosine of 30 degrees is 1 half and square root of 3 over 2, respectively. The sine and cosine of 45 degrees are both equal to square root of 2 over 2. The sine and cosine of 60 degrees, square root of 3 over 2 and 1 over 2. And the sine and cosine of 90 degrees is 1 and 0, respectively. Now there's several noteworthy patterns in this table, and one thing we can do to make these patterns a little bit more clear is to rewrite all of our numbers in a similar format. And notice two things, many of these numbers are over 2, and many of these numbers have square root of something in them. So let's rewrite them, 0 is the same as square root of 0 over 2, 1 half is the same as square root of 1 over 2, and 1, well that's really square root of 4 over 2. And so our sine values form this nice progression, square roots of 0, 1, 2, 3, or 4 over 2. And our cosine values go in the opposite direction, square roots of 4, 3, 2, 1, and 0 over 2. Now, you should remember you should memorize these sine and cosine values. You should know the tangent values as well, but since tangent is sine over cosine, you don't need to memorize the tangent values separately. Once you know the sine and cosine values, you can calculate them. Now 30, 45, and 60 are all angles in the first quadrant. What about other angles? For those, we can take a look at the reference angle. We look for the right triangle. The important thing to remember is that the reference angle determines the length of the opposite and adjacent sides of the corresponding right triangle. So it determines the magnitude of the sine, cosine, and tangent. However, the sines are determined by the sines of the x and y coordinates. And this suggests that we can find the sine, cosine, or tangent of an angle by finding the reference angle, determining the magnitude of the sine, cosine, or tangent, and adjusting the sine of the sine, cosine, or tangent. For example, let's try to find the cosine of 135 degrees. So the first thing we want to do is to find our reference angle. Since 135 is 180 minus 45, we can rotate by 135 degrees by making a half-turn counterclockwise, that's 180 degrees, and then rotating back by 45 degrees. So our reference angle is 45 degrees, and 135 degrees is a second quadrant angle. Now let's look for the right triangle. Now remember, cosine is our x-coordinate, the horizontal distance. And since our reference angle is 45 degrees, our horizontal distance will be the same as the cosine of 45 degrees. So pulling in our table of exact trigonometric values, we see the cosine of 45 degrees, square root of 2 over 2, is this length. And finally, since theta is an angle in the second quadrant where the x coordinates are negative, then the cosine of theta must be negative. So cosine of 135 must be negative square root of 2 over 2. How about tangent of 210 degrees? First, again, we can sketch the angle of 210 degrees, and we see that we can get to it by making a half-turn counterclockwise, 180 degrees, and continuing on for another 30 degrees. So the reference angle is 30 degrees, and the angle is in the third quadrant. So again, we look for the right triangle. Since tangent of theta is y over x, and both x and y are negative in the third quadrant, then tangent of theta will be positive. Our reference angle of 30 degrees has a tangent of 1 over square root 3, and so the tangent of 210 degrees will be the positive version of that number, 1 over square root 3. How about the sine of 660 degrees? And so we notice that 660 is more than 360. So we might say that 660 is 360 plus 300. Now, what can we do about this 300? Well, we could make a movie. More usefully, this is almost 360. In fact, it's 360 minus 60. And so this says we can rotate 660 degrees by making two full counterclockwise turns, and then turning back 60 degrees. So the reference angle is 60 degrees, and the angle is in the fourth quadrant. Again, we'll look for the right triangle. And remember, sine is the vertical distance. Since sine of theta is the y value, and y is negative in the fourth quadrant, sine of 660 degrees will also be negative. So our reference angle 60 degrees has a sine of square root of 3 over 2, and so the sine of 660 degrees will be the negative of that value minus square root of 3 over 2.