Is csc negative 225 degrees positive or negative?
Let’s begin by recalling the trigonometric identity that relates csc to either sine, cosine, or tangent. Csc 𝜃 is equal to one over sin 𝜃. So, csc of negative 225 degrees will be equal to one over sin of negative 225 degrees. So, the question we need to ask ourselves is whether sin of negative 225 degrees is positive or negative. There are a number of different ways we can answer this, but we’re going to use a CAST diagram. A CAST diagram looks like this. These allow us to remember the signs of the trigonometric functions in each of the quadrants. We label it as shown.
The A stands for all. All three trigonometric identities are positive in the first quadrant. That’s sine, cosine, and tan. S stands for sine. So, the sine function is positive in the second quadrant. T stands for tan. The tan function is positive in the third quadrant. And C, of course, stands for cosine. The cosine function is positive in the fourth quadrant. We’re trying to establish whether sin of negative 225 degrees is positive or negative. And we do have a bit of a problem.
Notice that our CAST diagram is labelled from zero to 360 degrees. So, one thing that we could do is relabel the angles. To do so, we travel in a clockwise direction, marking on intervals of 90 degrees as shown. This will allow us to work out where negative 225 degrees lies. And in fact, it will be somewhere in this quadrant. In this quadrant, sine is positive. So, that tells us sin of negative 225 degrees must be positive. That means, then, that csc of negative 225 degrees is one divided by some positive value. Well, one divided by a positive number will also be a positive number. So, csc of negative 225 degrees is positive.
Now, we will briefly look at the other methods we could’ve used. We go back to our CAST diagram and we look for 225 degrees. In this third quadrant, sine is not positive. So, it must be negative. We then recall that sine is an odd function. This means sin of negative 225 degrees is equal to negative sin of 225 degrees. We said that sin of 225 degrees itself was already negative. So, we end up with a negative negative value, which doesn’t make a lot of sense. In fact, sin of negative 225 degrees is therefore positive. We then continue the rest of the process as we did before. One divided by a positive number is positive.