Determine the value of y = sin ⁡ ( − π 2 ) + 50 y=\sin\left(-\frac{\pi}{2}\right)+50 y = sin ( − 2 π ) + 50 without using a calculator or the unit circle.

Let's see how we can solve this problem. So here we're asked to determine the value of y is equal to the sine of negative pi over 2 plus 50, And we're told that we cannot use a calculator, nor can we use the unit circle. So how could we really go about solving this problem? Well, it turns out there's another tool that we have in the toolbox, and that is going to be graphing the function for the sine of x. Because we know what these graphs look like, so we can use that graph to figure out what this value is, and thus determine the value of this whole thing. So what I'm first going to do is draw a graph for y is equal to the sine of x. Now recall that the sine of x starts right here in the middle of our graph. Then what we do is we go up here and we reach a peak when we get to pi over 2, and we're going to dip down and cross through pi on the x axis, then we're going to reach a valley at 3 pi over 2, and then we're gonna keep waving as we go to the right. Now likewise, this graph is also going to have this waving pattern as we go to the left. So we're going to go down here and reach a valley at negative pi over 2, we're going to cross through negative pi, we're going to reach a peak at negative 3 pi over 2, and then this graph is going to keep waving. So this is what the graph is going to look like for the sine of x, we should be familiar with this. And what we can do is use this graph to determine what this value is for the sine of negative pi over 2. Now what we can see here is that for negative pi over 2 we can just go to this on the x axis which is right there, and notice that the output that we have on this curve is right here at negative one. So because we're at negative one, that means this entire sign value is going to evaluate to negative one. So So we have negative one plus 50, and negative one plus 50 is 49. So the answer to this problem is y is equal to 49. And And notice how we were able to find this without using a calculator or the unit circle, we could just recognize how the graph behaves to evaluate this answer. So that's how you solve this problem. Thanks for watching.