Concept Check Suppose that ―90°

Question

Concept Check Suppose that ―90° < θ < 90° . Find the sign of each function value. sec θ/2

Accepted Answer

Welcome back. I am so glad you're here. We're told, determine whether the value of the following function is positive or negative. If A is between negative 90 degrees and degrees, our function is the C can of A divided by three. And our answer choices are answer choice. A positive and answer choice B negative. All right. So we've got a couple of things going on here. We're told that the value of A is between negative degrees and 90 degrees. So we could figure out what our function would be if we look at the limits. So if we take the of negative degrees for our, A divided by three, that's the lower limit. And then the C can of positive 90 degrees divided by three, that's the upper limit and 90 divided by three is 30. So we're talking about the of negative degrees to the ent of positive 30 degrees. And we recall from previous lessons, if we were to draw a really rough sketch of a coordinate plane, and then we've got a vertical y axis and a horizontal X axis. They come together at the origin in the middle, we have four quadrants up into the right of the origin is quadrant one, up into the left of the origin is quadrant two, down into the left of the origin is quadrant three and down into the right of the origin is quadrant four. And so where are we talking about here? Which quadrants are we in? Well, the sea can of negative 30 degrees. If we're talking about negative 30 degrees, that's going to have the initial side of an angle along the positive part of the X axis, it's going to have its vertex at the origin. And then because it's a negative 30 degrees, we are heading clockwise into the fourth quadrant and it's not too far for 30 degrees. It's closer to the positive part of the X axis than the negative part of the Y axis. There's our negative 30 degrees. And if it's between that and positive 30 degrees, well, that's going to be the same thing with an initial side along the positive part of the X axis vertex at the origin. But now this time because it's a positive 30 degrees, we're heading counterclockwise up into the first quadrant. And again, it doesn't get too far. It's still pretty close to the positive part of the X axis closer to that than the positive part of the Y axis. There's our positive 30 degrees. So we're talking about this space between negative 30 degrees and positive 30 degrees. And we can see that we're exclusively in quadrants one and four. Now, what's our function? Our function is the CCA for both of these here. And so we ask ourselves what's C can doing in quadrants one and four. So we can remember our pneumonic device that tells us where all of the different trigonometric functions are positive. That is all. So a in the first quadrant student, so s in the second quadrant take, so T in the third quadrant calculus. So C in the fourth quadrant and which ones C can't? Well, C can is the reciprocal function of cosine. That C in the fourth quadrant stands for cosine. So C is also positive in the fourth quadrant. And then up in the first quadrant where we have the A for all, all of them including secant are positive in the first quadrant. So C is positive in both the first and the fourth quadrants. Therefore, our function is positive for our C can of A divided by three. If A is between negative 90 degrees and positive degrees, we look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Concept Check Suppose that ―90° < θ < 90° . Find the sign of each function value. sec θ/2

Key Concepts

Domain and Range of the Angle θ

Definition and Sign of the Secant Function

Sign of Cosine in Quadrants I and IV

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