Suppose θ is in the interval (90°, 180°). Find the sign of each o...

Question

Suppose θ is in the interval (90°, 180°). Find the sign of each of the following.

cot(θ + 180°)

Accepted Answer

Hello, everyone. We are asked, what is the sign of the given expression? If theta is in the interval between 180 degrees and 270 degrees. The expression we are given is the sign of theta plus 180 degrees. Our answer choices are a positive b negative. So thinking about the angle and when we add 100 80 degrees to it, where will it land? So I'm gonna start with the minimum that is in our initial interval. So I'm gonna take the angle theta plus 180 degrees. And I'm going to substitute 180 degrees in for theta as that's our minimum value. So we have 180 degrees plus 180 degrees, which is 360 degrees. Since that's a full rotation, we can say that this is also equal to zero degrees. Now, for our maximum, I'm going to again use the plus 180 degrees. But now I'm substituting 270 degrees in for theta. So 270 degrees plus 180 degrees and that will equal 450 degrees. Since that's greater than one rotation, I want to subtract a full rotation to find out where would this land on the coordinate plane? 450 degrees minus 360 degrees leaves us with 90 degrees. So this means that we will land in the interval between zero degrees and 90 degrees, which is quadrant one. And since all of our trig functions are positive and quadrant one, I can say that the sign of theta plus 180 degrees will be answer choice. A positive. Have a nice day.

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Suppose θ is in the interval (90°, 180°). Find the sign of each of the following. cot(θ + 180°)