Identify the quadrant (or possible quadrants) of an angle θ that...

Question

Identify the quadrant (or possible quadrants) of an angle θ  that satisfies the given conditions. See Example 3. 
 
 csc θ > 0 , cot θ > 0

Accepted Answer

Welcome back. I am so glad you're here. We're told, considering the given conditions state the quadrant of angle theta or the possible quadrants, our given conditions are the cotangent of theta is less than zero and the of theta is less than zero. Our answer choices are answer choice. A quadrant one, answer, choice B quadrant two, answer choice, C quadrant three and answer choice D quadrant four. All right. If we recall from previous lessons, we can draw a rough sketch of a coordinate plane. We've got a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. We've got four quadrants. The upper right hand is quadrant one and then we're going to continue counterclockwise. The upper left hand is quadrant two. The lower left hand is quadrant three and the lower right hand is quadrant four. Now, we've got an acronym to help us remember which trigonometric functions are positive in which quadrants? That acronym is all students take calculus. And what does that stand for in quadrant one? The A for all stands for all, all of our trigonometric functions are positive in quadrant one in quadrant two the S for students, S stands for the sign function. And it's reciprocal, the co in quadrant three, the T for, take that stands for a tangent as well. It's as its reciprocal function, the cotangent and in quadrant four, the C in calculus, the C stands for cosine and its reciprocal function C can't. So what are, what are the conditions that we have? We're told first of all that the cotangent of theta is less than zero. What do we know about the cotangent of theta? Well, we know that cotangent is positive in the third quadrant and it's also positive in the first quadrant because they all are. So we're looking for where the co tangent of data is less than zero. Well, that's going to be where it's negative and that's going to be in the other two quadrants that's going to be in quadrant two and in quadrant four. All right, what's our other condition? We've got the CO C can of theta is less than zero. What do we know about the coy can of theta? Well, we know that the CO C can is positive in quadrant two, it's also positive in quadrant one because they all are. So where is it less than zero? It's going to be the other two, it's going to be in quadrant three and in quadrant four. So now our question is, what is the quadrant of angle theta? Where both of these things are true? Where are both cotangent and consequent sequent, less than zero or negative. And that's going to be in quadrant four. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

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Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3. csc θ > 0 , cot θ > 0