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

Question

Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3. cos θ > 0 , sec 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 that the cosine of theta is less than zero and the cotangent 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, we recall from previous lessons. If we draw a quick sketch of a coordinate plane, we have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle and to the upper right of the origin. That's quadrant one, then we continue counterclockwise 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 we have an acronym to help us remember which trigo trick functions are positive in which quadrants starting in quadrant. One. That pneumonic device that we've got is all so a students s take T calculus C. Now, what do those stand for? Well, the A for all that still stands for all because all of the trigonometric functions are positive in the first quadrant s in the second quadrant for students that stands for a sign and its reciprocal function Cosic can t in the third quadrant take that stands for tangent as well as its reciprocal function cotangent. And in the fourth quadrant C calculus C stands for cosine and its reciprocal function C can't. So that's where they're all positive. Now, what are our conditions here? First, we're told that the cosine of theta is less than zero. What do we know about the cosine of theta? Well, we know that the cosine of theta is positive in quadrant four, it's also positive and quadrant one because they all are, but we're looking for where the cosine of data is less than zero where it's negative. Well, that's going to be the other two quadrants. So quadrant two and quadrant three. All right, what is our other condition? We're told the cotangent of theta is less than zero. What do we know about the cotangent? Well, we know that it's positive in quadrant three, it's also positive in quadrant one because they all are. So where is it less than zero? Where is it negative that's going to be in the other two? So it's going to be in quadrant two and quadrant four. So now the question is, where is it true for angle theta that both conditions are met and both conditions are met. They're both negative in quadrant two. We look at our answer choices and that matches with answer choice. B well done. We'll catch you on the next one.

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

Key Concepts

Relationship Between Cosine and Secant

Signs of Trigonometric Functions in Quadrants

Quadrant Identification Using Inequalities

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