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. 
 
 tan θ < 0 , cos 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 cotangent of theta is less than zero and the sign 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 rough sketch of a coordinate plane, we have a vertical Y axis and a horizontal X axis. They come together in the middle, we have four quadrants. The upper right hand quadrant is quadrant one and then we continue counterclockwise. The upper left hand is quadrant two, the upper left, the lower left hand quadrant is quadrant three and the lower right-hand quadrant is quadrant four. And we've got an acronym to help us remember which trigonometric functions are positive in the different quadrants. That acronym is all students take calculus and the A in quadrant one stands for all because all of the trigonometric functions are positive there. The S for students in quadrant two that stands for sign and its reciprocal function Cosic the T and quadrant three for take that stands for a tangent and its reciprocal function cotangent and the C in quadrant four that stands for cosine and its reciprocal function C can't. So we know where they're all positive. Now, what do we have for our conditions? We're told that the cotangent of theta is less than zero. What do we know about the cotangent of theta? Well, we know that the cotangent is positive in quadrant three, it's also positive in quadrant one because they all are. But we're looking for where it's less than zero where it's negative. So that's going to be the other two quadrants quadrant two and quadrant four. All right. How about the sign of theta? What do we know about that one? We know that the sign is positive and quadrant two, it's also positive in quadrant one because they all are. But we're looking for where it's less than zero or where it's negative. And again, that's going to be the other two this time, that's quadrant three in quadrant four. So now what's the quadrant of angle theta where both of those conditions are true where both cotangent and sine are negative and that is going to be quadrant four. We look at our answer choices and that matches with answer choice. D 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. tan θ < 0 , cos θ < 0

Key Concepts

Quadrants of the Unit Circle

Signs of Trigonometric Functions

Inequalities in Trigonometry

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