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 , 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 an angle theta or the possible quadrants, our conditions are that the cosine of theta is less than zero and the of theta is less than zero. Our answer choices are answer choice. A quadrant one and quadrant three, answer choice B quadrant two and quadrant three. Answer choice C quadrant two and quadrant four and answer choice D quadrant one and quadrant four. All right, we recall from previous lessons that if we were to imagine a coordinate plane, we have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle, we have four quadrants. The first quadrant is the upper right-hand quadrant. And then we continue counterclockwise. The second quadrant is up into the left, the third quadrant is down into the left and the fourth quadrant is down and into the right. And we recall that we have an acronym to help remember which trigonometric functions are positive in which quadrants in the first quadrant. We have an A that stands for all, all of our trigonometric functions are positive in the second quadrant, we have an S that stands for sign and its reciprocal function. Coy can no, that's backwards. CS C. All right. In the third quadrant we have A T and that stands for tangent but also its reciprocal function cotangent. And in the fourth quadrant, we have C that stands for cosine and C can't, we can remember this acronym A STC we could think all students take calculus. All right. So the ones that were given are the cosine of theta and the second of theta. So where is cosine positive? Well, everything's positive in the first quadrant and cosine is also positive in the fourth quadrant. But we're looking for where cosine of theta is less than zero. We're looking for where it's negative. So that's going to be in the second quadrant and in the third quadrant. All right. The other condition is that the CC of theta is less than zero C can, is positive in the same places as cosine. It's positive in the first quadrant where they all are and it's positive in the fourth quadrant. So the places where C can is less the C can of theta is less than zero is also going to be the second and the third quadrant as well. So it's both of those two quadrants for our answer. We look at our answer choices and this 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. tan θ < 0 , cot θ < 0

Key Concepts

Trigonometric Functions

Quadrants of the Coordinate Plane

Sign of Trigonometric Functions

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