In Exercises 17–22, let θ be an angle in standard position. Name ...

Question

In Exercises 17–22, let θ be an angle in standard position. Name the quadrant in which θ lies.

sin θ < 0, cos θ < 0

Accepted Answer

Hello, today we're gonna be determining which quadrant theta lies in with the given conditions. So we are told that Sine theta is greater than zero and cosine theta is less than zero. In order to answer this problem, we're going to need to understand the unit circle. So if we draw a unit circle, the unit circle is defined as a circle with a radius of one. The direction of the circle is that the circle rotates counterclockwise in the positive direction. And the order of the quadrants goes counterclockwise as well. Quadrant one is located on the upper right quadrant two is located on the upper left quadrant three is located on the lower left and quadruple four is located on the lower right. Now, one thing to note is that any point or any value that's found on the unit circle itself is defined as the point cosine theta comma sine theta. What this means is that cosine represents any X value found on the unit circle and sine represents any Y value found on the unit circle. So if we take a look at our given conditions, we are told that Sine theta has to be greater than zero. What this means is that sine theta is going to be positive. And since sine represents any Y value on the unit circle, this tells us that Y is going to be positive. Now, let's take a look at the second condition we are given cosine theta is less than zero, cosine data represents any X value on the unit circle. And since cosine data is less than zero cosine data must be negative. Therefore, X is going to be negative. And so what this means is that for any value data that we plug into sign and cosign with the given conditions, that means that we are going to get a point negative X comma positive Y. This is a point in the form where the X value is negative and the Y value is positive and any point that's within this form can be found in quadrant two quadrant two contains any points with a negative X value and a positive Y value. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 17–22, let θ be an angle in standard position. Name the quadrant in which θ lies. sin θ < 0, cos θ < 0

Key Concepts

Quadrants of the Coordinate Plane

Sine and Cosine Functions

Standard Position of an Angle

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